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Mathematics (MT)

Professors:  R. J. Kolesar, L. J. Schneider, C. R. Spitznagel, P. L. Shick, M. Kirschenbaum, B. K. D’Ambrosia, T. H. Short; Associate Professors:  D. L. Stenson, F. J. Fuglister, D. A. Norris (Chair), P. B. Chen, B. Foreman

Major Programs
The Department of Mathematics and Computer Science offers two major programs in mathematics. The department also offers computer science programs described in the separate section on Computer Science (CS).

The major in mathematics leading to the Bachelor of Science degree prepares students for graduate study or for immediate employment after completion of the degree.  It is designed to give students a broad background in classical mathematics, while remaining flexible enough to allow students to tailor the program to meet the needs of their career objectives.  Graduates have entered graduate programs in mathematics, statistics, and operations research at many leading universities, while others have entered into a variety of employment situations—as computer programmers, systems analysts, actuarial trainees, statisticians, and teachers.  Other graduates have entered professional schools in law, medicine, and business.

The major in teaching mathematics leading to the Bachelor of Arts degree combines mathematics and education courses for licensure to teach Adolescent to Young Adult (AYA) mathematics.
Teaching Licensure
The mathematics courses necessary for licensure to teach Adolescent to Young Adult (AYA) mathematics are the same as those required for the Bachelor of Arts major in teaching mathematics.
For Middle Childhood (MC) licensure, the mathematics curriculum content courses are MT 122, 135, 160, 241, 251, and two courses chosen from CS 101, 128; MT 118, 120, 136, and 162.

For Early Childhood (EC) licensure, the mathematics curriculum content courses are MT 160, 171 and 171L.
In all cases, the content-area courses for licensure (mathematics and/or computer science) must be completed with a minimum average of 2.7 and a minimum grade of C in each course.

Major and Minor Requirements

Major in Mathematics:  50 semester hours.  CS 128, CS 128L; MT 135, 136, 233, 341, 342, 372, 420, 431, 441, 452, and four more MT courses numbered 400-479.
A comprehensive examination is required.
Major in Teaching Mathematics:  38 hours.  CS 128, CS 128L; MT 135, 136, 233, 330, 341, 342, 372, 420, 450, and 469.
Required Support Sequence:  34 hours.  ED 100, 186, 201, 253, 255, 337, 350, 427, 405C, 444C; PS/ED 262.
A comprehensive examination is required.

Minor in Mathematics:  24 hours.  MT 135, 136, 233, and four MT courses numbered 300-379 and/or 400-479.  MT 330 and 431 cannot both count toward the minor.

Minor in Probability and Statistics:  25 hours. MT 135, 136, 233, 342, 420, 421, 422.

NOTE:  Students who minor in probability and statistics cannot use MT 421 or MT 422 also to satisfy the requirements of the mathematics major or minor.

Interdisciplinary Concentration
The department offers a concentration in mathematics to those economics majors completing MT 233, 420, and two mathematics electives numbered above 300.  Similarly, B.S. mathematics majors may earn a concentration in economics by successfully completing EC 301, 302, 410, and one other upper-division economics elective.  Students seeking this interdisciplinary concentration should consult with the chair of the Department of Mathematics and Computer Science.

Other Programs
The department has a five-year program with Case Western Reserve University whereby a student can earn a B.S. in mathematics at John Carroll in four years and in one additional year earn an M.S. in operations research from Case.  Students interested in this program should consult with the chair of the Department of Mathematics and Computer Science at the end of their second year.

