# Department of Mathematics and Physical Sciences

*Mission: The Mathematics program serves a vital role in equipping students for a Spirit-empowered life of Christ-centered leadership and service and is particularly aligned with the Vanguard Mission by its contribution to providing Christian Educators with a missionary outlook and a spiritual mandate to care for God’s people through education. Moreover, the Department serves to enhance students’ problem-solving capacities and capabilities as well as improving their mental calculations and reasoning skills – all processes which eventually lead to lifelong success. Students are, therefore, prepared for careers in elementary or secondary education, for graduate studies in mathematics, or a career in applied mathematics. The department also provides background courses in mathematics necessary for other areas of study such as physics, chemistry, biology, or kinesiology. This subject matter preparation program is approved by the California Commission on Teacher Credentialing (CTC).*

All majors in Natural and Physical Sciences are expected to make satisfactory academic progress toward graduation. In addition to the requirements established by the University, students must maintain a 2.0 in all major coursework. Further, the Department also requires a minimum grade of C- as completion for pre-requisites in science-related coursework for all students enrolled in MATH, BIOL, ENVR, CHEM, and PSCI courses, regardless of major.

**Mathematics Placement Exam**

Courses in the Department of Mathematics at any institution are demanding, and Vanguard University is no exception. To help you prepare for college-level mathematics at Vanguard, the Department of Mathematics requires the ** placement** of all incoming students who want to take math (either MATH-116 College Algebra, MATH-170 Precalculus or MATH-180C Calculus 1), regardless of major. The Placement Exam is managed through a third party entity, called ALEKS Placement, Preparation and Learning (ALEKS PPL) which offers colleges and universities a complete solution for math placement and course preparedness. Combining a research-based, accurate placement assessment with personalized learning tools to help students refresh lost knowledge gives them the opportunity to succeed before they even begin class.

### VISTA: Mathematics Education Emphasis Minor Courses

The Vanguard Integrated STEM Teacher Achievement Program (VISTA) is one of several Integrated Teaching Education Programs (ITEP) offered at Vanguard. The ITEP programs allows for students to complete not only their bachelor’s degree but also the California Teaching Credential within four years. This is a great option for students preparing to teach mathematics at the secondary school level. The Bachelor of Science in Mathematics degree with this option is approved by the California Commission on Teacher Credentialing. Completion of this option will satisfy the subject matter requirement for the Single Subject Teaching Credential in Mathematics within four years. Students wishing to graduate on a four year plan are still eligible to enroll in the fifth year credential program at Vanguard University or some other accredited college or university to complete subject requirements for the California Clear Teaching Credential in Mathematics (see the Graduate Program in Education for these requirements). Students completing this option must include MATH-315 Teaching/Tutoring, MATH-330 Number Theory, MATH-380 Elementary Analysis I, MATH-410 Abstract Algebra and MATH-420 Higher Geometry as a part of their degree program. They should also take ENGL-453 Language, Culture, and Linguistics and POLS-155C American Democracy (see program requirements).

## Majors:

There are four possible degrees in Mathematics including: *Actuarial Science, Data Science, Mathematics*, and *Mathematics Teaching with an Education Minor* (the VISTA program). All four programs have a common core of foundational mathematics courses with additional specific courses that provide the academic expertise in the respective sub-discipline.

## Minors:

## California Teaching Credential:

## Computer Sciences

**CSCI-100C**

**Introduction to Computer Science**

**3 Credits**

Introductory course in computer science and the study of algorithms appropriate for students in data-intensive disciplines. Topics include how computers work, simple algorithms and their efficiency, networking, databases, artificial intelligence, graphics, simulation and modeling, security and the social impact of computing. Includes a choice of five programming language modules. The course also includes a gentle hands-on introduction to programming concepts with Python.

Terms Typically Offered: Spring.

**CSCI-100CL**

**Introduction to Computer Science Lab**

**1 Credit**

Introductory course in computer science and the study of algorithms appropriate for students in data-intensive disciplines. Topics include how computers work, simple algorithms and their efficiency, networking, databases, artificial intelligence, graphics, simulation and modeling, security and the social impact of computing. Includes a choice of five programming language modules. Students will gain a solid background in computing hardware and software networks, programming languages, and applications.

