Mathematics (MAT) Faculty Mary Hayward, Chair Charles B. Carey Garnet Hauger, Affiliate Tim Wegner

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1 Mathematics (MAT) Faculty Mary Hayward, Chair Charles B. Carey Garnet Hauger, Affiliate Tim Wegner About the discipline The number of applications of mathematics has grown enormously in the natural, physical and social sciences over the past century. In addition, the computer has transformed problem-solving in both pure and applied mathematics. The mathematics department at Spring Arbor University has responded by developing a program designed to prepare students for a variety of careers or graduate and professional schools. The department offers three majors and three minors. The department provides services for three types of students. 1. All graduates of the University must have at least one quantitative literacy course. The department takes seriously its responsibility to design and teach courses that prepare University graduates to be quantitatively literate. Our selection of quantitative literacy courses takes into consideration applications in a variety of areas. 2. Some students major in programs that require the study of specialized mathematics. The department supplies courses for students from computer science, biology, chemistry, physics, health and exercise science, education and business. 3. For students wishing to major or minor in mathematics, the department offers a program consistent with the recommendations of Mathematical Association of America ( org), National Council of Teachers of Mathematics ( and Conference Board of the Mathematical Sciences. Career opportunities Students who major or minor in mathematics pursue a variety of careers. Many certify to teach mathematics in elementary and secondary schools. A good proportion prepare for graduate school in mathematics or related fields. Still others combine their mathematics major or minor with other majors or minors (such as computer science, biology, chemistry, physics, business, accounting and health and exercise science) to prepare for entry-level positions in business and industry. A probability and statistics minor is available for students who want to enhance their career options within their academic discipline with a quantitative emphasis. An interdisciplinary actuarial science major is available to help students prepare for careers that use mathematics in business and industry. Actuarial Science majors would be positioned to pass the first two actuarial exams (P and MF) before graduation, to obtain employment using their skills, and to pursue other educational opportunities that will allow them to advance in the field (graduate school, passing additional exams, etc.). See Actuarial Science. The department has a strong commitment to prepare students for careers in specific areas. For example, the department conducts special problem seminars to help students prepare to take the GRE and actuarial exams. The GRE exams are needed for students going to graduate school and the actuarial exams are needed by those preparing for careers in insurance, pension planning and risk management. The department has information on graduate schools in mathematics and related areas. In addition, the department often arranges independent studies for advanced work in mathematics for those students who have taken all of our current mathematics offerings and wish to prepare for specific graduate or professional programs. The department also works with majors in arranging internships and practica for students preparing for entry-level jobs in businesses and industry. (See and 168

2 Program strengths and emphases The mathematics program at Spring Arbor University stresses two major strands in mathematics pure/ classical mathematics and applied mathematics. It is our belief that each complements the other, that the techniques of applied mathematics are based on the concepts and results of pure/classical mathematics, and that applications of mathematics often serve as an appropriate pedagogical segue into a study of pure/classical mathematics. The course offerings of the department have traditional pure/classical offerings like abstract algebra, linear algebra, real analysis and vector calculus. Within each of these courses, significant applications of concepts and results are considered. The department also offers courses more appropriately considered applied mathematics, such as differential equations, probability and statistics (see probability and statistics minor), numerical analysis, and mathematical modeling. Each of the topics studied in these courses is based firmly on theoretical aspects of pure/classical mathematics. The probability and statistics minor is intended for students who are not math majors or minors but who might want to strengthen their program with a quantitatively based minor. Students who take this minor with a major in biology or health and exercise science, are then prepared for graduate programs in biostatistics or other quantitatively oriented health related science programs. The combination of this minor with a major in finance or business, are then prepared for graduate programs in finance or more quantitatively oriented business programs. The faculty members in the mathematics department work closely with faculty from computer science, biology, chemistry, physics, health and exercise science, business, and education to provide mathematical experiences that serve their students. Requirements The hour mathematics major requires: MAT 232 Proof and Discrete Structures (3) Two additional courses from MAT level or above (excluding MAT 330, 331 and 342) OR Two courses from CPS level or above OR Two courses from PHY level or above The 30-hour mathematics major for those certifying to teach elementary requires: Electives from the following to reach 30 hours: MAT 322 Abstract Algebra II (3) The 35-hour mathematics major for those certifying to teach secondary requires: 169

