Department of Mathematics and Computer Science

Santa Clara University
Department of Mathematics and Computer Science

Course Goals and Objectives

• Mission Statement
• General Departmental Goals and Objectives of Courses
• Core Curriculum Learning Goals and Objectives for Mathematics Courses
• General Information for All Syllabi
• Specific Course Goals and Objectives

Mission Statement: The Department of Mathematics and Computer Science promotes the methods and benefits of rigourous, objective mathematical thought, theoretical computer science and algorithmic and logical understanding, both for their intrinsic beauty and for their applications to other disciplines. These principles are incorporated into the larger University program of educating the whole person in the Jesuit tradition. Our aims are focused not only on our students, but also on our professional communities and the community at large.

General Departmental Goals and Objectives of Courses

Link to Complete List of Goals and Objectives for the Department.
Link to Chart Relating Course Goals and Objectives to Specific Courses.

Goal 1: Connect the study of mathematics and computer science to other disciplines.
Objective:

a) Students will obtain skills and logical perspectives in our introductory (core) courses that prepare them for subsequent courses inside and outside our department. Specifically, students will develop proficiency with the techniques of mathematics and/or computer science, the ability to evaluate logical arguments, and the ability to apply mathematical methodologies to solving real world problems.

a) Students must be able to understand and write rigorous arguments (i.e., proofs) for theorems.

b) Students must show mastery in the three basic areas of mathematics: analysis, algebra, and topology /geometry on a basic level in lower division courses and at an advanced level in upper division courses.

a) Students will develop a strong foundation in programming, software development and data manipulation and become familiar with theoretical aspects of computer science.

b) Students will acquire a strong facility for developing, analyzing, and applying algorithms.

Core Curriculum Learning Goals and Objectives for Mathematics Courses (primarily for Math 6, 7, 11, 30). Goals: Critical Thinking, Mathematics and Quantitative Reasoning
Objectives: Students will:

1.1 Demonstrate their problem solving skills, including their ability to interpret problem situations, choose among several potentially appropriate mathematical methods of solution, persist in the face of difficulty, and present full and cogent solutions that include appropriate justification for their reasoning.
1.2 Understand and be able to articulate the differences between inductive and deductive reasoning. In particular, students will appreciate the role of mathematical proof in formalizing deductive reasoning and as a means of conveying mathematical knowledge, and to understand the differences between proofs and other less formal arguments.
1.3 Utilize and describe mathematical ideas from multiple perspectives, including the internal connections between geometry, algebra, and numerical computation, as well as the connections between theory and applications. This flexibility should be evident in students' approach to problem solving as well as their ability to communicate their solutions and methods.
1.4 Demonstrate an understanding of mathematical content (including the limits to its application) that goes beyond mere fluency in using mathematical symbols, language and formulas.

General Information for All Syllabi

Academic Integrity: The penalty for cheating may be a failing grade for the course, and the University may take further disciplinary action. All of the work that you turn in should be your own, and not that of a classmate or copied from another source. Please see http://www.scu.edu/studentlife/resources/academicintegrity/index.cfm for further information.

Disability accommodation policy: To request academic accommodations for a disability, students must contact the Disability Resources Office located in Benson room 216, (408) 554-4111; TTY (408) 554-5445. Students must provide documentation of a disability to Disability Resources prior to receiving accommodations.

Specific Course Goals and Objectives

Prenotes:

(1) In addition to providing you with a good foundation in a fundamental area of mathematics, all mathematics courses, especially those designated as fulfilling the Santa Clara Core, will also contribute to a student's skills and logical perspective that will be applicable to many other courses requiring mathematical methods and careful reasoning.
(2) In most mathematics courses, all assignments, quizzes, and exams foster all core learning goals, the departmental goals, and the course goals.

