M - Mathematics

M 005 Co-Req Support for M 105Q: 1 Credits (1 Lec)

(F, Sp) This co-requisite support course allows students who do not meet the prerequisites of M105Q to enroll in specific sections of M105Q. This course will provide an additional day of instruction and will present additional topics to support student success and understanding in M105Q. Course does not earn college-level credit.

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Use strong skills in critical and logical thinking to make wise personal decisions, navigate the media, and be an informed citizen.
Be competent in estimation so that they can put numbers from the news into context that makes them understandable.
Apply the mathematical tools needed to make basic financial decisions.
Read news reports of statistical studies in a way that will allow them to evaluate them critically and decide whether and how the studies should affect their personal beliefs.
Be familiar with basic ideas of probability and be aware of how it affects their lives.
Describe and explain how mathematics helps us study important social issues, such as global warming and the growth of populations.
Interpret mathematics topics to help them develop quantitative reasoning skills they will need for college, career, and life.

M 021 Co-Requisite Support for M121Q College Algebra: 2 Credits (2 Lec)

PREREQUISITE: M 090 or MPLEX 30 or ACT 21 or SAT 530 or 26 or old SAT 500 or M 065 A- or A. (F, Sp, Su) This course serves as a co-requisite for M121Q (College Algebra). Upon completing this course along with the co-requisite M121Q, students will be prepared to take M151Q or M161Q (depending on major). This course is intended to allow some students placing into developmental math an opportunity to enroll in M121Q while providing the additional time and support associated with developmental courses. Course does not earn college-level credit

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This course provides support in achieving the M121 learning outcomes. This support will be provided through extra instruction of basic algebraic concepts at the beginning of the semester as well as a more detailed and in depth look at M121Q topics throughout the semester.

M 063 Foundations of Mathematics: 1 Credits (1 Lec)

COREQUISITE: This course will be linked to a M 090 Introductory Algebra Course with the same section number. (F, Sp) This instructor-taught course covers basic concepts relating to whole numbers, integers, fractions, decimals, percent and selected geometry topics. This course is to be taken during the same semester as M 090. The course is offered as a review and/or preparation for further studies in Mathematics. Offered by Gallatin College. Course does not earn college-level credit

View Course Outcomes:

Add, subtract, multiply, and divide whole numbers, integers and decimals
Convert between decimals, fractions and percent.
Simplify, add, subtract, multiply and divide fractions
Evaluate exponential expressions with signed numbers and fractions
Solve real-world application problems involving signed numbers and fractions
Simplify expressions containing signed numbers and fractions using order of operations
Apply geometric formulas, including area and perimeter

M 065 Pre-Algebra: 4 Credits (4 Lec)

This instructor-taught course covers basic concepts relating to fractions, decimals, ratios, proportions, percent, selected geometry topics, topics of signed numbers, and 1-variable linear equations. The course is offered as a review and/or preparation for further studies in Mathematics. Common final.

M 066 Pre-Algebra Lab and Study: 1 Credits (1 Other)

Students enrolled in M 065 co-enroll in this course for additional instruction and practice with M 065 curriculum and Math study skills. This course will help students understand Math concepts, practice course material, and prepare for Math tests. Course is offered pass/fail.

M 090 Introductory Algebra: 4 Credits (4 Lec)

PREREQUISITE: M063 or ACT 17 or ACT 15/16 and HS GPA greater than 3. (F, Sp, Su) Offered by Gallatin College. Intended for students pursuing majors requiring the M 121Q track and/or chemistry. This course serves as an introduction to algebra, which includes the study of basic operations with algebraic fractions and polynomials, linear equations and inequalities in one and two variables, systems of linear equations, and linear applications in one and two variables including percent applications. Course does not earn college-level credit

View Course Outcomes:

Perform arithmetic operations with real numbers;
Add, subtract, multiply and divide polynomials and algebraic fractions;
Simplify radical expressions;
Apply the rules of exponents to algebraic expressions and scientific notation; ;
Solve linear equations and inequalities in one variable;
Recognize and determine equations of lines;
Graph linear equations and inequalities in two variables;
Set up and solve linear application problems in one and two variables, including percent applications;
Use appropriate language and notation to clearly communicate mathematical ideas.

M 091 Special Topics: 1-4 Credits (1-4 Lec)

Repeatable up to 12 credits.

M 105Q Contemporary Mathematics: 3 Credits (3 Lec)

PREREQUISITE: Math Level 290. (F, Sp, Su) Formerly M 145Q. Designed to give liberal arts students the skills required to understand and interpret quantitative information that they encounter in the news and in their studies, and to make numerically-based decision in their lives. Topics include working with large numbers and units, linear and exponential relations, financial mathematics, and essentials of probability and statistics. Common final

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attain a degree of mathematical literacy, including an ability to read mathematical material and write using mathematical notation correctly. To develop skills to think and reason mathematically in order to function more effectively in the modern world.\\n
examine ways in which mathematics is used, to follow and understand logical arguments, and to solve applied quantitative problems. This includes learning to formulate a problem precisely, to interpret solutions, and to make critical judgments in the face of competing formulations and solutions.\\n
understand elementary probability concepts and phenomena including: sample spaces with equally likely outcomes, basic parameters (mean, standard deviation), the normal distribution, and a qualitative view of the Central Limit Theorem.\\n
understand elementary statistical concepts such as data description, statistical estimation, randomization, and statistical inference.\\n
explore and examine other aspects of contemporary mathematics including but not limited to: management science (e.g. graph models for network problems), social choice and decision making (e.g. elections, voting, fair division, Congress apportionment), or applied geometry (e.g. symmetry, tilings, growth rates).\\n

M 108 Business Mathematics: 3 Credits (3 Lec)

(F, Sp) Students of this course will examine the mathematics of business ownership and will demonstrate an understanding of business decisions. Concepts to be covered include cash flow, simple and compound interest, inventory valuation, purchasing discounts, cost markup, business and consumer loans, and analysis of financial statements. Additional topics which may be covered include payroll, depreciation, and bonds and annuities.

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Understand banking services and record keeping; prepare a bank reconciliation.
Compute gross earnings, payroll deductions, and employer payroll taxes.
Calculate and apply trade and cash discounts.
Determine markups and markdowns.
Describe and employ methods for valuation of inventory.
Understand the structure of promissory notes, simple discount notes, and simple interest.
Process compound interest, present value and future value.
Define and analyze installment buying.
Understand mortgages.
Compute depreciation expense and book value of assets.
Read, analyze and interpret financial reports.
Utilize business statistics to evaluate financial data.

M 111 Technical Mathematics: 3 Credits (3 Lec)

(F) Offered by Gallatin College. This course presents basic mathematical topics as they are applied in a trades program. Topics covered include: use of measuring tools, measurement systems, dimensional arithmetic, percent, proportion, applied geometry, basic trigonometry. NOTE: This course is intended for specific programs and does NOT provide sufficient Pre-Algebra material to serve as a prerequisite for students wanting to take additional mathematics.

M 121Q College Algebra: 3 Credits (3 Lec)

PREREQUISITE: Math Placement Level 300. (F, Sp, Su) Intended for students preparing for precalculus or calculus. Further development of algebraic skills through the study of linear, quadratic, polynomial, exponential, and logarithmic functions

View Course Outcomes:

Simplify, factor, and perform any of the basic arithmetic operations on polynomials and rational expressions.
Perform arithmetic operations and simplify algebraic expressions with rational exponents.
Solve linear, quadratic, and rational, exponential and logarithmic equations and be able to use each of these to model and solve applied problems.
Identify relations vs. functions; use function notation; identify domain, range, intervals of increasing/decreasing/constant values.
Find zeros, asymptotes, and domain of rational functions.
Evaluate and sketch graphs of piecewise functions and find their domain and range.
Use algebra to combine functions and form composite functions, evaluate both combined and composite functions and their graphs, and determine their domains.
Identify one-to-one functions, find and verify inverse functions, and sketch their graph.
Write logarithms as exponentials and vise versa
Solve exponentials and logarithms using the one to one property or inverse properties.
Expand and condense logarithmic expressions.

M 132 Numbers & Operations for K-8 Teachers: 3 Credits (3 Lec)

PREREQUISITE: M 121Q or Math Placement Level 300. (F, Sp) The study of number and operations for prospective elementary and middle school teachers, including whole numbers, decimals, fractions, percents, integers, operations, numeration systems, and problem solving

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Explain the meanings of whole numbers, integers, fractions, and decimals, as well as representations of those numbers;
Explain the meanings of operations and the connections between those operations, concepts, and procedures in doing computations (using both standard and nonstandard algorithms), interpreting story problems, and writing story problems;
Evaluate the efficiency of and use various representations of numbers and operations, as well as their applications to problem solving;
Employ various modeling strategies to solve problems in real world contexts;
Recognize some common misconceptions and be able to understand the faulty reasoning behind those misconceptions;
Explain their reasoning, both verbally and in writing, while solving problems.