3 The 20-hour mathematics minor requires: Six additional hours in MAT level or above (excluding MAT 330,331 and 342). The 23-hour mathematics minor for those certifying to teach elementary requires: The 26-hour mathematics minor for those certifying to teach secondary requires: MAT 342 Foundations of Secondary School Mathematics (3) Note: MAT 342 and one other upper level course in this minor must be taken with SAU. The 24-hour probability and statistics minor requires: CPS 201 Foundations of Computing I (4) MAT 363 Theory of Statistics I (3) MAT 364 Theory of Statistics II (3) See Physics for the Physics/Mathematics major. Students planning to attend graduate school in mathematics or a related area Recommended courses: MAT 422 Seminar: Topics in Advanced Mathematics (1-3) Students planning on entry level careers in business and industry Recommended courses: At least two courses in computer science beyond CPS 150. Students maximize their chances for employment and career advancement by obtaining a minor or second major in biology, business, chemistry, computer science, economics, accounting, health and exercise science or physics. Notes for general education: MAT 201 is required for the major and also meets the requirement for one of two natural science liberal arts courses. Mathematics majors who minor in biology, chemistry or physics will take courses in those programs that meet the requirement for the lab natural science liberal arts course. Notes for students certifying to teach: The mathematics for elementary major and minor meet certification requirements for elementary (complete Option I). The mathematics for secondary major and minor meet certification requirements for secondary. See Education for details. 170

4 Course descriptions: MAT 100 Basic College Mathematics (3) Using number concepts and computation skills, solving word problems involving integers, fractions, or decimals including percents, ratios, and proportions, interpreting information from graphs, tables and charts, graphing number and number relationships, solving one- and twovariable equations, solving word problems involving one and two variables, understanding operations with algebraic expressions, solving problems involving geometric figures and applying reasoning skills. This course is only available to students with an ACT math score less than 18. (Offered in interim.) MAT 101 Algebra (3) Operating with algebraic and geometric properties of linear, quadratic, polynomial, rational, radical, exponential, and logarithmic functions; linear inequalities in one and two variables. Applications in many areas. Students with ACT math score greater than or equal to 24 may not take this course for credit. Does not count toward major or minor. Prerequisites: One year high school algebra, one year high school geometry and ACT math score between 12 and 24. (Offered in fall and MAT 112 Precalculus (3) The study of elementary functions to prepare students for the calculus sequence or courses in the sciences. Topics include properties of the real number system, inequalities and absolute value, functions and their graphs, solutions of equations; polynomial, rational, radical, trigonometric, exponential, and logarithmic functions. Prerequisite: MAT 101 or three years of high school mathematics including two years of algebra. (Offered in The theory and techniques of differentiation and integration of the elementary, and other functions of one variable, as well as functions defined by parametric equations or polar coordinates, and applications of the calculus in the physical and social sciences. Includes continuity, infinite series and sequences. Prerequisite: MAT 112 or high school math through pre-calculus with at least a B average. (Offered in fall.) The theory and techniques of differentiation and integration of the elementary, and other functions of one variable, as well as functions defined by parametric equations or polar coordinates, and applications of the calculus in the physical and social sciences. Includes continuity, infinite series and sequences. Prerequisite: MAT 201. (Offered in MAT 222 Applied Calculus (4) Mathematical models of linear equations and inequalities, differential and integral calculus of algebraic, exponential, and logarithmic functions, maxima and minima, curve sketching, exponential growth and decay, and differential equations; focus on applications in business, economics, life sciences, and social sciences. Prerequisite: MAT 112 or high school math through pre-calculus with at least a B average. (Offered in fall.) MAT 232 Proof and Discrete Structures (3) Functions, relations, sets, basic logic, proof techniques, counting theory, graph and tree concepts, discrete probability, recurrence relations, difference equations, algorithm development, and linear programming. (Offered in fall.) Calculus of functions of two variables: limits, continuity, partial derivatives, chain rule, directional derivatives and the gradient, tangent planes and normals to surfaces, iterated integrals, cylindrical and spherical coordinates, curves in space, polar coordinates, line integrals, surface integrals, Green s theorem, Stoke s theorem, change of variable, Taylor s expansion, implicit function theorem, and Divergence theorem. Prerequisite: MAT 202. (Offered in fall.) Fundamental methods of solving differential equations of the first order and first degree, higher ordered differential equations with constant coefficients, systems of equations and applications of all. Prerequisite: MAT 202. (Offered in Structure of our number system. Topics such as rings, fields and integral domains are covered. Prerequisite: MAT 201 and 232. (Offered in fall.) The structure of our number system with emphasis on group theory. Prerequisite: MAT 321. (Offered in spring of even academic years.) MAT 330 Foundations of School Mathematics I (3) Fundamental concepts of mathematics for prospective teachers of grades K-8. Topics include strands I, IV and V from Michigan Curriculum Framework: Patterns, Relationships and Structures; Number Sense and Numeration; Numerical and Algebraic Operations and Analytical Thinking. Prerequisite for education credit: EDU 202, 262. Does not count toward mathematics major or minor. Also listed as EDU 330. (Offered in fall and 171

