Course Learning Objectives - Mathematics Department Handbook

University Senate guidance regarding Course Learning Objectives.
“After completing this course, a student should be able to…”

MAT 117

Have a deep understanding of numeration, operations, number theory, probability and statistics concepts and processes that are foundational for PreK-6 school mathematics.
Engage in mathematical thinking, reasoning, communication, and problem solving.
Use technology as a tool to explore, learn, and communicate mathematics
Reflect on their learning of mathematics.

MAT 118

Have a deep understanding of geometry, measurement, and rational number concepts and processes that are foundational for PreK-6 school mathematics.
Engage in mathematical thinking, reasoning, communication, and problem solving.
Use technology as a tool to explore, learn, and communicate mathematics
Reflect on their learning of mathematics.

MAT 121

Make and interpret diagrams that illustrate data.
Compute and interpret numerical descriptions of data including mean, median, mode, quartiles, percentiles, standard deviation, and variance.
Compute and interpret elementary probabilities.
Compute and interpret probabilities based on normal distributions.
Compute from data and interpret confidence intervals for mean, proportion, and standard deviation.
Use the statistical package Statdisk to do all the previous computations.

MAT 122

Perform and interpret parametric hypothesis tests for one proportion, one mean, one standard deviation, and linear correlation.
Construct and interpret scatter plots.
Compute and interpret linear regression lines.
Perform and interpret hypothesis tests involving goodness-of-fit and contingency tables.
Perform and interpret non-parametric hypothesis tests for matched pairs, one median, and nominal data.
Construct and interpret control charts for variation, mean, and proportion.
Use the statistical package Statdisk to make the preceding calculations.
Compute probabilities using Bayes’ Theorem.
Solve systems of linear equations using the echelon method and the Gauss-Jordan method.
Add, subtract, multiply, and find inverses of matrices.
Compute probabilities with Markov chains.
Find long range stable states for regular and absorbing Markov chains.

MAT 183

Effectively use appropriate mathematical technology.
Determine the basic properties of matrices and solve simple matrix equations.
Use the basic properties and formulas of probability and statistics to compute simple probabilities in a statistical setting and to interpret the results.
Recognize and work with elementary Markov processes.
Apply basic formulas from the mathematics of finance in a variety of settings that arise in personal finance, such as interest, annuities and amortization.

MAT 193 and 194

Demonstrate strategic competence in using and understanding the usage of mathematical notation to solve problems involving functions, expressions, and equations;
Demonstrate conceptual understanding of functions, the short- and long-run behavior of functions, and equivalent expressions;
Demonstrate adaptive reasoning in selecting and justifying the choice of an appropriate mathematical model for a given real world problem, representing the model in symbols, graphs, and numeric forms with and without graphing technology;
Demonstrate procedural fluency in accurately performing algebraic manipulations, applying skills appropriately in the problem context;
Demonstrate productive disposition in their ability to make sense of problems with mathematics.

MAT 221

Demonstrate understanding of the key principles of descriptive statistical analyses
Summarize data numerically and graphically
Recognize particular sampling and statistical experimental designs, as well as their usage and purpose
Demonstrate understanding of basic concepts of mathematical probability.
Compute probabilities and work with key random variable distributions based on specific contexts and using various rules
Demonstrate understanding of the importance of Central Limit Theorem and the associated approximations
Demonstrate understanding of basic statistical inference procedures (confidence intervals and statistical tests of hypotheses)

MAT 222

Demonstrate understanding of basic ideas of statistical inference.
Calculate, estimate, and manipulate probabilities and random variable distributions commonly used.
Conduct, with and without the aid of technology, statistical model determinations and assessments.
Choose appropriate statistical models to solve applied problems.
Apply statistical software and interpret their output for real data analysis.

MAT 284

Perform algebraic manipulations with the polynomials, exponential, and logarithmic functions
Compute basic indeterminate forms of limits using algebraic manipulations
Find the derivatives of rational, exponential, and logarithmic functions, and their compositions
Recast a word problem in terms of mathematical optimization and then solve it
Find indefinite integrals of the powers of the variable and the exponential function
Compute and interpret the marginal cost and revenue functions and the elasticity of demand

MAT 285

Use the properties of linear, polynomial, exponential, logarithmic and trigonometric functions.
Evaluate limits and determine the continuity of a function at a given point.
Compute the derivatives of polynomials, rational functions, trigonometric functions, logarithmic and exponential functions
Differentiate implicitly.
Use derivatives to find the rates of change of related quantities
Use derivatives to solve maximum/minimum problems, including in applied contexts.
Solve problems involving antiderivatives and areas
State and apply the Fundamental Theorem of Calculus