M 133Q Geometry & Measure K-8 Teachers: 3 Credits (3 Lec)

PREREQUISITE: A grade of C or better in M 132. (F, Sp) The study of geometry and geometric measurement for prospective elementary and middle school teachers, including synthetic, transformational, and coordinate geometry, constructions, congruence and similarity, 2-dimensional and 3-dimensional measurement, and problem solving

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Students will be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Students will be able to apply transformations and use symmetry to analyze mathematical situations
Students will be able to use visualization, spatial reasoning, and geometric modeling to solve problems
Students will be able to describe and apply measurable attributes of objects and the units, systems, and processes of measurement
Students will be able to apply appropriate techniques, tools and formulas to determine measurements for length, area, and volume.
Students will develop a deep understanding of the mathematical concepts needed for effective teaching by developing the ability to examine and explain underlying mathematical structure in using multiple geometric representations and tools for solving problems.

M 140 College Math for Healthcare: 3 Credits (3 Lec)

PREREQUISITE: Math Placement Level 290 or M 090 or M 105Q. (F, Sp) This course is designed to provide students with a solid mathematical foundation necessary to succeed in a health care profession. This course will review algebra, systems of measurement, ratio and proportions, basic probability and statistics concepts, and ionic solutions and pH calculations. This course will apply mathematical reasoning and problem solving as it applies to the health care field and is a suitable prerequisite for STAT 216Q

View Course Outcomes:

Apply knowledge of decimals, fractions, and percents to solve algebraic linear equations in the healthcare field.
Use knowledge of rational equations to solve problems involving ratios and proportions (including but not limited to volume, mass, weight and temperature).
Use the fundamental units of the metric system (SI), household units, and the apothecary system in making measurements and doing calculations related to allied health applications.
Interpret the meaning of range, standard deviation, and the coefficient of variation in applied situations.
Use and apply the basic probability concepts: probability models (Venn diagrams, two-way tables), sample spaces with equally likely outcomes (counting), probability distributions.
Use and apply the rudiments of statistics: measures of center and spread, the normal distribution.
Use and interpret exponential and logarithmic functions and graphs.
Apply knowledge of logarithmic functions to solve problems in healthcare.
Apply mathematical and statistical reasoning to a variety of applied or theoretical healthcare problems.

M 151Q Precalculus: 4 Credits (4 Lec)

PREREQUISITE: M 121Q or Math Level 400. (F, Sp, Su) Functions, graphs, and the use of symbols for expressing mathematical thoughts. Polynomials, rational, exponential, logarithmic, and trigonometric functions

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Analyze properties of functions (domain, range, intercepts, end behavior, asymptotes, and holes) from graphical, symbolic, numerical, and contextual representations \\n
Describe functional relationships between changing quantities, and where applicable use amounts of change to write explicit equations (progressing from average rate of change towards instantaneous)
Relate properties of polynomial, rational, exponential, logarithmic, and trigonometric functions between symbolic and graphical representations\\n
Explain relationships between inverse functions and consequences of truncated domains for solving problems involving inverse trigonometric functions\\n
Apply basic methods of symbolic manipulation and properties of functions necessary for studying calculus\\n

M 161Q Survey of Calculus: 4 Credits (4 Lec)

PREREQUISITE: M 121Q or Math Placement Level 400. (F, Sp, Su) A survey of basic calculus including limits, differentiation, and integration with applications to business, biology, and social science problems. COMMON FINAL ONLY

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solve simple problems involving limits, infinite limits, limits at infinity, asymptotes
differentiate simple functions from the derivative definition, the power, product, quotient and chain rules\\n
differentiate exponential and logarithmic functions
differentiate implicitly and logarithmically\\n
graphically analyze functions including how to find local and global extrema, concavity, and inflection points
use the derivative to solve related rate, optimization and other application word problems
evaluate simple integrals and know its relationship to area\\n
state the Fundamental Theorem of Calculus
compute partial derivatives of simple function

M 165Q Calculus for Technology I: 3 Credits (3 Lec)

PREREQUISITE: M 151Q or Math Level 500. (F, Sp) Calculus with emphasis on problems of interest to engineering technologists. Includes analytic geometry, differentiation, and introduction to integration

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Explain and understand the basic concepts of limits, derivatives, and integrals
Turn word problems into mathematical problems and work with the basic mathematical symbolism of calculus
Apply calculus to solve mathematical problems in engineering and physics applications

M 166 Calculus for Technology II: 3 Credits (3 Lec)

PREREQUISITE: M 165Q or M 171Q. (F, Sp) Calculus with emphasis on problems of interest to engineering technologists. Includes integration, infinite series, and differential equations

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Explain and understand the basic concepts of integrals
Use integrals to calculate geometric quantities such as areas, volumes, and arc lengths
Use integrals to calculate physical quantities such as center of mass, fluid pressure, and work
Explain and understand the basics of differential equations, as well as solve simple first-order differential equations
Apply integrals and differential equations to solve mathematical problems in engineering and physics applications

M 170 Supplemental Instruction in Trigonometry for Calculus: 1 Credits (1 Lec)

PREREQUISITE: Math Placement Level 450
COREQUISITE: M 165Q or M 171Q. (F) For students concurrently enrolled in M 171Q or M 165Q. Provides supplemental instruction in concepts and procedures in trigonometry that are necessary for success in calculus. Students with credit for M 151Q are not eligible to also earn credit for M 170
.

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Define the relationship between degree and radian angle measure, and compute conversions between them.
Define sine, cosine, tangent, secant, arcsine, and arctangent in terms of right triangle ratios and on the unit circle.
Use with fluency the unit circle values for sine, cosine, and tangent for the reference angles 0°, 30°, and 45°, and derive other values from these.
Describe and use the features of the graphs of sine and cosine, including characterization as even or odd.
Use similar triangles to reason about proportional relationships.
Recognize and apply the identity (sin x)^2 + (cos x)^2 = 1.
Recognize problems in which other trigonometric identities are useful and locate and use those identities efficiently.
Apply the Law of Sines and Law of Cosines in problem solving

M 171Q Calculus I: 4 Credits (4 Lec)

PREREQUISITE: M 151Q or Math Level 500. (F, Sp, Su) Functions, elementary transcendental functions, limits and continuity, differentiation, applications of the derivative, curve sketching, and integration theory. COMMON FINAL EXAM

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Explain the definition of limit, how to compute it in elementary cases, and how to determine the limits of transcendental, rational and piecewise defined functions;\\n
Define infinite limits, limits at infinity, asymptotes, indeterminate forms and how to use L’Hopital’s Rule;
Explain the limit definition of continuity;\\n
Explain the limit definition of the derivative of a function, how it related to the function itself, and how to use it to compute derivatives;
Use derivatives to find tangent lines to curves and velocity for particle motion;\\n
Apply the power, sum, product, quotient and chain rules of differentiation;
Use the derivatives of exponential, logarithmic , trigonometric and hyperbolic functions;\\n
Explain implicit and logarithmic differentiation;
Apply the Intermediate and Mean Value Theorems;\\n
Graphically analyze functions including using continuity and differentiation to determine local and global extrema, concavity, and inflection points;
Use the derivative to solve challenging related rate and optimization word problems;\\n
Explain Newton’s Method for estimating zeros of a function;\\n
Explain the Riemann integral, areas under graphs, antiderivatives the Fundamental Theorem of Calculus;
Apply integration using the method of substitution

M 172 Calculus II: 4 Credits (4 Lec)

PREREQUISITE: M 171Q. (F, Sp, Su) Methods of integration, applications of the integral, infinite sequences and series including Taylor series, parametric and polar equations. COMMON FINAL ONLY

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Evaluate integrals using a variety of methods (substitution, integration by parts, partial fraction decomposition, trigonometric substitution)\\n
Set up and compute integrals in applied situations, such as finding volumes of solids of revolution, arc length, surface area, work, fluid pressure, and centers of mass\\n
Determine convergence/divergence of series via various tests (recognizing geometric series, root test, ratio test, alternating series test)\\n
Be familiar with power series (including interval and radius of convergence) and Taylor series expansions of common functions\\n

M 181Q Honors Calculus I: 4 Credits (4 Lec)

PREREQUISITE: M 151Q with an "A" grade, 700 on the SAT Math exam, 31 on the ACT Math exam, 4 on an AP AB Calculus exam, or consent of the instructor. (F) Honors section of M 171Q. Topic coverage parallels M 171Q but with a greater emphasis on theory and more difficult problems