• CS 3 -- Introduction to Computing and Applications
• Math 4 (formerly 41) -- The Nature of Mathematics
• Math 6 -- Finite Mathematics for Social Science
• Math 7 -- Calculus for Social Science
• Math 8 -- Introduction to Statistics
• Math 9 -- Precalculus
• CS 10 -- Introduction to Computer Science
• Math 11 -- Calculus and Analytic Geometry I
• Math 12 -- Calculus and Analytic Geometry II
• Math 13 -- Calculus and Analytic Geometry III
• Math 14 (formerly 21) -- Calculus and Analytic Geometry IV
• Math 21 (to be renumbered 14 in Fall 2009) -- Calculus and Analytic Geometry IV
• Math 22 -- Differential Equations
• Math 30 -- Calculus for Business I
• Math 31 -- Calculus for Business II
• Math 32 -- Mathematical Logic
• Math 41 (to be renumbered 4 in Fall 2009) -- The Nature of Mathematics
• Math 44 -- Mathematics for Elementary Teachers I
• Math 45 -- Mathematics for Elementary Teachers II
• Math 51/COEN 19 -- Discrete Mathematics
• Math 52 -- Introduction to Abstract Algebra
• Math 53 -- Linear Algebra
• CS 60 -- Object-oriented Programming
• CS 61 -- Data Structures
• Math/CS 90 -- Lower-Division Seminars
• Math 100 -- Writing in the Mathematical Sciences
• Math 101 -- A Survey of Geometry
• Math 102 -- Advanced Calculus
• Math 103 -- Linear Algebra II
• Math 105 -- Theory of Functions of a Complex Variable
• Math 111 -- Abstract Algebra I
• Math 112 -- Abstract Algebra II
• Math 113 -- Topology
• Math 122 -- Probability and Statistics I
• Math 123 -- Probability and Statistics II
• Math 125 -- Mathematical Finance
• Math 133 -- Logic and Foundations
• Math 134 -- Set Theory
• Math 144 -- Partial Differential Equatiions
• Math 153 -- Intermediate Analysis I
• Math 154 -- Intermediate Analysis II
• Math 155 -- Ordinary Differential Equations
• CS 161 -- Theory of Automata and Languages I
• CS 162 -- Theory of Automata and Languages II
• CS 163/COEN 179 -- Theory of Algorithms
• CS 164 -- Computer Simulation
• Math/CS 165 -- Linear Programming
• Math/CS 166 -- Numerical Analysis
• CS 167 -- Switching Theory and Boolean Algebra
• CS 168 -- Computer Graphics
• CS 169 -- Programming Languages
• Math 170 -- Development of Mathematics
• Math 172 -- Problem Solving
• Math 174 -- Differential Geometry
• Math 175 -- Theory of Numbers
• Math 176 -- Combinatorics
• Math 177 -- Graph Theory
• Math 178 -- Cryptography
• CS 181 -- Applied Cryptography
• CS 182 -- Digital Steganography
• Math/CS 190 -- Upper-Division Seminars
• Math/CS 196 -- Advanced Topics
• Math/CS 198 -- Internship/Practicum
• Math/CS 199 -- Independent Study

• will practice effective communication and interaction skills.
• as part of classroom learning, will interpret problem situations, choose among several appropriate mathematical methods of solution, persist to a solution and present solutions verbally and in written form that include appropriate justification for your reasoning.
• on assignments and tests, will need to be able to formalize deductive reasoning, make connections between theory and applications, and demonstrate an understanding of mathematical content beyond symbols and formulas.

Math 7 -- Calculus for Social Science
General Departmental Goals and Objectives: 1a; 2b
Specific Goals and Objectives: In this course, students will apply the methods of differential and integral calculus to solve a variety of problems particularly applicable to the social sciences and to explain their reasoning.

• Students will use the derivative as an important problem-solving tool for real-world situations. Students will also briefly look at indefinite and definite integrals.
• Students will predict properties of curves from their derivative.
• Students will be able to explain how differential calculus combines geometric ideas of slope, real-world concepts of rates and analytical concepts of derivatives to give a unified perspective of mathematics.
• Beyond computational proficiency, students will be led to understand the meaning of results, as well as central theorems of mathematics.