MAT 286

Use integration by parts
Find volumes of solids using integrals
Evaluate improper integrals and double integrals
Compute partial derivatives
Use partial derivatives to solve maximum/minimum problems in two variables
Solve differential equations of certain types: separable, first-order linear, etc.
Use differential equations to solve applied problems such as the mixing of fluids

MAT 295

Estimate and calculate limits proficiently
Use limits to define and calculate basic derivatives
Interpret derivatives in various ways (algebraically, graphically, physically)
Use shortcuts to calculate derivatives proficiently
Apply derivatives to analyze functions and solve related rates and optimization problems
Explain what integrals are and use them to find areas and total change
Use the Evaluation Theorem and u-substitution to evaluate integrals proficiently

MAT 296

Apply integrals to solve area, volume, arc length, and surface area problems
Use advanced integration techniques to evaluate integrals proficiently
Determine convergence or divergence of sequences and series
Use Taylor or Maclaurin series to represent and analyze functions
Use parametric equations and polar coordinates to represent and analyze curves
Apply polar coordinates to find areas bounded by polar curves

MAT 331

Solve and analyze systems of linear equations
Use the properties of vectors and matrices
Determine the linear independence of a set of vectors
Translate geometric notions into matrix and vector problems
Represent linear transformations by matrices
Use the concepts of vector subspaces, bases, and linear independence to investigate properties of matrices
Find eigenvalues and eigenvectors of a matrix and determine its diagonalizability
Construct an orthogonal basis for a subspace

MAT 375

Understand the nature and role of deductive reasoning in mathematics
Perform basic calculations with propositional logic and the theory of sets
Use and understand the usage of mathematical notation
Read, summarize, and evaluate proofs and other mathematical discourse
Construct and write rigorous proofs of mathematical statements using mathematical induction and other standard proof techniques
Communicate mathematical reasoning and proof through clear writing and oral communication
Distinguish between finite, countable, and uncountable sets.

MAT 397

Apply vectors to describe and study the geometry of space
Extend the notion of a function to vector and multivariable functions
Calculate partial derivatives and apply them to analyze multivariable functions
Evaluate double and triple integrals and apply them to analyze three-dimensional objects
Evaluate line and surface integrals of scalar functions and vector fields
Apply Green’s, Stokes’, and divergence theorems to evaluate line and surface integrals proficiently

MAT 412

Analyze mathematical statements.
Use mathematical notation correctly and efficiently.
Follow proofs and other mathematical discourse.
Apply the ideas of known proofs to new problems.
Write proofs of mathematical statements related to the topics covered in this course (real numbers, limits, continuity, derivatives, Riemann integral).

MAT 414

Solve first-order separable, exact, and linear differential equations and associated initial value problems.
Find equilibrium solutions to autonomous differential equations and determine their stability.
Solve second-order linear differential equations and their associated initial value problems using methods including the characteristic equation, undetermined coefficients, reduction of order, and variation of parameters.
Compute Laplace transforms and apply them to the solution of initial value problems for differential equations.
Solve systems of linear differential equations.
Apply computational and analytical techniques to formulate, solve, and interpret mathematical models involving differential equations.
Employ critical thinking and mathematical reasoning skills when analyzing solutions, existence and uniqueness theorems, and the connections between ordinary differential equations and other areas of mathematics, such as linear algebra and calculus.
Utilize technology, including computational software and graphing utilities, to help solve ordinary differential equations, analyze solutions, and visualize solution behavior.

MAT 422

Access, organize, and re-format data,
Understand and utilize general programming concepts for statistical computing purposes,
Perform data summary and basic statistical analyses,
Generate data visualization and interpret such outputs.

MAT 485

Solve separable differential equations and linear differential equations.
Solve systems of linear differential equations.
Find equilibrium solutions to autonomous systems and determine their stability.
Compute Laplace transforms and apply them to the solution of initial value problems for differential equations.
Use the tools of matrix algebra to solve linear systems, compute determinants, find the inverse, the eigenvalues and eigenvectors for a given matrix.
Determine the dimension and find a basis for the kernel and the image of a linear transformation.