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Explain the definition of limit, how to compute it in elementary cases, and how to determine the limits of transcendental, rational and piecewise defined functions;
Define infinite limits, limits at infinity, asymptotes, indeterminate forms and how to use L'Hospital Rule;
Explain the limit definition of continuity;
Explain the limit definition of the derivative of a function, how it related to the function itself, and how to use it to compute derivatives;
Use derivatives to find tangent lines to curves and velocity for particle motion;
Apply the power, sum, product, quotient and chain rules of differentiation;
Use the derivatives of exponential, logarithmic , trigonometric and hyperbolic functions;
Explain implicit and logarithmic differentiation;
Apply the Intermediate and Mean Value Theorems;
Graphically analyze functions including using continuity and differentiation to determine local and global extrema, concavity, and inflection points;
Use the derivative to solve challenging related rate and optimization word problems;
Explain Newton's Method for estimating zeros of a function;
Explain the Riemann integral, areas under graphs, antiderivatives the Fundamental Theorem of Calculus;
Apply integration using the method of substitution

M 182 Honors Calculus II: 4 Credits (4 Lec)

PREREQUISITE: M 171Q with an "A" grade or M 181Q with a "B" grade or 5 on an AP AB exam or consent of instructor. (Sp) Honors section of M 172. Topic coverage parallels M 172 but with a greater emphasis on theory and more difficult problems

View Course Outcomes:

Using the integral to find the area between two curves, volumes of revolution, work and the average value of a function;\\n
integration by direct and trigonometric substitution, parts, and partial fractions. Trigonometric integrals;\\n
Simpson's rule for approximating integrals, improper integrals and indeterminate forms of limits;\\nusing the integral to find arc length, surface areas of revolution, moments, centers of mass and hydrostatic pressure. Pappas's Theorem;\\n
infinite sequences of real numbers, their monotonicity and boundedness, and the Montonic Sequence Theorem;
convergent series of real numbers, geometric series, telescoping series, and the basic test for divergence;\\n
the integral, comparison, limit comparison, and alternating series tests for series convergence;
absolute convergence and the ratio and root tests;\\n
power series, radius of convergence, and the integration and differentiation of power series;
Taylor series and Taylor polynomial approximation of functions;\\n
parametrized curves in rectangular and polar coordinates, their derivatives, arc lengths and enclosed areas.

M 194 Introduction to Mathematical Sciences: 1 Credits (1 Lec)

(F) For first-year students. Integration into the department and campus community. Development of mathematical and statistical habits of mind and introduction to software relevant to the mathematical sciences.

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identify perspectives, habits of mind, and tools used in the disciplines of mathematics, applied mathematics, statistics, and mathematics teaching.

M 221 Introduction to Linear Algebra: 3 Credits (3 Lec)

PREREQUISITE: M 166 or M 172. (F, Sp, Su) Matrix algebra, systems of linear equations, determinants, vector algebra and geometry in Euclidean 3-space, eigenvalues, eigenvectors

View Course Outcomes:

Explain the basic linear algebra topics: \\nthe theory of linear equations, matrix algebra, determinants, vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors;\\n\\n
Explain some applications of linear algebra,
Write and read simple proofs.

M 234 Higher Math for K-8 Teachers: 3 Credits (3 Lec)

PREREQUISITE: A grade of C or better in both M 132 and M 133Q. (F, Sp) The study of algebra, number theory, probability and statistics for prospective elementary and middle school teachers, including proportional reasoning, functions, elementary number theory, statistical modeling and inference, and elementary probability theory

View Course Outcomes:

apply algebra in many forms (e.g., as a symbolic language, as generalized arithmetic, as a study of functions, relations, and variation) and use algebra to model physical situations and solve problems. \\n
describe proportionality and its invariant properties.
apply number theory concepts and theorems, including greatest common factors, least common divisor, properties of prime and composite numbers, and tests for divisibility.
represent, analyze and interpret data.
simulate random events and describe expected features of random variation.
distinguish between theoretical and experimental probability and describe how to use one or both to determine a probability in a given situation.

M 242 Methods of Proof: 3 Credits (3 Lec)

PREREQUISITE: M 172. (F, Sp) Reasoning and communication in mathematics, including logic, generalization, existence, definition, proof, and the language of mathematics. Topics include functions, relations, set theory, recursion, algebra, number theory, and other areas of mathematics

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understand the definitions of various terms used in mathematical logic including: logical equivalence, quantifiers, conjecture, generalization, existence statement, open sentence, contrapositive, converse, mathematical induction, counter example;\\n\\n
identify and classify mathematical statements as conditional statements, existence statements, or generalizations;
manipulate various mathematical statements to produce forms more easily examined for meaning and truth using logical tools such as negation and logical equivalences; \\n
evaluate the truth of a mathematical generalization and construct a counterexample if it is false and prove it if it is true;
take a mathematical statement in casual conversational language and rewrite it in an equivalent, mathematically correct logical form so that its meaning and truth can be examined; \\n
know the difference between beliefs, intuition, informal justifications (heuristics), and formal mathematical proof;
construct mathematical proofs using any of the basic different types of proofs, including direct and indirect proofs and proofs by mathematical induction;
evaluate the validity of mathematical arguments based on their logical correctness;
read mathematical definitions and theorems they have not seen before well enough to use them properly;
develop an understanding of what mathematicians do as professionals and how mathematicians determine truth.

M 273 Multivariable Calculus: 4 Credits (4 Lec)

PREREQUISITE: M 172. (F, Sp, Su) Topics in two and three dimensional geometry. Manipulation and application of vectors. Functions of several variables, contour maps, graphs, partial derivatives, gradients, double and triple integration, vector fields, line integrals, surface integrals, Green's Theorem, Stokes' Theorem, the Divergence Theorem. COMMON FINAL ONLY

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Explain three-dimensional coordinate systems, dot and cross products, equations of lines and planes, cylinders and quadric surfaces;\\n
Explain vector-valued functions and space curves, their derivatives, arc length and curvature, and motion in space;
Explain limits, continuity and partial derivatives of functions of several variables;\\n
Explain tangent planes to surfaces and linear approximations;\\n
Explain the chain rule, directional derivative and gradient vector, extreme values and Lagrange Multipliers;\\n
Explain double and triple integrals over general regions, and their applications;\\n
Explain triple integrals in cylindrical and spherical coordinates;\\n
Explain vector fields, line integrals and the Fundamental Theorem of Line Integrals;\\n
Define Greenâ€™s Theorem;\\n
Explain curl and divergence of vector fields;\\n
Explain surface integrals, Stokes Theorem, and the Divergence Theorem.

M 274 Introduction to Differential Equation: 4 Credits (4 Lec)

PREREQUISITE: M 172. (F, Sp, Su) An introduction to qualitative, quantitative, and numerical methods for ordinary differential equations. Topics include modeling via differential equations, linear and nonlinear first order differential equations and systems, elementary phase plane analysis, forced oscillations, and Laplace transform techniques. COMMON FINAL ONLY

M 283 Honors Multivariable Calculus: 4 Credits (4 Lec)

PREREQUISITE: M 182 with a 'B' grade, M 172 with an 'A' grade, AP Calculus BC exam with a 5, or consent of the instructor. (F) Honors section of M 273. Topic coverage parallels M 273 but with a greater emphasis on theory and more difficult problem solving

View Course Outcomes:

Three-dimensional coordinate systems, dot and cross products, equations of lines and planes, cylinders and quadric surfaces;
vector-valued functions and space curves, their derivatives, arc length and curvature, and motion in space;\\n
limits, continuity and partial derivatives of functions of several variables;\\ntangent planes to surfaces and linear approximations;
the chain rule, directional derivative and gradient vector, extreme values and Lagrange Multipliers;\\n
double and triple integrals over general regions, and their applications;\\ntriple integrals in cylindrical and spherical coordinates;
vector fields, line integrals and the Fundamental Theorem of Line Integrals;\\nGreen's Theorem;\\n
curl and divergence of vector fields;
surface integrals, Stokes Theorem, and the Divergence Theorem.