• explain the basic concepts of statistics
• summarize numeric data by computing descriptive statistics
• apply basic probability theory
• explain confidence intervals
• test hypotheses applying probability theory
• explain the differences among various statistical techniques and identify an appropriate technique for a given set of variables and research questions
• use a calculator as a tool in statistical analysis

CS 10 -- Introduction to Computer Science
General Departmental Goals and Objectives: 1a; 3a; 3b
Specific Goals and Objectives: In this course, students will develop:
1. proficiency in basic algorithm extraction. Given a problem statement, be able to extract the given data as input, formulate a clear goal or desired output, and lay out a set of instructions to go from input to output.
2. competence with the fundamental flow control structures of branching, looping, subroutine, and recursion as well as the data structures of variables and arrays. Be able to demonstrate an ability to read and write code using these structures.
3. understanding of the moral issues involved in computing and the role it plays in our society.
4. a capacity to work with object oriented programming at the introductory level as demonstrated by the ability to read and write code involving classes and objects.

Math 11 -- Calculus and Analytic Geometry I
Fulfills Mathematics Core Requirement
General Departmental Goals and Objectives: 1a; 2b
Specific Goals and Objectives: In this course students will apply the methods of differential calculus to solve a variety of problems and to explain their reasoning.

• (CG 1.1) Students will use the derivative as an important problem-solving tool.
• (CG 1.2) Students will predict properties of curves from their derivative.
• (CG 1.3) Students will be able to explain how differential calculus combines geometric ideas of slope, real-world concepts of rates and analytical concepts of derivatives to give a unified perspective of mathematics.

• emphasis will be placed on the use of the definite and indefinite integral as an important problem-solving tool.
• students will learn to predict properties of functions from their antiderivatives.
• integral calculus will be seen to combine geometric ideas of area and analytic concepts of the indefinite integral to give a unified perspective of mathematics.
• beyond computational proficiency, students will be led to understand the meaning of results, as well as central results such as the Fundamental Theorems of Calculus.

• emphasis will be placed on using results from previous calculus courses to solve more complex problems.
• students will learn to find infinite power series to approximate functions and also some non-elementary integrals involving them.
• students will have multiple opportunities to analyze problems from analytical, geometric, and numerical perspectives.
• beyond computational proficiency, students will strive to understand the meaning of our results, as well as encountering some central theorems of mathematics.

• emphasis will be placed on the use of integrals and multivariable calculus as an important problem-solving tool.
• students will learn to analyze surfaces, using partial derivatives, and their volumes by studying properties of their antiderivatives.
• students will combine geometric visualization (of 2 dimensional curves and 3 dimensional surfaces) with careful analytical reasoning to solve problems and connect our ideas to other disciplines.
• beyond computational proficiency, students will strive to understand the meaning of our results, as well as encountering some central theorems of mathematics.

• will be able to categorize Ordinary Differential Equations (ODEs): order, linear vs. nonlinear, homogeneous vs. nonhomogeneous, etc.
• will, within each category, master techniques to solve ODEs;
• will connect material both to other courses within the department and to phenomena from other disciplines.

• (CG 1.1) Students will use the derivative as an important problem-solving tool for real-world situations.
• (CG 1.2) Students will predict properties of curves, and thus of rates of change, from their derivative.
• (CG 1.3) Students will be able to explain how differential calculus combines geometric ideas of slope, real-world concepts of rates and analytical concepts of derivatives to give a unified perspective of mathematics.

Math 44 -- Mathematics for Elementary Teachers I
General Departmental Goals and Objectives: 1a; 2b; 3b
Specific Goals and Objectives: In this course, students will:

• learn problem-solving techniques and become proficient and confident in solving problems;
• learn to communicate effectively about mathematics, both verbally and in writing;
• reaffirm and strenghten their foundation in arithmetic, learning why things work the way that they do.

• learn problem-solving techniques and become proficient and confident in solving problems;
• learn to communicate effectively about mathematics, both verbally and in writing;
• reaffirm and strengthen your foundation in the various mathematics content standards for the elementary level, emphasizing the whys as well as alternatives to standard approaches.

• Prove (or disprove) that two propositions are logically equivalent.
• Prove that two sets are equal.
• Determine if a function is invertible.
• Determine the next term in a sequence, total complicated summations, and encrypt and decrypt secret messages.
• Count how many passwords are possible given certain constraints, count how many ways one can get particular poker hands, etc.