MAT 495/695

Articulate the key methods of data science and the contexts for which they apply
Use the software implementations of the methods covered in the course, and interpret their output
Understand the mathematical and statistical background behind the methods
Recognize the limitations and nuances of the methods based on the theory
Follow the current developments related to the methods

MAT 511

Determine whether a given subset of Euclidean space is open, closed, compact, or connected.
State and apply the definitions of continuity and uniform continuity for functions of several variables.
Use the definition of differentiability and the Chain Rule for functions of several variables.
Find the extreme values for a function of several variables.
State and apply the Inverse Function Theorem and Implicit Function Theorem
Define and evaluate the Riemann integral over a domain in Euclidean space.
Define and evaluate line integrals and surface integrals of vector fields.
Apply the fundamental theorems of vector analysis (Green’s, Stokes’, and divergence theorem)

MAT 512

Analyze mathematical statements.
Use mathematical notation correctly and efficiently.
Follow proofs and other mathematical discourse.
Apply the ideas of known proofs to new problems.
Write proofs of mathematical statements related to the topics covered in this course (Riemann integral, infinite series, uniform convergence).

MAT 513

Demonstrate algebraic identities and inequalities involving complex numbers.
Determine the topological properties of a given subset of the complex plane.
Determine the complex differentiability of a given function.
Evaluate and estimate path integrals
Represent a given function by a power series (Taylor or Laurent series)
Find the radius(es) of convergence of a given power series
Find and classify the singularities of a function
Compute the residues of a function and use them for evaluating integrals

MAT 517

Classify partial differential equations by their order, linearity, and homogeneity
Solve the wave and diffusion equations in one spatial dimension
Find the eigenvalues and eigenfunctions for boundary value problems (Dirichlet, Neumann, Robin boundary conditions)
Use the separation of variables and the theory of Fourier series to solve boundary value problems for the diffusion and wave equations
Solve Laplace’s equation in a rectangle or a disk

MAT 518

Compute the Fourier series of a given function.
Use the properties of Fourier series, such as the convolution property.
Define and evaluate the Fourier transform and inverse Fourier Transform of a given function.
Define and evaluate the Discrete Fourier transform of a given periodic sequence.
Define and evaluate the Haar wavelet decomposition of a given signal.

MAT 521

understand basic ideas of mathematical probability and its applications;
calculate, estimate, and manipulate probabilities and random variable distributions commonly used, with and without the aid of technology;
choose appropriate mathematical models to solve given word problems;
understand and apply mathematical notations relating to theoretical basis of probability.

MAT 524

describe basic ideas of least squares fitting;
explain concepts, theory and methods of linear and logistic regression analysis;
estimate and make inferences on the model parameters with and without the aid of statistical software;
identify appropriate regression models using various model selection methods;
conduct data analysis using R and interpret results in software output;
accurately and effectively summarize results.

MAT 525

understand basic ideas of statistical inference;
calculate, estimate, and manipulate probabilities and random variable distributions commonly used;
conduct, with and without the aid of technology, probability model determinations and assessments;
choose appropriate mathematical models to solve statistical problems;
understand theoretical basis of fundamental inferential procedures;
apply statistical software and interpret their output for real data analysis.

MAT 526

calculate, estimate, and manipulate probabilities and random variable distributions commonly used;
conduct, with and without the aid of technology, probability model determinations and assessments;
understand the role of stochastic modeling
gain practice developing and analyzing specific stochastic models
learn and master some of the basic mathematical tools and techniques of stochastic modeling
understand the relevant mathematical concepts, methods, and theoretical basis of fundamental stochastic models
choose appropriate mathematical models to solve specific problems

MAT 527

describe basic ideas of experimental design
explain concepts, theory and methods of analysis of variance
estimate and make inferences on the model parameters with and without the aid of statistical software
identify appropriate statistical models and methods to analyze data
conduct data analysis using R and SAS and interpret results in software output
accurately and effectively summarize results.

MAT 528 tbd

MAT 529 tbd*

MAT 531

State the definitions for the concepts of abstract vector space, inner product space, linear transformation, linear operator, eigenvalue, eigenvector, and diagonalization.
Provide proofs for many of the basic theorems involving the previously mentioned concepts.
Solve problems involving these concepts and provide rigorous proof for these solutions.

MAT 532

Solve systems of linear equations using Gaussian elimination with back substitution and Gauss-Jordan elimination.
Analyze algorithms for solving linear systems and understand error due to rounding and floating point arithmetic and the implications for the conditioning of the linear system.
Construct linear models for a variety of applied problems.
Compute solutions to linear systems accurately and efficiently by hand and with the aid of software when appropriate.
Understand and use ideas of linear independence, basis, and dimension in vector spaces to draw conclusions about applied problems in linear algebra.
Understand and compute factorizations and decompositions that aid in the computation of solutions to linear systems (LU factorization, QR factorization, singular value decomposition).