M 284 Honors Introduction to Differential Equations: 4 Credits (4 Lec)

PREREQUISITE: M 182 with a grade of B or higher, M 172 with a grade of A, AP Calculus BC exam with a 5, or consent of the instructor. (Sp) Honors section of M 274. Topic coverage parallels M 274 but with a greater emphasis on theory and more difficult problem solving

View Course Outcomes:

Classifications of ordinary and partial differential equations, linear and nonlinear differential equations;
solutions of differential equations and initial value problems, and the concepts of existence and uniqueness of a solution to an initial value problem;
using direction fields and the method of isoclines as qualitative techniques for analyzing the asymptotic behavior of solutions of first order differential equations;
using the phase line to characterize the asymptotic behavior of solutions for autonomous first order differential equations;
classification of the stability properties of equilibrium solutions of autonomous first order differential equations;
separable, linear and exact first order differential equations;
substitution and transformation techniques for first order linear differential equations of special forms. These include Bernoulli and homogeneous equations;
mathematical modeling applications of first and second order differential equations;
methods for solving second order, linear, constant coefficient differential equations. (includes both homogeneous and nonhomogeneous equations)
some techniques for solving second order, linear, variable coefficient differential equations. (includes Variation of Parameters, Reduction of Order and Variable Substitutions for Euler equations);
the principal of superposition for linear differential equations;
basic theory of nth order linear, constant coefficient ordinary differential equations;
the method of Laplace Transforms for solving first and second order, linear ordinary differential equations;
using Unit Step (Heaviside) and Dirac Delta functions to model discontinuous, periodic and impulse forcing functions for first and second order, linear ordinary differential equations;
using Laplace Transforms to solve linear differential equations containing Unit Step (Heaviside) and Dirac delta functions;
basic matrix methods for linear systems of ordinary differential equations;
phase planes for linear systems of ordinary differential equations;

M 290R Undergraduate Research: 1-8 Credits (1 Other)

PREREQUISITE: Consent of the department head. (F, Sp, Su) Directed undergraduate research. Course will address responsible conduct of research
Repeatable up to 8 credits.

M 291 Special Topics: 1-4 Credits (1-4 Lec)

Offered on demand. Courses not required in any curriculum for which there is a particular one-time need, or given on a trial basis to determine acceptability and demand before requesting a regular course number.
Repeatable up to 12 credits.

M 328 Higher Math for Sec Teachers: 3 Credits (3 Lec)

PREREQUISITE: M 242. (F) Offered every other fall; offered in even-numbered years. Concepts, processes, and proof relevant to school mathematics, including number theory, abstract algebra, combinatorics and probability; a focus on standards-based content for teachers in secondary schools

View Course Outcomes:

Solve problems with and reason about functional relationships and algebraic structures.
Apply fundamental ideas of number theory and combinatorics in the exploration, solution, and formulation of problems.
Analyze alternate definitions, language, and approaches to mathematical ideas
Analyze common problems of high school mathematics from a deep mathematical level
Demonstrate alternate ways of approaching problems, including use of technology

M 329 Modern Geometry: 3 Credits (3 Lec)

PREREQUISITE: M 242. (Sp) A study of Euclidean and non-Euclidean geometries, chosen from; hyperbolic, spherical, projective, finite, transformational, and fractal geometries; computer tools for geometry; a focus on standards-based content for teachers in secondary schools

M 330 History of Mathematics: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 274 or consent of instructor. () Offered on demand. Topics will be selected from the entire span of history from Egyptian, Babylonian, and Greek times through the 20th century. The course may focus on milestones that lead to the development of modern mathematics as well as the contributions of great mathematicians from ancient times until today. Some ideas will require mathematical sophistication at the upper division level

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Upon completion of this course, a student will be able to demonstrate understanding of:
demonstrate understanding of the contributions of various mathematicians to the body of mathematics, from ancient to modern times
demonstrate understanding of the evolution of mathematical ideas, and how the significance of these ideas ebbed and flowed during the past 3000 years
demonstrate understanding of the varied mathematical contributions of different cultures and peoples
demonstrate understanding of how the development of mathematics continues today

M 333 Linear Algebra: 3 Credits (3 Lec)

PREREQUISITE: M 221 and M 242. (F) Vector spaces, subspaces, bases, and dimension. Linear transformations, representation by matrices, nullity, rank, isomorphism. Eigenvalues, eigenvectors, and diagonalizability of linear transformations. Inner products, and vector, matrix, and operator norms. Singular value decomposition. The Perron-Frobenius theorem

View Course Outcomes:

knowledge of the formal development of the theory of abstract vector spaces, their subspaces, and dimension
ability to deal with linear transformations on various spaces and their matrix representations in different bases
understanding of inner product spaces and normed spaces, including different norms of linear transformations
understanding of the theoretical basis for selected applied techniques
as exemplified by the least squares approximation, the singular value decompostion, Perron-Frobenius theory
ability to explore and prove simple conjectues and comprehend moderately complex proofs in the above areas

M 348 Techniques of Applied Math I: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 274. (F) An introduction to advanced analytical techniques frequently used by scientists and engineers to study ordinary differential equations and two-point boundary value problems. Topics include series solution techniques, method of Frobenius, Laplace transforms, Fourier series, and boundary value problems

M 349 Techniques of Applied Mathematics II: 3 Credits (3 Lec)

PREREQUISITE: M 348. (Sp) Science and engineering majors often encounter partial differential equations in the study of heat flow, vibrations, electric circuits, and similar areas. Topics include Sturm-Liouville theory, partial differential equations boundary value problems, and Laplace Transform methods

M 362 Linear Optimization: 3 Credits (3 Lec)

PREREQUISITE: M 221. Introduction to linear programming and modeling techniques with applications. Topics may include basic convex geometry, geometry of linear programs, simplex method, duality, sensitivity analysis, game theory, transportation, assignment, and network models

View Course Outcomes:

Demonstrate the techniques of linear optimization and their applications
Formulate a linear program for an appropriate "real-world" problem\\n
Solve linear programs using appropriate software packages
Explain the beautiful theoretical underpinnings of linear programming

M 383 Introduction to Analysis I: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 242, or consent of instructor. (F) A rigorous development of calculus with formal proofs. Functions, sequences, limits, continuity, differentiation, and integration

View Course Outcomes:

Upon completion of this course, a student will be able to:\\nApply the basic proof techniques, including direct proofs, indirect proofs, and mathematical induction;\\n
Construct counterexamples, in each of the following areas, to conjectures that are actually false even though they look plausible and resemble given results;
Define the terms associated with sequences and be able to prove the major results about their limits
Define the terms (for example, continuity and uniform continuity) associated with limits in the context of functions and be able to prove the major results;
Define the terms associated with differentiation and be able to prove some of the major results about derivatives (for example, the Product Rule, the Mean Value Theorem, and the First Derivative Test).

M 384 Introduction to Analysis II: 3 Credits (3 Lec)

PREREQUISITE: M 383. (Sp) A rigorous development of multivariate calculus. Differentiable functions, inversion theorem, multiple integrals, line and surface integrals, infinite series

M 386R Software Applications in Mathematics: 3 Credits (3 Lec)

PREREQUISITE: M 221, M 273, and M 274. (Sp) An introduction to modern mathematical and scientific computing. Software such as MAPLE and MATLAB will be used to explore, solve, and visualize solutions of standard mathematical problems as well as simple models of various physical and/or biological systems

View Course Outcomes:

know the types of mathematical models that exist;
decide which modeling approach is appropriate for a given scientific question;
formulate a mathematical model for an appropriate “real-world” problem;
simulate mathematical models using appropriate software packages;\\n
validate and estimate parameters for a given mathematical model.

M 419 Ratio and Proportion in School Mathematics: 3 Credits (3 Lec)

PREREQUISITE: A grade of C or better in M 242 and junior standing; or, admission to the MAT program. (Sp) Develop knowledge of ratio and proportion necessary to teach standards-based school mathematics. Connect ratio, rate, and proportion to elementary, middle, and high school topics. Explore use of manipulative materials and technologies, and discuss related pedagogical issues and national standards. Note: This course is not appropriate for students in the undergraduate Elementary Education program

View Course Outcomes:

Students will understand that reasoning with ratios involves attending to and coordinating two quantities.
Students will understand and apply two perspectives on ratio: a multiplicative comparison of two quantities and a joining of two quantities in a composed unit.
Students will understand ratio as a measure of an attribute in a real-world situation.\\n
Students will understand the mathematical connections that link multiplication, fractions and ratios.\\n
Students will distinguish between proportional and non-proportional relationships and situations. \\n
Students will understand how rate is related to proportional reasoning.\\n
Students will understand how proportional reasoning supports student understanding of topics typically taught in middle and high school, including similarity and scaling, slope, rate of change, probability, and variation.

M 420 Geometry, Measurement, and Data in the Middle Grades: 3 Credits (3 Lec)

PREREQUISITE: A grade of C or better in M 234, or M 242, and junior standing. () Offered Fall, odd years. Develop content knowledge necessary to teach standards based middle school mathematics. Investigate the underlying conceptual structure of topics in geometry, measurement and data analysis appropriate to middle school. Explore the use of manipulative materials and technologies, and discuss related pedagogical issues and national standards

View Course Outcomes:

describe the development of geometric thinking in the middle grades as it relates to local and national standards and as preparation for formal geometry
apply geometric thinking (e.g., identifying invariants, finding relationships, generalizing) in a variety of problem situations and applications
demonstrate appropriate use of manipulatives, computer software, and Web-based tools that support standards-based instruction in geometry, measurement, and data analysis
identify geometric concepts, patterns, and properties that support visualization and spatial reasoning as well as analytic thinking appropriate for middle grades
identify strategies and contexts that develop skills in data collection, representation, analysis, interpretation, and reporting appropriate for middle grades
recognize opportunities within student work to emphasize and extend geometric habits of mind.