Math 53 -- Linear Algebra
General Departmental Goals and Objectives: 1a; 2b; 3b
Specific Course Goals and Objectives: In this course, students will learn to:

• solve problems, including choosing and developing appropriate methods, as well as communicating mathematical ideas effectively. Emphasis will be placed on the use of matrix theory as an important problem-solving tool;
• use mathematical reasoning and deduction to draw valid conclusions from given information; for example, students will learn to predict the nature of solution sets of linear systems using matrix theory;
• use and understand mathematical ideas from multiple and interconnected perspectives, including algebraic, geometric, analytical and numerical points of view. Linear algebra combines geometric ideas of vectors spaces, solutions to systems of linear equations, and fundamental concepts of eigenvectors and eigenvalues ini pure and applied mathematics to give a unified perspective of mathematics;
• understand significant mathematical ideas and results in addition to mastering computational techniques. In addition to knowing how to carry out a certain procedure, students will learn when that procedure is appropriate and, most importantly, why it works. Students will strive to understand the meaning of results, as well as encounter some central theorems of mathematics.

• understand and practice the basic tenets of object-oriented programming;
• use, design, and implement abstract data types; and
• develop graphical user interfaces usiing C++ and Qt.

CS 61 -- Data Structures
General Departmental Goals and Objectives: 3a; 3b
Specific Course Goals and Objectives: By the end of this course students will be able to

• specify, design, and implement basic data structures such as stacks, queues, binary trees, heaps, hash tables, graphs; and
• understand and implement basic searching and sorting algorithms using these data structures.

• Be proficient in reading and writing with a critical point of view that demonstrates depth of thought and a thorough understanding of the rhetorical situation. (Critical Thinking, Complexity, Communication)
• Write essays that contain well-supported, arguable theses and that demonstrate personal engagement and clear purpose. (Critical Thinking, Complexity, Communication)
• Be proficient in independently and deliberately locating, selecting, and appropriately using and citing evidence that is ample, credible, and smoothly integrated into intellectually honest writing appropriate to a particular discipline. (Complexity, Communication; Meta-Goal: Information Literacy)
• Use writing processes as modes of thinking and learning that can be generalized across a range of writing and thinking tasks. (Critical Thinking, Complexity; Meta-Goals: Intentional Learning)

• Demonstrate general understanding of Euclidean, hyperbolic, and spherical geometry
• Show mastery in one area chosen from these
• Understand and write rigorous arguments, reasoning from axioms to theorems

• understand the interplay between the geometry of the plane and the arithmetic of complex numbers.
• Integrate previous notions of derivatives, logarithms, vectors, line integrals, and power series from the calculus sequence with their complex counterparts, and understand how the complex versions generalize and sometimes simplify the earlier notions.

• learn, understand, and communicate definitions, examples, fundamental theorems and applications relevant to the study of groups, and
• analyze, develop, and communicate rigorous mathematical proofs of statements concerning groups.

• learn, understand, and communicate definitions, examples, fundamental theorems and applications relevant to the study of rings and fields, and
• analyze, develop, and communicate rigorous mathematical proofs of statements concerning rings and fields.

• learn, understand, and communicate definitions, examples, fundamental theorems and applications relevant to the study of topological spaces, and
• analyze, develop, and communicate rigorous mathematical proofs of statements in topology and related fields.

• learn, understand, and communicate definitions, examples, fundamental theorems and applications relevant to the study of probability, and
• analyze, solve, and communicate solutions to problems in probability.

• learn, understand, and communicate definitions, examples, fundamental theorems and applications relevant to the study of mathematical statistics, and
• analyze, solve, and communicate solutions to problems in mathematical statistics.

• understand the basic principles of finance and economic investments, which they can apply to their own investing or a companies.
• see how probability and differential equations combine to understand processes with white noise. This generates the noise not only in stock market prices, but also the noise in radar signals, movements of microscopic particles in physics, chemistry, and biology, etc.
• understand the principles of control theory/ dynamic optimization, which means how to continuously change a control to optimize an outcome. This is used not only in finance, but also throughout engineering and is central to the field of operations research.
• appreciate how their background prepares them for tackling new mathematics and computional methods as applied to financial applications they have likely not thought about before.