5 MAT 331 Foundations of School Mathematics II (3) Fundamental concepts of mathematics for prospective teachers of grades K-8. Topics include strands II, III, VI from Michigan Curriculum Framework: Geometry and Measurement; Data Analysis and Statistics; Probability and Discrete Mathematics. Prerequisite for education credit: MAT/EDU 330. Does not count toward mathematics major or minor except for those certifying for elementary education. Also listed as EDU 331. (Offered fall and MAT 342 Foundations of Secondary School Mathematics (3) Topics unique to the teaching of mathematics at the secondary level. Emphasis on recent curriculum developments in mathematics including reform curricula, use of technology and cooperative groups. Prerequisites: SOE acceptance; MAT 201, MBST and grade of 2.67 in EDU 202 and 262. (Offered in Descriptive and inferential statistics including z- and t-tests for means and proportions, Chi-Square, ANOVA and regression. Topics from probability: sample space, events, additive rules, conditional probability, multiplicative rules, Bayes rule, random variables, discrete and continuous probability distributions, expected value, Chebyshev s theorem. Prerequisite: MAT 101. (Offered in This course deals with the study of vectors, vector spaces, matrices and linear transformations, and multilinear vector functions. Prerequisite: MAT 202. (Offered in A course in mathematical modeling emphasizing model construction and selection, as well as specific model types (proportionality, optimization, fitting experimental data, dimensional analysis, similitude and simulation), especially as applied to problems in physics, chemistry, biology, economics and computer science. Prerequisites: MAT 201 or MAT 222. (Offered in spring of odd academic years.) MAT 363 Theory of Statistics I (3) General Probability (Set functions including set notation and basic elements of probability, Mutually exclusive events, Addition and multiplication rules, Independence of events, Combinatorial probability, Conditional probability Non Bayes Theorem, Bayes Theorem/ Law of total probability); Univariate probability distributions including binomial, negative binomial, geometric, hypergeometric, Poisson, uniform, exponential, chisquare, beta, Pareto, lognormal, gamma, Weibull, and normal (Probability functions and probability density functions, Cumulative distribution functions, Conditional probability, Mode, median, percentiles, and moments, Variance and measures of dispersion, Moment generating functions, Transformations. Prerequisites: MAT 202 and MAT 351. (Offered in fall of odd years). MAT 364 Theory of Statistics II (3) Multivariate probability distributions including the bivariate normal (Joint probability functions and joint probability density functions, Joint cumulative distribution functions, Central limit theorem, Conditional and marginal probability distributions, Moments for joint, conditional, and marginal probability distributions, Joint moment generating functions, Variance and measures of dispersion for conditional and marginal probability distributions, Covariance and measures of dispersion for conditional and marginal probability distributions, Transformations and order statistics, Probabilities and moments for linear combinations of independent random variables. Prerequisite: MAT 363. (Offered in spring of odd years.) A rigorous treatment of the calculus of functions of one real variable. Formal proofs are given for the familiar theorems dealing with limits, continuity, differentiation, integration, sequences and series. Special topics may include vector analysis, gamma functions and Fourier analysis. Prerequisite: MAT 232. (Offered in Solution of a single nonlinear equation; linear systems of equations; interpolation and approximation; numerical integration and differentiation; initial value problems in ordinary differential equations. Also listed as CPS 402. Prerequisites: MAT 201 or MAT 222 and CPS/MAT 232. (Offered in spring of even academic years.) A general overview of elementary geometry appropriate for all math majors and required of secondary education math majors. Primary emphasis on Euclidean and non- Euclidean geometries with some discussion of finite, neutral, transformational and projective geometry. Consideration of historical development. Prerequisite: MAT 232. (Offered in spring of even academic years.) MAT 422 Seminar: Topics in Advanced Mathematics (1-3) Designed to meet the current needs of the mathematics majors. Examples of topics that may be selected are: Introduction to topology, projective geometry, complex variables and partial differential equations. Prerequisite: MAT 232. (Limited offering.) 172