Terms Typically Offered: Spring.

**CSCI-400**

**Data Visualization**

**3 Credits**

Visualization is increasingly important in this era where the use of data is growing in many different fields. Data visualization techniques allow people to use their perception to better understand this data. The goal of this course is to introduce students to data visualization including both the principles and techniques. Students will learn the value of visualization, specific techniques in information visualization and scientific visualization, and understand how to best leverage visualization methods.

## Data Sciences

**DSCI-100C**

**Introduction to Data Science**

**3 Credits**

This course will introduce students to this rapidly growing field and equip them with some of its basic principles and tools as well as its general mindset. Students will learn concepts, techniques and tools they need to deal with various facets of data science practice, including data collection and integration, exploratory data analysis, predictive modeling, descriptive modeling, data product creation, evaluation, and effective communication. The focus in the treatment of these topics will be on bredth, rather than depth, and emphasis will be placed on integrtion and synthesis of conecepts and their applicaiton to solving problems. To make the learning contextual, real datasets from a variety of disciplines will be used.

Co-Requisite: DSCI-100CL

Terms Typically Offered: Fall.

**DSCI-100CL**

**Introduction to Data Science Laboratory**

**1 Credit**

Co-Requisite: DSCI-100CL This course will introduce students to this rapidly growing field and equip them with some of its basic principles and tools as well as its general mindset. Students will learn concepts, techniques and tools they need to deal with various facets of data science practice, including data collection and integration, exploratory data analysis, predictive modeling, descriptive modeling, data product creation, evaluation, and effective communication. The focus in the treatment of these topics will be on breadth, rather than depth, and emphasis will be placed on integration and synthesis of concepts and their application to solving problems. To make the learning contextual, real datasets from a variety of discipliens will be used.

Co-Requisite: Must take DSCI-100C at same time.

Terms Typically Offered: Fall.

**DSCI-300**

**Database Management**

**3 Credits**

Developing and managing efficient and effective database applications requires understanding the fundamentals of database management systems, techniques for the design of databases, and principles of database administration. This course emphasizes database concepts, developments, use and management in three main sections: database concepts, practice, and emerging trends. Relational database systems are the main focus, but other types, including object-oriented databases, are studied. Practical design of databases and developing database applications using modern software tools will be emphasized.

**DSCI-350**

**Data Mining**

**3 Credits**

Data Mining studies algorithms and computational paradigms that allow computers to find patterns and regularities in databases, perform prediction and forecasting, and generally improve their performance through interaction with data. It is currently regarded as the key element of a more general process called Knowledge Discovery that deals with extracting useful knowledge from raw data. The knowledge discovery process includes data selection, cleaning, coding, using different statistical and machine learning techniques, and visualization of the generated structures. The course will cover all these issues and will illustrate the whole process by examples. Special emphasis will be given to the Machine Learning methods as they provide the real knowledge discovery tools. Important related technologies, as data warehousing and online analytical processing (OLAP) will be also discussed. The students will use recent Data Mining software.

Terms Typically Offered: Fall.

**DSCI-410**

**Data Visualization**

**3 Credits**

Visualization is increasingly important in this era where the use of data is growing in many different fields. Data visualization techniques allow people to use their perception to better understand this data. The goal of this course is to introduce students to data visualization including both the principles and techniques. Students will learn the value of visualization, specific techniques in information visualiziation and scientific visualization, and understand how best to leverage visualization methods.

Terms Typically Offered: Fall, even years.

## Mathematics

**MATH-104**

**Math for Liberal Arts**

**3 Credits**

A non-technical course emphasizing the ideas and concepts of mathematics. Algebra, number theory, set theory, geometry, statistics, probability, and analysis of the ideas and methods involved.