MAT 534

State the definition of and give examples of groups and rings.
Provide proofs for many of the basic theorems in groups.
Solve problems involving many of the basic properties of groups and provide rigorous proofs for these solutions.
Provide proofs for a few of the elementary theorems in rings.
Solve problems involving a few of the elementary properties of rings and provide rigorous proofs of these solutions.

MAT 541

Use the elementary facts and computational methods of number theory.
Solve problems related to number theory, be able to define and work with the important number theoretic functions.
Be able to state the central theorems, understand the methods of proof used, and be able to make similar arguments.
Model situations from everyday life and other branches of science.

MAT 545

Master the definitions and elementary theorems of graph theory.
Construct graphs to model complicated relationships between the object in an arbitrary collection of concrete or abstract objects and apply basic graph theory to better understand these relationships.
Master each of the counting techniques covered in the course.
Identify and apply the appropriate counting technique for a wide variety of counting problems.

MAT 551

State the axioms for Euclidean and some non-Euclidean geometries.
Explain the difference between Euclidean and non-Euclidean geometries.
Provide proofs for many of the basic theorems in Euclidean and non-Euclidean geometries.
Solve problems involving many of the basic properties of Euclidean and non-Euclidean geometries and provide rigorous proofs for these solutions.

MAT 554

Compute the Frenet-Serret frame of a space curve. State the fundamental theorem of space curves and understand its relationship to the Frenet-Serret frame.
Compute the curvature and torsion of a space curve.
Determine whether a given curve is a geodesic on a given surface.
Compute the first and second fundamental forms and Gaussian and mean curvatures of a surface
Determine whether a given surface is a minimal surface.
State the Gauss-Bonnet theorem and enunciate its implications for the geometry and topology of surfaces.

MAT 562

Define the fundamental concepts of topology, including metric and topological spaces, continuous maps, homeomorphisms, compactness, convergence, connectedness, and separation axioms.
Evaluate and construct mathematical proofs involving topological concepts.
Construct examples of topological spaces with required properties or verify that a given topological space has such properties.
Effectively communicate topological arguments.

MAT 581

Articulate the difficulty of obtaining analytic solutions and why numerical methods are needed
Write programs in MATLAB, use common built-in functions, write & test new functions, and visualize functions and data.
Articulate the meaning and importance of error analysis.
Implement basic numerical methods for nonlinear equations, interpolation, differentiation, integration, and ordinary differential equations.
Perform error analysis for these methods.
Compare the accuracy and computational efficiency of different methods.
Choose an appropriate numerical method to solve a given word problem.

MAT 593

Solve certain mathematical problems by methods used at different periods in history and in different cultures even though these methods might be different from those currently used.
Explain how many mathematical ideas have remained constant over time and in different cultures while others have changed and varied.
Explain how what happens in mathematics is often related to what is happening in the general culture.
Describe how approaches to certain mathematical problems have evolved over time.

MAT 598

Identify a real-world problem for analysis.
Collect and sample data using various methods.
Provide reviews of technical literature.
Conduct statistical modeling and data analysis.
Write statistical reports.
Give presentations and interpreting results in a professional manner.
Plan and manage a project.

MAT 599

Present mathematics to a mathematical audience in a clear understandable manner.
Think creatively about how to solve a mathematical problem and explain these thoughts to others.
Exhibit deep understanding of covered mathematical topics.

Notes

tbd* = course is yet to run

Generic objectives

These objectives are a fallback option to be used when specific objectives are not available.

For all Math courses:

Students will be able to use and understand the usage of mathematical notation
Students will be able to select an appropriate mathematical model for a given real world problem
Students will be able to do hand calculations accurately and appropriately
Students will be able to do calculations with the aid of appropriate hardware and/or software

For all Math courses MAT 295 and above:

Students will understand the nature and role of deductive reasoning in mathematics
Students will be able to follow proofs and other mathematical discourse
Students will be able to write simple proofs in the major proof formats (direct, indirect, inductive), and, more generally, to engage in mathematical discourse
Students will be able to apprehend and enunciate the limitations of conclusions drawn from mathematical models

For all Math majors:

Students will have a basic knowledge of the contributions and significance of important historical figures in mathematics
Students will have a basic knowledge of the major modern theories of analysis, abstract algebra, geometry, and applied mathematics
Students will be able to effectively use mathematical word processing software
Students will have a basic understanding of career options available to mathematics majors
Students will be able to locate and use sources and tools that aid mathematical scholarship