M 424 Algebraic Thinking and Number Sense in the Middle Grades: 3 Credits (3 Lec)

PREREQUISITE: A grade of C or better in either M 234 or M 172. (Sp) Offered spring, even years. Develop algebraic knowledge necessary to teach standards-based middle school mathematics. Investigate the underlying conceptual structure of topics in algebra and number appropriate to middle school. Explore the use of manipulative materials and technologies, and discuss related pedagogical issues and national standards

View Course Outcomes:

Describe the development of algebraic thinking in the middle grades as it relates to local and national standards and as preparation for formal algebra.
Apply algebraic thinking strategies and habits of mind (e.g., pattern-finding, symbol sense, doing-undoing, generalizing) in a variety of problem situations and applications.
Compare and contrast standards-based problems, activities, and curriculum materials in terms of their support for algebraic thinking in the middle grades.
Demonstrate appropriate use of manipulatives, computer software, and Web-based tools that support standards-based instruction in algebraic thinking and number sense.

M 428 Mathematical Modeling for Teachers: 3 Credits (3 Lec)

PREREQUISITE: M 242 and STAT 216Q. (F) Offered every other fall; offered odd-numbered years. Senior capstone course. Applications of the modeling process in key areas of mathematics and statistics. Simulation and other activities, use of relevant technology, modeling in the secondary curricula, and the classroom assessment of modeling activities. Emphasis on technology and authentic applications using pre-college mathematics

View Course Outcomes:

By the end of the course, the successful student will be able to \\nModel, analyze, and interpret situations using data analysis, statistics, and probability. \\n
Develop, apply, and validate mathematical models using current and emerging technologies.
Provide an example of a high school level problem that addresses each of the modeling content standards in CCSS.

M 430 Mathematical Biology: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 274 or consent of the instructor. (Sp) Mathematical modeling of basic biological processes in ecology, physiology, neuroscience, epidemiology and molecular biology using difference equations, differential equations, and partial differential equations

View Course Outcomes:

Have an enhanced knowledge and understanding of mathematical\\nmodeling methods in the analysis of biological systems.\\n
Be familiar with discrete and continuous models of\\nbiological phenomena.
Be aware of the use of computers to assist them in studying\\nmathematical models.
Be able to analyze data from experiments and draw sound conclusions about the underlying processes using their understanding of mathematics.

M 431 Abstract Algebra I: 3 Credits (3 Lec)

PREREQUISITE: M 333. (Sp) Senior capstone course. The integers, integers modulo n, the Euclidean algorithm. Groups, subgroups, normal subgroups, quotient groups, homomorphism and isomorphism theorems, and abelian groups. Rings, ideals, homomorphism and isomorphism theorems. Integral domains, fields, and fields of quotients

M 441 Numerical Linear Algebra & Optimization: 3 Credits (3 Lec)

PREREQUISITE: M 221 and M 273. (F) Numerical solution of nonlinear equations. Numerical solutions of linear systems and eigenvalue problems. Least squares, data smoothing, and optimization techniques

View Course Outcomes:

Apply the basic concepts used in numerical linear algebra to answer mathematical questions related to the construction and analysis of numerical algorithms. These basic concepts include (but may not be limited to) inner products, vector and matrix norms, symmetric matrices and orthogonal matrices, along with various factorizations of matrices including the singular value decomposition and the QR factorization of a matrix.\\n\\n
Compute the Singular Value Decomposition of a matrix and apply it to Least Squares problems.
Compute the QR factorization of a matrix and apply it to Least Squares problems.
Apply and evaluate concepts of conditioning and stability to matrix problems.
Numerically solve a system of linear equations by direct and iterative methods.
Numerically compute the eigenvalues and eigenvectors of a matrix.
Numerically solve nonlinear equations.
Apply some basic numerical techniques for numerically solving optimization problems.

M 442 Numerical Solution of Differential Equations: 3 Credits (3 Lec)

PREREQUISITE: M 221 and M 274. (Sp) Senior capstone course. Numerical integration, numerical solutions of initial and boundary value problems in ordinary differential equations. Numerical solutions of partial differential equations

M 450 Applied Mathematics I: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 274. (F) An introduction to modern methods in applied mathematics. Topics include introductions to dimensional analysis and scaling, perturbation and WKB methods, boundary layers, calculus of variations, stability, and bifurcation analysis

View Course Outcomes:

Upon completion of the course students will have an understanding of the following: Dimensional analysis and its application to real world problems
Regular perturbation techniques as they apply to algebraic and differential equation problems
Elementary singular perturbation methods as they apply to algrebraic problems
Lindstedt's method for approximating perturbed oscillatory solutions
Method of matched asymptotics for simple boundary value problems
Introductory theory for functional optimization
Simple examples of real world calculus of variation problems
Euler Lagrange equation derivations and solutions including natural boundary conditions and Lagrangians having several unknown dependent variables
Isoperimetric problems
How to write simple compartmentalized solutions for mathematical problems which involve multiple techniques.

M 451 Applied Mathematics II: 3 Credits (3 Lec)

PREREQUISITE: M 450. () Offered Spring, even years. This is the second semester of a course that introduces modern methods in applied mathematics. Topics involve methods for linear and nonlinear partial differential equations, including introductions to Green's functions, Fourier analysis, shock waves, conservation laws, maximum and minimum principles, and integral equations

M 454 Introduction of Dynamical Systems I: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 274. (F) Existence and uniqueness of solutions to ordinary differential equations, linearization, phase portraits, stability theory, and the analysis of specific examples

View Course Outcomes:

Find fixed points and low period periodic points for simple one-dimensional maps both graphically and analyticaly;
Analyze dynamics of one dimensional maps using symbolic dynamics;
Understand and be able to reproduce construction of the Smale's horseshoe;
Have an understanding of simple models of chaotic dynamics.

M 455 Introduction to Dynamical Systems II: 3 Credits (3 Lec)

PREREQUISITE: M 454. () Offered Spring, odd years. Gradient systems, Poincare'-Bendixson theory, Poincare' maps, structural stability and chaotic systems

M 472 Introduction to Complex Analysis: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 242. (Sp) An introduction to the techniques of complex analysis that are frequently used by scientists and engineers. Topics include complex numbers, analytic functions, Taylor and Laurent expansions, Cauchy's theorem, and evaluation of integrals by residues

View Course Outcomes:

Explain the basics of complex analysis (definitions, terminology, concepts, techniques, methods);
Explain the different ways in which analyticity can be defined;
Explain Cauchy's theorem and integral formula and some of their applications;
Apply complex analytic methods to evaluate real integrals;
Write a clear proof involving above items;
Think independently and write clearly.

M 476 Introduction to Topology: 3 Credits (3 Lec)

PREREQUISITE: M 221 and M 242 or consent of instructor. () Offered Fall, odd years. Provides an intuitive and rigorous introduction to this important and broad-ranging discipline of modern mathematics. Students will learn to recognize those properties which are topological, i.e., stable under small perturbation. Course participants will compute and see the utility of various topological invariants which arise in a variety of fields from data science, to particle physics, to advanced mathematics

View Course Outcomes:

Define and recognize (with proof) topological properties.
Provide examples of topological spaces, such as surfaces, moduli spaces, and polyhedra.
Give examples of and compute topological invariants, including the winding number of a closed path.
Understand and manipulate the notions of continuity, connectedness, and compactness.

M 490R Undergraduate Research: 1-6 Credits (1-6 Other)

PREREQUISITE: Junior standing in mathematics and consent of department head. (F, Sp, Su) Directed undergraduate research which may culminate in a research paper, journal article, or undergraduate thesis. Course will address responsible conduct of research. May be repeated
Repeatable up to 12 credits.

M 491 Special Topics: 1-4 Credits (1-4 Lec)

PREREQUISITE: Course prerequisites as determined for each offering. Offered on demand. Courses not required in any curriculum for which there is a particular one-time need, or given on a trial basis to determine acceptability and demand before requesting a regular course number
Repeatable up to 12 credits.

M 492 Independent Study: 1-3 Credits (1-3 Other)

PREREQUISITE: Junior standing, consent of instructor, and approval of department head. (F, Sp, Su) Directed research and study on an individual basis
Repeatable up to 6 credits.