**MATH-105**

**Essential Mathematics**

**4 Credits**

This course prepares students to understand the essential mathematical concepts in number sense, elementary algebra, sets of numbers, problem solving, ratios, proportions, percentages, and graphing linear equations and inequalities. The course is designed to introduce students to practical mathematical skills necessary for courses in business and statistics. Emphasis will be placed on the structural and logical foundations of business. (Meets Elective requirement only; not applicable to Natural Science/Math requirement)

**MATH-106**

**Business Math**

**4 Credits**

This course applies the principles and practices of mathematics to everyday business problems and situations. The course prepares students to understand the mathematical and business concepts in problem solving, ratios and proportions, percentages, simple and compound interest, graphing linear functions, and inventory valuation. The course introduces students to common mathematical skills necessary for courses in business. (Meets Natural Science/Math requirement)

**MATH-109**

**Mathematics for Statistics**

**4 Credits**

This course prepares students to understand the mathematical and statistical concepts in problem solving, critical thinking, ratios and proportions, algebraic equations, sets and logic, probability and statistics, including frequency of distribution, statistical graphs, measures of central tendency, and measure of position and dispersion. The course introduces students to common mathematical skills necessary for coursework in statistics. (Meets Natural Science/Math requirement)

**MATH-116**

**College Algebra**

**3 Credits**

Sets and real numbers, linear equations and inequalities, polynomials, functions, graphing linear and polynomial functions, exponential and logarithmic functions, systems of equations, matrices and determinants, sequences and series.

**MATH-116C**

**College Algebra**

**3 Credits**

Sets and real numbers, linear equations and inequalities, polynomials, functions, graphing linear and polynomial functions, exponential and logarithmic functions, systems of equations, matrices and determinants, sequences and series. Fulfills the Math CORE requirement.

**MATH-117**

**College Trigonometry**

**3 Credits**

Complex numbers, trigonometric functions and applications.

**MATH-120**

**Math for Elementary Teachers**

**3 Credits**

Introductory set theory, problem solving, basic algorithms, elementary number theory, geometry and coordinate geometry. Emphasis will be on the structural and logical foundations of mathematics.

**MATH-145C**

**Data Analysis**

**3 Credits**

The use of mathematics as a thinking and problem-solving tool, emphasizing data interpretation, graphs, tables, statistical arguments, probability, statistics, and the use and misuse of numbers. Spreadsheet applications include variables, conditionals, and statistical functions. Students majoring in Mathematics are exempt from taking the core curriculum requirement in mathematics.

**MATH-170**

**Precalculus**

**3 Credits**

Equations and inequalities; systems of linear equations: functions, graphs, exponential, logarithmic, and trigonometric functions and their limits; polynomial and rational functions; analytic geometry.

**MATH-170C**

**Precalculus**

**3 Credits**

Equations and inequalities; systems of linear equations: functions, graphs, exponential, logarithmic, and trigonometric functions and their limits; polynomial and rational functions; analytic geometry. This course fulfills the Math CORE requirement.

**MATH-180C**

**Calculus 1**

**4 Credits**

Graphing of functions, logarithmic functions and their inverses; limit of functions and derivatives; rules of differentiation and their application; definite and indefinite integrals. Fulfills Math CORE requirement.

Prerequisite: MUST COMPLETE: MATH-170c

**MATH-181C**

**Calculus II**

**4 Credits**

PREREQUISITE: MATH-180 or Consent of the Instructor Application of integration for volumes, work and areas; advanced techniques of integration; advanced application of integration; first order differential equations; parametric equations and polar coordinates; infinite sequences and series.

Prerequisite: Must take MATH-180 or have consent of the instructor

**MATH-207**

**Computer Programming**

**3 Credits**

Open to all majors, this course offers the basic concepts of programming, problem solving, programming logic, as well as the design techniques. Programming language is chosen from languages such as Visual Basic, C++ and MatLab. Special fee for computer lab use.

**MATH-210**

**Technical Writing**

**3 Credits**

Expository writing on technical subjects dealt with in industry, science, and education. Long and short forms including reports, proposals, journal articles, and research papers.

**MATH-235**

**Statistics for Health Professionals**

**3 Credits**

Limited to those enrolled in Health Sciences. Meets statistics prerequisite for NURS 405. This course introduces the conceptual background of statistical techniques and reasoning with an emphasis on application relevant to identifying outcomes. Provides a framework for understanding and applying commonly used data analysis techniques in health science research. Includes selecting, applying, and interpreting univariate and bivariate statistical methods in answering research questions from a health science perspective.