The department also offers programs leading to the M.A. and M.S. in mathematics.  Program requirements and course descriptions are published in The Graduate Studies Bulletin.
118. APPLIED MATHEMATICS 3 cr.  Introduction to the use of mathematics to model various aspects of everyday life.  Topics include application of graphs and networks to urban services and business efficiency, planning and job scheduling, interpreting data for decision making, digital information representation, growth, voting systems, fair division.
120. MATHEMATICS AND COMPUTING 3 cr.  Mathematical modeling using computer spreadsheets and programs.  The course includes computer familiarization.
122. ELEMENTARY STATISTICS I 3 cr.  Describing data by graphs and measures, sampling distributions, confidence intervals and tests of hypotheses for one and two means and proportions, Chi-square tests, correlation and regression.  Use of appropriate statistical software.
123. ELEMENTARY STATISTICS II 3 cr.  Prerequisite:  MT 122.  Power analysis, factorial and repeated measures analysis of variance, nonparametric procedures, contingency tables, introduction to multiple regression.  Use of appropriate statistical software.
133-134. CALCULUS AND ANALYTIC GEOMETRY IA-IB 3 cr. each.  Placement by the Math Department.  Sequence covers the same calculus topics as MT 135 with algebra review integrated into the course as needed.  The MT 133-134 sequence will count as one course in Division IV of the Core, but neither MT 133 nor MT 134 will count as a Core course individually.  Note:  MT 133-134 will satisfy the MT 135 or equivalent prerequisites and requirements listed throughout the Bulletin.  Academic credit will not be given for both MT 134 and MT 135.
135. CALCULUS AND ANALYTIC GEOMETRY I 4 cr.  Prerequisite:  placement by the Math Department.  Functions, limits, continuity, differentiation, differentiation rules, optimization, antiderivatives, definite integrals.  Fundamental Theorem of Calculus, improper integrals, applications of integrals, including probability.  (See “Note” under MT 133-134 above.)
136. CALCULUS AND ANALYTIC GEOMETRY II 4 cr.  Prerequisite:  MT 135 or equivalent.  Second course in a three-semester calculus sequence.  Parametric curves, trigonometric functions and their inverses, differentials, related rates, techniques of integration, additional applications of integrals, introduction to differential equations, polar coordinates, sequences, and series of numbers.
160. MATHEMATICS AND CREATIVITY 3 cr.  Mathematics and the men and women who have contributed to it.  Topics in modern mathematics and examples of mathematical creativity are emphasized, as well as the student’s reaction to and thoughts on selected readings concerning the nature of mathematics and mathematics as a creative art.
162. MATHEMATICS FROM NON-WESTERN CULTURES 3 cr.  Introduction to mathematics developed in non-Western and Native American societies and illustrations of modern mathematical ideas within non-Western cultures.
167. THE MATHEMATICS OF CHANGE AND CHANCE 3 cr.  Prerequisite:  placement by the Math Department.  Topics from discrete and continuous probability; introduction to statistics; calculus topics from a conceptual viewpoint.  Mathematical software (spreadsheets, and computer algebra systems) will be used throughout the course.
171. FOUNDATIONS OF EARLY CHILDHOOD MATHEMATICS 3 cr.  Prerequisite:  MT 160.  Corequisite:  MT 171L.  Focus on understanding, from an advanced standpoint, the mathematics taught in elementary school.  Curriculum issues, methods, instructional resources, and assessment strategies for grades pre-K through 3 will be addressed.
171L. FOUNDATIONS OF EARLY CHILDHOOD MATHEMATICS LAB 0 cr.  Corequisite:  MT 171.  Math teaching methods lab for students in the pre-K through 3 licensure program.
199. SPECIAL TOPICS IN MATH 1-3 cr.  Subject announced in schedule of classes.
228. STATISTICS FOR THE BIOLOGICAL SCIENCES 3 cr.  Prerequisite:  MT 135 or equivalent.  Exploratory data analysis, probability fundamentals, sampling distributions and the Central Limit Theorem, estimation and tests of hypotheses through one-factor analysis of variance, simple linear regression, and contingency tables using appropriate statistical software.   Course content in biology context.
233. CALCULUS AND ANALYTIC GEOMETRY III 4 cr.  Prerequisite:  MT 136.  Calculus of vector-valued functions, infinite series, partial differentiation, multiple and line integrals.
241. FOUNDATIONS OF MIDDLE SCHOOL MATHEMATICS 3 cr.  Prerequisites:  MT 135, 160.  For students seeking the license to teach mathematics in grades 4-9.  Reasoning and proof in mathematical sets, number systems, functions, and binary operations.  Students will learn to communicate mathematics, to make connections among mathematical systems, and to construct valid arguments and proofs.