M 494 Seminar: 1 Credits (1 Other)

PREREQUISITE: Junior standing and as determined for each offering. () Offered on demand. Topics offered at the upper division level which are not covered in regular courses. Students participate in preparing & presenting material
Repeatable up to 4 credits.

M 497 Educational Methods: Teaching Fellowship: 1-3 Credits (1-3 Other)

PREREQUISITE: Junior standing, consent of instructor, and approval of department head. (F, Sp) As co-teachers of a Mathematics course, students will learn and have the opportunity to practice classroom teaching strategies as well as mentoring skills. Does not satisfy upper division elective for Math-Teaching option
Repeatable up to 4 credits.

View Course Outcomes:

As co-teachers of a Mathematics course, students will learn and have the opportunity to practice classroom teaching strategies as well as mentoring skills.

M 498 Internship: 2-12 Credits (2-12 Other)

PREREQUISITE: Junior standing, consent of instructor, and approval of department head. (F, Sp, Su) An individualized assignment arranged with an agency, business, or other organization to provide guided experience in the field
Repeatable up to 12 credits.

M 501 Intermediate Probability & Statistics: 3 Credits (3 Lec)

PREREQUISITE: STAT 422 or M 384. (F) Families of probability distributions, distributions of functions of random variables, limiting distributions, order statistics. Cross-listed with STAT 501

M 502 Intermediate Mathematical Statistics: 3 Credits (3 Lec)

PREREQUISITE: STAT 501 or M 501. (Sp) Estimation, likelihood inference, statistical hypothesis tests, sufficient statistics, exponential families, Bayesian statistics. Cross-listed with STAT 502

M 503 Advanced Linear Algebra: 3 Credits (3 Lec)

PREREQUISITE: M 333 or consent of instructor. (Sp) Topics include abstract vector spaces, diagonalization, Schur's Lemma, Jordan canonical form and spectral theory for finite dimensional operators

M 504 Abstract Algebra: 3 Credits (3 Lec)

PREREQUISITE: M 431 or consent of instructor. (Sp) The theory of groups, rings and fields with particular emphasis on finite groups, polynomial rings and fields of characteristic zero

M 505 Principles of Mathematical Analysis: 3 Credits (3 Lec)

PREREQUISITE: M 384 or consent of instructor. (F) Principles of analysis in Euclidean spaces and metric spaces

M 507 Mathematical Optimization: 3 Credits (3 Lec)

PREREQUISITE: M 273, M 441. () Offered Fall, odd years. Introduction to mathematical optimization at the graduate level. Overview of computational methods for solving linear and nonlinear optimization problems. Fundamental concepts in optimization, simplex method, duality theory, methods for unconstrained optimization, optimality conditions for constrained problems, and penalty and augmented Lagrangian methods for solving nonlinear constrained problems

View Course Outcomes:

construct optimization models for real-world problems;
understand fundamental optimization concepts, including convex set/function, feasible solution/set/direction, local/global optimal solution, primal/dual problem, and weak/strong duality;
apply the simplex method to solve linear programs;
solve unconstrained optimization problems using the 1st and 2nd optimality conditions, gradient methods, and Newton's method;
solve constrained optimization problems using the optimality conditions;
apply the penalty and augmented Lagrangian methods to solve nonlinear constrained optimization problems.

M 508 Mathematics of Machine Learning: 3 Credits (3 Lec)

PREREQUISITE: M 273 and M 441. () Offered Spring, even years. Mathematical models for pattern recognition and machine learning. Fundamental concepts of parametric and non-parametric probability distributions and dimensionality reduction. Data classification and clustering, regression, kernel methods, artificial neural networks, and Markov-based models. Practical examples drawn from practical data science problems
Repeatable up to 3 credits.

View Course Outcomes:

Describe and understand the mathematics of basic models used in machine learning, and their training
Explain various mathematical approaches to dimensionality reduction with PCA (minimum error, maximum variance, probabilistic)
Understand the mathematical underpinnings of linear models for regression and classification, and kernel-based extensions
Understand and apply basic artificial neural network structure and training, from perceptron to multilayer networks
Build, train, and use basic graphical models such as Hidden Markov models (fields and chains)

M 509 Stochastic Processes: 3 Credits (3 Lec)

PREREQUISITE: STAT 421. () Offered Spring, on demand. Conditional probability theory, discrete and continuous time markov chains including birth and death processes and long run behavior; Poisson processes; queuing systems; system reliability. Cross-listed with STAT 509

M 511 General Topology: 3 Credits (3 Lec)

PREREQUISITE: M 384 or consent of instructor. (F) Definition of a topology, relative topology, metric topology, quotient topology, and the product topology. Connectedness, local connectedness, components and path components. Compactness and local compactness, countability and separation axioms, the Urysohn Lemma, metrization and compactification

M 512 Geometry & Algebraic Topology: 3 Credits (3 Lec)

PREREQUISITE: M 511 or consent of instructor. (Sp) Topics in continua theory, topics in dimension theory, covering spaces and the fundamental group, simplicial complexes, topics in homology and cohomology theory

M 516 Language of Mathematics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Offered Summer, on demand. Features of the language of mathematics, including syntax, vocabulary, and structure. Logic, proof and mathematical communication for high school classrooms

View Course Outcomes:

Understand interpret mathematical concepts relevant to secondary-level mathematics and address misconceptions underlying mathematical language.
Explore uses and abuses of mathematical language in geometry, algebra, arithmetic, and logic/reasoning.
Communicate mathematical concepts and justifications in clear and concise ways.
Review and apply fundamentals of proof and logic.

M 517 Advanced Mathematical Modeling for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Offered Summer, even years. Focus on the use of modeling to solve real-world problems. Topics include the modeling process, an overview of relevant technology, strategies to engage students in modeling in the secondary classroom, and classroom assessment of modeling activities. Extensive use of mathematics to explore application areas, leading to the construction of original models

View Course Outcomes:

Model, analyze, interpret situations using deterministic, stochastic, and empirical models.
Develop, apply, and validate mathematical models using current and emerging technologies.

M 518 Statistics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics or science education, teaching endorsement in mathematics or science, or consent of instructor. (Su) Stochastic concepts including probabilistic underpinnings of statistics, measures of central tendency, variability, correlation, distributions, sampling, and simulation. Exploratory data analysis including experiments, surveys, measures of association and inferential statistics. Discussion of methods for teaching statistics in secondary mathematics and science

View Course Outcomes:

Understand the nature of randomness and how this concept influences statistical interpretation and testing.
Model, analyze, and interpret data in a variety of forms and distributions.
Design experiments and simulations, both random and otherwise.
Critically examine statistical situations.

M 519 Ratio and Proportion in School Mathematics: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Offered Summer, on demand. Develop knowledge of ratio and proportion necessary to teach standards-based school mathematics. Connect ratio, rate, and proportion to elementary, middle, and high school topics. Explore use of manipulative materials and technologies, and discuss related pedagogical issues and national standards

View Course Outcomes:

Students will understand that reasoning with ratios involves attending to coordinating two quantities.
Students will understand and apply two perspectives on ratio: a multiplicative comparison of two quantities and a joining of two quantities in a composed unit.
Students will understand ratio as a measure of an attribute in a real-world situation.
Students will understand the mathematical connections that link ratios to fractions.
Students will be able to reinterpret ratios as quotients.
Students will understand proportion as a relationship of equality between two ratios.
Students will distinguish between proportional and non-proportional relationships.
Students will understand how rate is related to proportional reasoning
Students will understand and explain the reasoning behind common algorithms for solving proportion problems.
Students will distinguish between proportional and non-proportional situations.
Students will understand how proportional reasoning supports student understanding of topics typically taught in high school, including similarity and scaling, slope, rate of change, probability, and variation.
Students will develop and implement a teaching and learning project that connects course content to their classroom settings.

M 520 Access and Equity in Mathematics Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Study of the social context of schooling in the U.S. through the lens of access and equity in mathematics education. Key content themes and connections in algebra, geometry, probability/data analysis, number, and measurement with a focus on mathematical practices. Exploring, extending, designing, and teaching equity-oriented classroom activities for middle/high school students and reflecting on issues of access, equity, and student outcomes

View Course Outcomes:

Define, recognize, and embed equity-oriented mathematical practices in teaching and learning. ;
Design lessons that engage all students in equity-oriented mathematical practices in the context of specific mathematical content domains.
Explore the social context of schooling in the U.S. through the lens of access and equity in mathematics education. ;

M 521 Mathematics Learning Theory for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing, teaching endorsement in mathematics, or consent of instructor. (F) Offered Fall, even years. Examine theories of learning as they apply to the mathematics classroom. The course focuses on theories and research about learning and human development. These are used (a) to understand mathematics learning among students of all cultural, linguistic and socioeconomic backgrounds, and (b) to formulate effective, equitable teaching and learning strategies

View Course Outcomes:

Recognize a variety of classical popular educational learning theories.
Articulate the forms and influences of constructivist theory.
Understand how mathematics learning is influenced by culture, language, and socioeconomic background.
Apply theories of learning to instruction in various classroom contexts.