**MATH-265C**

**Intro to Statistical Methods**

**3 Credits**

This course is cross listed with PSYC-265C and SOC-265C. A course in basic statistical concepts and methods of collecting, summarizing, presenting, and interpreting data in the behavioral sciences; including descriptive statistics (use of graphs and charts), normal distribution curve, measures of central tendency, deviation and dispersion, hypothesis testing, statistical fallacies, correlation, and topics in probability. Students majoring in Accounting, Anthropology, Psychology, or Sociology must take MATH-265C, PSYC-265C, or SOC-265C to fulfill the core curriculum requirement in mathematics. Students majoring in Mathematics are exempt from taking the core curriculum requirement in mathematics.

**MATH-270C**

**Health Professions Statistical Methods**

**3 Credits**

The focus of this course is on exploring the statistical methods used in health professions. Students review parametric and nonparametric techniques and explore the purpose, assumptions, selection, and interpretation of descriptive and inferential statistics. As part of the course, students use Microsoft Excel to organize and analyze data sets. Open to all undergraduate health science students.

**MATH-281**

**Multivariable Calculus**

**4 Credits**

Vectors and geometry of space including cylindrical and spherical coordinates; vector functions and space curves with vector calculus, partial derivatives with directional derivatives and applications; multiple integrals with various coordinate systems with application; and second-order differential equations.

Prerequisite: MATH-181 or MATH-181C

**MATH-285**

**Intro to Adv Mathematics**

**3 Credits**

A course for both mathematics and mathematics education majors. Covers the fundamentals of axiomatic proof theory including laws of inference, set theory, induction, cardinality, relations and functions.

Prerequisite: MATH-180 or MATH-180C

**MATH-291**

**Special Topic:**

**1 Credit**

Study of a special topic in mathematics. May be repeated for credit.

**MATH-292**

**Special Topic:**

**2 Credits**

Study of a special topic in mathematics. May be repeated for credit.

**MATH-293**

**Special Topic:**

**3 Credits**

Study of a special topic in mathematics. May be repeated for credit.

**MATH-294**

**Sp Top: Into to Comp Science**

**4 Credits**

Study of a special topic in mathematics. May be repeated for credit.

**MATH-300**

**Linear Algebra**

**3 Credits**

Systems of linear equations, vector spaces, linear mappings, matrices and matrix algebra, eigenvectors and eigenvalues, Cayley-Hamilton theorem, quadratic forms, and applications.

Prerequisite: MATH-285

**MATH-310**

**Differential Equations**

**3 Credits**

Methods of solution of ordinary differential equations and applications.

**MATH-315**

**Teaching/Tutoring**

**3 Credits**

This course is designed to give the student introductory experiences in both tutoring and planning classroom observations. These early field experiences will link together mathematical content along with classroom management and interaction with students from diverse populations represented in the California public schools. Lab fee.

**MATH-330**

**Number Theory**

**3 Credits**

Divisibility properties of integers, prime numbers, the Euclidean algorithm, the unique factorization theorem, congruences, Fermat's theorem, Wilson's theorem, Diophantine equations, number-theoretic functions, and the quadratic reciprocity theorem.

**MATH-365**

**Probability and Statistics**

**3 Credits**

Frequency interpretation of probability, axioms of probability theory, discrete probability and combinatorics, random variables, distribution and density functions, sampling theory and limit theorems.

**MATH-370**

**History of Mathematics**

**3 Credits**

An introduction to the history of mathematics from ancient times to the twentieth century, with applications to elementary mathematics through calculus.

Prerequisite: MATH-180

**MATH-375**

**Discrete Mathematics**

**3 Credits**

This course is designed to study graph theory including networks, voting systems including game theory and fair apportionment, and patterns.

**MATH-380**

**Elementary Analysis I**

**3 Credits**

Rigorous analysis of the calculus and its foundations. Continuous and differentiable functions, and topological properties of the real number line.