251. TOPICS FROM MIDDLE SCHOOL MATHEMATICS 3 cr.  Prerequisites:  MT 135, 160.  For students seeking the license to teach mathematics in grades 4-9.  Metric geometry, synthetic and transformational geometry with the use of dynamic geometry software; topics from discrete mathematics such as counting techniques, probability, recursive processes, graphs and networks.
330. INTRODUCTION TO ANALYSIS 3 cr.  Prerequisite:  MT 233.  Rigorous mathematical treatment of the fundamental ideas of calculus:  sequences, limits, continuity, differentiation, and integration.
341. INTRODUCTION TO ABSTRACT ALBEGRA 3 cr.  Prerequisite:  MT 136.  Sets, equivalence relations, permutations, polynomial rings, and groups, with emphasis on proof techniques.
342. INTRODUCTION TO LINEAR ALGEBRA 3 cr.  Prerequisite:  MT 136.  Algebra of matrices, linear systems, vector spaces, linear transformations, eigenvectors, applications.
372. DISCRETE AND CONTINUOUS MODELING 3 cr.  Prerequisite:  MT 233.  Prerequisite or corequisite:  MT 342.  Topics include graphical models, discrete dynamical systems, curve fitting, linear programming, simulation, differential equations, graph theory.
420. PROBABILITY AND STATISTICS I 4 cr.  Prerequisite: MT 136. Probability, discrete and continuous distributions, sampling distributions and the Central Limit Theorem, introduction to data analysis, estimation, and hypothesis testing; simple linear regression and correlation; use of appropriate statistical software.
421. PROBABILITY AND STATISTICS II 3 cr.  Prerequisites: MT 233, 420. Moment generating functions, transformations, properties of estimators, foundations of hypothesis tests, one and two-factor analysis of variance, and nonparametric analyses.
422. APPLIED STATISTICS 3 cr.  Prerequisites: MT 342, 420.  Categorical data analysis, multiple regression, analysis of variance of various designs, introduction to design of experiments.  Use of statistical software.
425. OPERATIONS RESEARCH 3 cr.  Prerequisite: MT 342.  Linear programming,  sensitivity analysis and duality, queuing theory, topics from networks, decision making, game theory, Markov chains, dynamic programming, and simulation.
431. ADVANCED CALCULUS OF ONE VARIABLE 3 cr.  Prerequisites: MT 233, 341. Real-number system, limits, continuity, differentiability, Riemann integral, properties of continuous and differentiable functions, sequences and series of functions.
432. ADVANCED CALCULUS OF SEVERAL VARIABLES 3 cr.  Prerequisites: MT 233, 342.  Development of and motivation for vector‑valued functions, calculus of functions of several variables, implicit functions and Jacobians, multiple integrals, line integrals.
436. INTRODUCTION TO COMPLEX ANALYSIS 3 cr.  Prerequisite: MT 341 or 342 or permission of department chair. Complex number plane, analytic functions, integration of complex functions, sequences and series. Residue theorem, evaluation of real integrals.
438. ORDINARY LINEAR DIFFERENTIAL EQUATIONS 3 cr.  Prerequisites: MT 233, 342. Linear equations and systems, existence and uniqueness theorems, oscillation theory.  Autonomous equations and systems, their solutions and qualitative properties.
441. ABSTRACT ALGEBRA 3 cr.  Prerequisite: MT 341. Groups, rings, domains, fields, extension fields, and polynomial rings.
450. EUCLIDEAN AND NON‑EUCLIDEAN GEOMETRY 3 cr.  Prerequisite: MT 341 or 342 or permission of department chair. Alternative ways of investigating the Euclidean plane, including transformational geometry; examination of the parallel postulate and how it can be changed to create new geometries; hyperbolic geometry.
452. ELEMENTARY TOPOLOGY 3 cr.  Prerequisite: MT 341. Topological spaces, homeomorphisms, connected spaces, compact spaces, regular and normal spaces, metric spaces, and topology of surfaces.
456. FRACTAL GEOMETRY 3 cr.  Prerequisites: MT 233, 341. Topics from metric spaces, transformations, iterated function systems, dynamical systems, fractal dimension, Julia sets, and Mandelbrot sets.
468. THEORY OF NUMBERS 3 cr.  Prerequisite: MT 341. Divisibility theorems, number‑theoretic functions, primitive roots, quadratic congruences and reciprocity, partitions.
469. HISTORY OF MATHEMATICS 3 cr.  Prerequisite: MT 341.  Study of mathematics from its origins to its present state.  Topics include the development and impact of geometry, algebra, number theory, irrational numbers, analytic geometry, calculus, non‑Euclidean geometry, and infinite sets.
479. COMBINATORICS AND GRAPH THEORY 3 cr.  Prerequisite: MT 341 or 342. Pigeonhole principle, inclusion and exclusion, recurrence relations and generating functions, combinatorial designs, the theory of graphs, graphical optimization problems.
480. SPECIAL TOPICS cr. TBA.  Readings about, reports on, and investigation of selected material and topics.
499. INDEPENDENT STUDY IN MATHEMATICS 1-4 cr.  Prerequisite:  consent of chair and instructor.  Designed for the student who wants to undertake a research project supervised by a faculty member.

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