M 522 Assessment of Mathematics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered on demand. Connects assessment theory and models to teachers' practice through classroom observations and hand-on activities. Focus on assessment practices consistent with standards-based mathematics and classroom assessment of student learning

View Course Outcomes:

Investigate compare standards-based approaches to formative and summative assessment in mathematics.
Explore the range and scope of assessment practices designed for the mathematics classroom.
Discuss relationships among standards, curriculum, instruction, and assessment.
Connect assessment theory and models to practice through classroom observations and activities.
Develop knowledge of effective assessment items, instruments, and strategies and develop an action plan to implement them.

M 523 Number Structure for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered for two consecutive years; alternates with M 526. Develop the relationship and distinction between the mathematics that underlies the structure of number and the learning and teaching of number structure in schools. Explore representation, abstraction, and basic proof in the context of number and operations. Develop foundations of the real number system and examine relevant research about students' understanding of number

View Course Outcomes:

Characterize the structure and representation of numbers as used in school mathematics.
Apply mathematical concepts and principles to explain operations foundational to high school mathematics.
Understand and apply number theory concepts and theorems, including greatest common factors, least common divisors, tests for divisibility, and the binomial theorem, and explain their connections to high school mathematics.
Analyze K-12 students’ understandings, thinking, and learning of number structure.
Construct and validate mathematical arguments in number structure.

M 524 Linear Algebra for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Sp) Algebraic systems, special matrices, determinants, vector spaces, and linear programming. Includes applications relevant to industry and business and connections to topics in secondary mathematics

View Course Outcomes:

Use the Four Fundamental Subspaces associated with an mxn matrix to find the least squares solution to an inconsistent system.
Represent transformations of the coordinate plane with matrices.

M 525 Analysis for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (F) A study of calculus concepts and processes from graphical, numerical and algebraic perspectives. Technology is incorporated throughout the course. Includes connections to topics in secondary mathematics

View Course Outcomes:

Characterize the real number system.
Demonstrate key features of Cantor Theorem.
Prove convergence and divergence of sequences and series.
Prove results about limits and continuity.

M 526 Discrete Mathematics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Su for two consecutive years; alternates with M 523. A study of classical topics in discrete mathematics, chosen from combinatorics, probability, graph theory, & other areas relevant to secondary mathematics. Emphasis on problem solving and justification

View Course Outcomes:

Understand and apply methods of discrete mathematics such as proofs, counting principles, number theory, logic and set theory relevant to secondary mathematics.
Explain the historical importance of discrete mathematics and its relevance to modern practical situations, such as in cryptography.

M 527 Geometry for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. (Su) Offered Summer, odd years. Explorations of special topics in geometry, such as geometry of transformations including Euclidean motions and similarity, projective geometry, geometric topology and geometry of inversion. Technology in incorporated throughout the course

View Course Outcomes:

Solve geometric problems using transformational, projective, coordinate geometry
Compare Euclidean and Non-Euclidean geometries.

M 528 Curriculum Design: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered Spring, even years. Focuses on the design, implementation, and evaluation of curricula in mathematics. Includes historical changes and trends in mathematics curriculum and an examination of current research

View Course Outcomes:

Students will be able to describe the historical development of mathematics in the United States and place current trends in that context.
Students will be able to use the backward design process to create a unit of study in mathematics.
Students will be able to design and analyze simple curriculum evaluation studies.

M 529 Assessment Models and Issues: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered Fall, odd years. Examines critical K-12 issues including; alignment and interaction of assessment with standards, curriculum, and instruction; role of assessment systems at local, state, and national levels; evaluation of assessment tools and programs; equity considerations in assessment

View Course Outcomes:

Describe the range, scope, and purpose of assessment practices designed for classrooms, districts, and states.
Describe standards-based approaches to formative and summative assessment in mathematics.
Use formative and summative assessment as a tool to inform instruction and improve student learning.
Explain relationships among standards, curriculum, assessment and accountability.
Analyze issues of equity, accountability, and the achievement gap in relation to assessment.

M 533 History of Mathematics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered on demand. Focus on the history of mathematics as a context for classroom instruction. Includes the changing nature of mathematics, classical problems, proofs and mathematical processes, and the development of teaching units that incorporate the history of mathematics

View Course Outcomes:

Demonstrate knowledge of, interest in, enthusiasm for the history of mathematics.
Recognize and appreciate the history and rigor of mathematics as a field.
Apply strategies for using history as a context for teaching mathematics.
Write lesson plans to incorporate the history of mathematics in the classroom.

M 534 Research in Mathematics Education: 3 Credits (3 Lec)

PREREQUISITE: Consent of instructor. () Offered on demand. Examination of quantitative and qualitative research findings and methodology in mathematics education. Review of current trends and literature. Writing for publication and proposals

View Course Outcomes:

Describe current practices, paradigms, and trends in mathematics education research.
Analyze and apply standard research methods and methodologies used in mathematics education research.
Demonstrate awareness of significant bodies of research-based knowledge in mathematics education research.

M 535 Technology and Mathematics for Teaching: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics, or consent of instructor. () Offered Summer, on demand. Calculator, computer and Web-based technologies for K-12 mathematics education. Analysis of the influence of technology on the K-12 mathematics curriculum, instruction, and assessment

View Course Outcomes:

An overarching conception about the purposes for incorporating technology in teaching mathematical topics
Knowledge of K-12 students understanding, thinking, and learning in mathematics when taught with technology
Knowledge of curriculum and curricular materials that integrate technology in learning and teaching mathematics
Knowledge of instructional strategies and representations for teaching and learning mathematics with technology.

M 540 Introduction to Calculus on Manifolds: 3 Credits (3 Lec)

PREREQUISITE: M 503 and M 505 or consent of instructor. (F) Offered Fall of odd years. An introduction to: manifolds and their atlases, fiber bundles, vector fields, tensor fields and differential forms, the exterior and Lie derivatives, Stokes Theorem, & de'Rham cohomology

M 544 Partial Differential Equations I: 3 Credits (3 Lec)

PREREQUISITE: M 384 and M 451, or consent of instructor. () Offered Fall, odd years. An extended survey of the origins of a large number of scientific and mathematical partial differential equations and an overview of the theoretical techniques which are available to solve them

M 545 Partial Differential Equations II: 3 Credits (3 Lec)

PREREQUISITE: M 544 and M 547. () Offered Spring, even years. Linear partial differential equations and the function spaces and functional analysis which one uses to study them. Topics include: Holder and Sobolev functions, Sobolev and Poincare inequalities, embedding density, semigroup theory for evolution equations

M 547 Measure Theory: 3 Credits (3 Lec)

PREREQUISITE: M 384 or M 505. (F) Lebesgue measure, and the Lebesgue integral of functions of a real variable. General measure and integration theory. Lebesgue-Stieltjes integral and product measures

M 551 Complex Analysis: 3 Credits (3 Lec)

PREREQUISITE: M 505. (Sp) Analytic functions and conformal maps, contour integrals, Cauchy's theorem, Cauchy's integral formula, the maximum modulus theorem, harmonic functions, Taylor's theorem and Laurent series. Classification of singularities, the residue theorem and evaluation of definite integrals, Rouche's theorem and the argument principle

M 554 Abstract Algebra II: 3 Credits (3 Lec)

PREREQUISITE: M 504. (F) A second graduate-level course in Abstract Algebra, building on M 504 Abstract Algebra, covering further topics in groups, rings, and modules. Particular topics include: multilinear algebra, homological algebra, commutative algebra, and representation theory

View Course Outcomes:

Develop a deeper facility with groups, rings, fields, and modules, building on the introductory understanding from M 504.
Competency with further fundamental topics in algebra not covered in M 504, such as multilinear algebra, homological algebra, commutative algebra, and representation theory.
Ability to put forth conjectures, explore them via examples, and formulate sensible proof strategies.
Ability to recognize and apply algebraic techniques in advanced mathematics courses, such as algebraic topology, functional analysis, representation theory, number theory, and differential geometry.