• see how physics and PDEs are fundamentally related. PDE are the natural result of modeling certain phenomena in physics. Solving PDEs can tell us about the behavior of physical problems. Physics can tell us which solution of a PDE is desired.
• understand the difference between the behavior of solutions to the Laplace (elliptic), Heat (parabolic), and Wave (hyperbolic) equations.
• learn to interpret, and manipulate Fourier Series.
• understand the difference between linear and nonlinear equations and begin to understand the behavior of some nonlinear equations such as some image processing PDE and first order nonlinear hyperbolic equations.

CS 161 -- Theory of Automata and Languages I
General Departmental Goals and Objectives: 2a; 3a; 3b
Specific Goals and Objectives: In this course, students will learn how to

• understand formal machine models for recognizing strings;
• formulate finite descriptions for many (infinite) sets of strings using these machine models;
• convert among equivalent descriptions;
• prove that no formal descriptions exist for certain sets of strings.

• generate random variates for various standard probability distributions;
• evaluate randomness of observed data using the chi-square and Kolmogorov-Smirnov tests;
• implement discrete-event simulations of queueing systems and integral evaluations;
• estimate distribution properties of simulation outcomes; and
• speed up simulations using variance-reduction techniques.

• learn about the power of various numerical methods, the pros and cons of different numerical algorithms, and the sources of numerical errors.
• find numerical roots of non-linear equations and solutions of systems of linear equations, both directly and iteratively, and learn about convergence rates and conditioning problems.
• perform numerical integration and find solutions to differential equations numerically.
• approximate data by various functions and fit functions to data points via cubic splines and other functions.
• study other numerical problems (e.g., finding eigenvalues, orthogonal polynomials) as time permits.

CS 169 -- Programming Languages
General Departmental Goals and Objectives: 3a
Specific Goals and Objectives: In this course, students will be exposed to a variety of programming languages, in addition to studying the theoretical foundations for programming languages. This course helps students overcome fears and aversions to learning additional programming languages. Among the various topics presented, students will

• learn about classifications of programming languages (e.g., procedural, functional, logic, imperative) and learn an example from different paradigms.
• examine the differences in languages in terms of availability of data types, program structures, variable bindings, parameter passing schemes, and other aspects.
• study an overview of formal language and automata theory to learn about the abstract background for programming languages.
• learn about issues raised by parallel computing and languages and languages structures needed for parallel programming.

• first, how to understand and state the relevant parts of a given problem;
• second, how to find a connection between the given data and the mathematics that may be used to find the problem's solution and then make a plan for obtaining the solution;
• third, carry out the plan;
• fourth, examine the solution to see if it makes sense, if its correctness can be verified, if the ideas used can help you to solve similar more complicated problems, if you could have done the problem in an easier way, etc.

Math 174 -- Differential Geometry
General Departmental Goals and Objectives: 2b
Specific goals and objectives: Through this course, students will

• Develop and demonstrate the ability to apply techniques from calculus to prove statements about curves and surfaces.
• Acquire knowledge of the fundamental vocabulary of curvature for one- and two-dimensional objects in three-space.
• Practice and demonstrate mastery of techniques and theories of analysis and geometry.

• ability to use theory to solve concrete problems
• write proofs
• understand applications of abstract algebra
• know the theory behind the mathematics used for cryptography
• solve recreational problems, many of which are rooted in number theory.

• enable students to see the beauty of number theory
• learn the history and basics concepts of number theory
• give students the ability to solve problems in elementary number theory.

• permutations
• combinations
• generating functions
• recursion relations
• (PIE) the principal of inclusion exclusion
• (PET) the Pólya Enumeration Theorem
• a selection of topics from combinatorial geometry
• graph enumeration
• algebraic combinatorics.

• Appreciate and develop facility with mathematical structures. We will connect the different representations and properties of graphs and develop facility in their use in algorithms. We will learn to write graph-theoretic proofs by studying existing proofs and writing our own. We will understand the place of graph theory in the larger structure of discrete mathematics.
• Understand the foundations of Computer Science and appreciate some of its theoretical and applied uses. We will learn ways to represent graphs as data structures, and develop graph algorithms for classical problems in graph theory. We will become adept at applying these algorithms and proving their correctness.

This page last updated 30 March 2013.