**MATH-390**

**Numerical Analysis**

**3 Credits**

Numerical solution of algebraic equations. Approximate numerical solutions of systems of linear and nonlinear equations, interpolation theory, numerical differentiation and integration, and numerical solution of ordinary differential equations.

Prerequisite: MATH-281or equivalent

**MATH-400**

**Casualty & Actuarial Mathematics**

**3 Credits**

The course covers concepts from calculus and probability as they pertain to actuarial sciences. The calculus part covers limits of functions, derivative and integration and their applications, power series and polar coordinates, multivariate differentiation and integration. The probability part covers basic concepts of probability, conditional probability and Bayes' theorem, discrete and continuous random variables and distributions, bivariate distributions, conditional expectation and variance, transformations of random variables and moment generating function. The pace will be fast and the stress will be on the SOA/CAS Course 1 exam-type of word problems.

**MATH-410**

**Abstract Algebra**

**3 Credits**

An introduction to modern ideas of algebra, set theory, groups, rings, and fields.

**MATH-420**

**Higher Geometry**

**3 Credits**

Euclidean geometry from an advanced standpoint, and topics in non-Euclidean geometry.

**MATH-450**

**UG Research and Internship Program**

**1-4 Credits**

Prerequisite: Open to juniors and seniors. This course may be taken for a maximum of 4 units in one semester. A maximum of 6 combined units credit for MATH-450 or MATH-485 apply to graduation. This course is designed with the purpose of providing students the opportunity to conduct research off-campus at universities or STEM companies in the community. This course promotes early entry into the workplace for the student through part-time employment. This course requires actual work experience be sought in a biotech or STEM-focused buisness firm providing an opportunity to integrate classroom teaching in practical application under the direct supervision of the assigned intructor. Students are responsible for completing a project report and presenting their research results in MATH-499C

Terms Typically Offered: Summer.

**MATH-455**

**Mathematics Teaching Internship**

**1-3 Credits**

Must have the consent of the instructor and department chair. Regular hours each week for classes and/or meetings are established at the beginning of the semester with the supervising instructor. The intern assists an instructor in planning and conducting a course and/or laboratory session. This course may be taken for 1-3 units per semester. May be repeated for a maximum of six units.

**MATH-470**

**Special Topic:**

**1-3 Credits**

Study in a special topic in mathematics. May be repeated for credit.

**MATH-480**

**Individual Studies:**

**1-3 Credits**

May be repeated for credit.

**MATH-485**

**Undergraduate Research**

**2 Credits**

This course provides the student with an empirically-based research experience in the mathematical, biological, or chemical sciences. This course is structured around addressing problems in advanced laboratory research with emphasis on research techniques including: project managment, safety, instrumentation, computer programming, and notebook documentation. Research is carried out under the supervision of theinstructor with weekly conferences to discuss results and direction. A written proposal and report emphasizing the literature background of the problem and the experimental results are required. The results of the research project will also be presented in an oral format in MATH-499C, BIOL-499C, CHEM-499C. This course is a variable credit course. At least two units are recommended for all mathematics, environmental sciences, and biology majors and are required for all chemistry, engineering physics, and biochemistry majors. A minimum of 50 hours of laboratory work is required per unit. May be repeated. Lab fee.

Terms Typically Offered: Summer.

**MATH-488**

**Senior Project**

**2 Credits**

An advance course providing the opportunity for a student to create a novel and independent intellectual work by comparing, contrasting and synthesizing recent research and his/her cumulative knowledge and understanding in the biological sciences. The precise nature, scope and format of the project must be developed and approved under the guidance of the instructor and in collaboration with the student's academic advisor. Senior projects are typically initiated in the Fall. The project resuts must also be presented in MATH-499C.

Prerequisite: MATH-285

Terms Typically Offered: Summer.

**MATH-499C**

**Mathematics Capstone**

**2 Credits**

This course includes analysis and evaluation of current research in STEM and the integration of faith and learning in the mathematical scienes. An oral presentation of MATH-485 or MATH-488 is required. In-class presentations by faculty and guests are a part of the course. Attendance of both on-campus and off-campus seminars are required. This course fulfills the Core Curriculum Captstone requirement for Mathematics majors.