M 560 Methods of Applied Mathematics I: 3 Credits (3 Lec)

PREREQUISITE: M 451. () Offered Fall, even years. Finite dimensional vector spaces, spectral theory, Fredholm theorem of matrices, pseudo-inverses. Integral equations, Fredholm alternative and resolvent kernels, singular integral equations. Differential equations and Green's functions, eigenvalue expansions for differential operators

M 561 Methods of Applied Mathematics II: 3 Credits (3 Lec)

PREREQUISITE: M 560. () Offered Spring, odd years. Calculus of variations, Hamilton's principle, asymptotic and perturbation methods, transform techniques and scattering theory. Partial differential equations, Green's functions, separation of variables and transform methods

M 570 Individual Problems: 1-3 Credits (1-3 Other)

PREREQUISITE: Graduate standing, consent of instructor, approval of department head and Dean of Graduate Studies. (F, Sp, Su) Directed research and study on an individual basis
Repeatable up to 6 credits.

M 575 Professional Paper and Project: 1-4 Credits (1 Other)

PREREQUISITE: Graduate standing. (F, Sp, Su) A research or professional paper or project dealing with a topic in the field. The topic must have been mutually agreed upon by the student and his or her major advisor and graduate committee
Repeatable up to 6 credits.

M 576 Internship: 1-12 Credits (1-12 Other)

PREREQUISITE: Graduate standing, consent of instructor and approval of department head. (F, Sp, Su) An individualized assignment arranged with an agency, business or other organization to provide guided experience in the field
Repeatable up to 99 credits.

M 577 Conducting Action Research in Mathematics Education: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing in mathematics education, teaching endorsement in mathematics and consent of instructor. (Sp) Offered Spring, odd years. With guidance from faculty, students conduct action research addressing a problem in the context of their classroom, school or district that influences student success in mathematics. Students work with a faculty advisor to implement an intervention, collect and analyze data resulting, and summarize results. Findings are presented orally to peers and faculty

View Course Outcomes:

Identify plan action to address a problem relevant to improving mathematics education
Demonstrate expertise in a specific mathematics teaching and learning knowledge base
Plan and carry out a structured process of implementing an innovation or intervention
Plan and carry out a structured process of data collection and analysis to learn from your innovation or intervention
Interpret and reflect on findings and present conclusions to advance the knowledge base

M 580 Special Topics: 4 Credits (4 Lec, 4 Other)

PREREQUISITE: Upper division courses and others as determined for each offering. () Offered on demand. Courses not required in any curriculum for which there is a particular one time need, or given on a trial basis to determine acceptability & demand before requesting a regular course number
Repeatable up to 12 credits.

M 581 Numerical Solution of Partial Differential Equations I: 3 Credits (3 Lec)

PREREQUISITE: M 442. (F) Finite difference and finite element solution techniques for elliptic, parabolic, and hyperbolic partial differential equations, numerical linear algebra

M 582 Numerical Solution of Partial Differential Equations II: 3 Credits (3 Lec)

PREREQUISITE: M 581. (Sp) A continuation of topics from M 581

M 584 Functional Analysis I: 3 Credits (3 Lec)

PREREQUISITE: M 547. (F) Offered Fall, even years. Banach spaces, fixed point theorems, Hilbert spaces, the Dirichlet principle, generalized Fourier series, & spectral theory

M 585 Functional Analysis II: 3 Credits (3 Lec)

PREREQUISITE: M 584. (Sp) Offered Spring, odd years. The Hahn Banach theorem, variational principles, weak convergence, uniform boundedness theorem, the open mapping theorem & the implicit function theorem

M 586 Probability Theory: 3 Credits (3 Lec)

PREREQUISITE: M 547. (Sp) Offered Spring, on demand. Combinatorial probability and measure theoretic foundations of probability; axioms for probability spaces. Borel-Cantelli Lemmas, weak & strong laws of large numbers, and the central limit problem

M 587 Lie Groups: 3 Credits (3 Lec)

PREREQUISITE: M 504, M 511. (F) Offered fall, even years. Lie groups, Lie algebras, representation theory

View Course Outcomes:

Gain familiarity with standard techniques and concepts in differential geometry and representation theory.
Tangent bundles and Lie algebras, Riemannian metrics and exponential maps, representations, maximal tori and Weil data_
Acquire access to a host of rich examples in the subject of differential geometry.
Acquire ability to recognize mathematical problems as ones concerning Lie theory; notably as continuous symmetries and representations thereof.
Become aware of, and empowered by, the effectiveness of infinitesimals and symmetry as they manifest in other parts of mathematics and physics.
Develop ability to ask articulate questions, and to develop mathematics to solve those questions.
Develop mathematical communication skills, especially concerning writing.

M 588 Professional Development: 1-3 Credits (1-3 Lec)

PREREQUISITE: Graduate standing, teaching experience or current employment in a school organization, consent of instructor and Dean of Graduate Studies. () Offered on demand. Courses offered on a one time basis to fulfill professional development needs of in-service educators. A specific focus is given to each course which is appropriately subtitled. May be repeated
Repeatable up to 3 credits.

M 589 Graduate Consultation: 3 Credits (3 Other)

PREREQUISITE: Master's standing. (F, Sp, Su) This course may be used only by students who have completed all of their course work (and thesis, if on a thesis plan) but who need additional faculty or staff time

M 590 Master's Thesis: 1-10 Credits (1 Other)

PREREQUISITE: Master's standing
Repeatable up to 99 credits.

M 591 Topics in Applied Math I: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing and consent of instructor. On demand. Topics may include numerical solution of linear and nonlinear problems, eigenvalue problems, continuation methods, numerical optimization, computational mechanics, spectral methods, bifurcation theory, invariant manifold theory, index theory, nonlinear analysis, reaction-diffusion equations, nonlinear oscillations, asymptotic methods and perturbation methods

View Course Outcomes:

Master the basic theory of hyperbolic PDEs and nonlinear conservations laws
Understand the development of high-resolution shock-capturing finite volume methods for solving these equations
Demonstrate a knowledge of some physical applications of hyperbolic problems
Gain experience in using the Clawpack software for solving these equations, including how to set up a new problem
Learn the basics of Python programming and use of Jupyter notebooks (and a bit about Fortran)

M 592 Topics in Applied Math II: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing and consent of instructor. () On demand. Topics may include numerical solution of linear and nonlinear problems, eigenvalue problems, continuation methods, numerical optimization, computational mechanics, spectral methods, bifurcation theory, invariant manifold theory, index theory, nonlinear analysis, reaction-diffusion equations, nonlinear oscillations, asymptotic methods & perturbation methods

M 594 Seminar: 1 Credits (1 Other)

PREREQUISITE: Graduate standing or seniors by petition. () On demand. Course prerequisites as determined for each offering. Topics offered at the graduate level which are not covered in regular courses. Students participate in preparing & presenting discussion material
Repeatable up to 6 credits.

M 595 Dynamical Systems I: 3 Credits (3 Lec)

PREREQUISITE: M 503. (F) Offered Fall, odd years. Topics in differential equations including existence and uniqueness, continuous dependence on parameters, extendibility, the existence and stability of equilibria and limit cycles & the Poincare-Bendixon theorem

M 596 Dynamical Systems II: 3 Credits (3 Lec)

PREREQUISITE: M 595. (Sp) Offered Spring, even years. Topics include Hartman's theorem, invariant manifold theory, Smale-Birkhoff theorem, horseshoe chaos, & the Melnikov method. Topics in discrete dynamical systems may also be covered

M 597 Topics in Math i: Character Varieties and 3-manifolds: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing or consent of instructor. () Offered on demand. Topics include the theory of representations of finitely generated groups into matrix groups and applications of this theory to the study of low-dimensional topology. Our primary tools will be SL(2, C) and PSL(2, C) character varieties. Culler-Shalen theory and its applications will be discussed in depth. We will also cover some of the basics of low-dimensional topology along with classical affine and projective algebraic geometry. Computational techniques in algebraic geometry and commutative algebra will be highlighter to encourage experimentation and exploration

View Course Outcomes:

Master the basics of representation theory, classical affine projective algebraic geometry, and 3-manifold theory needed to define and understand the applications of character varieties.
Understand the Culler-Shalen machine which uses character varieties to produce essential surfaces in 3-manifolds.
Understand the A-polynomial, the famous ``boundary slopes are boundary slopes'' theorem, and the root-of-unity phenomenon.
Learn some basic computational techniques related to the study of character varieties.

M 598 Topics in Math II: 3 Credits (3 Lec)

PREREQUISITE: Graduate standing, consent of instructor. () Offered on demand. Topics selected from: continuum theory, symbolic dynamics, ergodic theory and low dimensional topology

M 689 Doctoral Reading & Research: 3-5 Credits (3 Other)

PREREQUISITE: Doctoral standing. (F, Sp, Su) This course may be used by doctoral students who are reading research publications in the field in preparation for doctoral thesis research
Repeatable up to 15 credits.

M 690 Doctoral Thesis: 1-10 Credits (1 Other)

PREREQUISITE: Doctoral standing
Repeatable up to 99 credits.