# Master of Science in Mathematics, Master of Science inAdolescence Education Mathematics

Director, Graduate Program

## MS in Mathematics

This program is designed for in-service secondary Mathematics teachers who desire further study in the discipline, teachers who aspire to become master teachers in their own school district, and for those who hope to teach Mathematics in a Community College. Students in the program will expand their understanding of tertiary Mathematics in four key areas: Algebra, Analysis, Statistics/Data Analysis and Number Theory. An undergraduate concentration in Mathematics is required for admission. This preparation should include courses in Linear and Abstract Algebra, Analysis and Probability Theory.

The program requires 30 credits of graduate mathematics including a 3 credit Statistics/Data Analysis elective, a 3 credit Algebra elective, a 3 credit Analysis elective, 3 credits of Number Theory, a 3 credit Research Seminar, and three terms of the Graduate Colloquium. Twelve additional elective credits round out the program. The student must pass Masters Comprehensive exams in 4 areas chosen from courses selected in the content areas of statistics, algebra, analysis, and number theory.

This program may be completed on either a full-time or part time basis. Qualified undergraduates may begin graduate classes as upper class-men, thereby enabling completion of the MS degree in a single postgraduate year.

### Required Graduate Courses for the MS in Mathematics:

MATG 632 or MATG 633 Statistics/Data Analysis Elective; MATG 642 Number Theory; MATG 688 or MATG 690 , Analysis Elective; either MATG 671 or MATG 678 Algebra elective , MATG 791 and MATG 792 Graduate Colloquium, MATG 699 Research Seminar plus 12 additional elective credits of graduate mathematics (MATG 600-799).

## BS-MS in Adolescence Education Mathematics

This program is designed for the undergraduate student seeking certification for grades 7-12. The program is a seamless 5-year BS-MS program with a major in Adolescence Education Mathematics and an MS in Mathematics which leads to professional certification in Adolescence Education, Mathematics upon completion of three years of teaching experience. The 5-year program is grounded in a deep knowledge of college mathematics and its connection to secondary mathematics and to the common core standards. Qualified students may use undergraduate electives to begin graduate course-work, earning nine graduate credits during the first four years. Upon satisfactory completion of specific program requirements for the bachelor’s degree, and successful completion of LAST, ATS-W, and CST (Multi-subject and Students with Disabilities), students will be recommended for initial certification in each area.

After completion of the requirements for initial certification, students pursue a 5^{th} year of graduate mathematics. Required courses include a two-term sequence in the roots of high school Geometry and Trigonometry, and a course on Contemporary Issues in Teaching Mathematics. Students also participate in departmental seminars and colloquia. An additional elective rounds out the program.

Graduates of the Manhattan College masters program in Adolescence Education Mathematics will gain deep knowledge in five content areas: Algebra, Number Theory, Calculus, Probability and Statistics, and Geometry and make connections from these content areas to the Secondary Curriculum and to the common core standards. They will understand how to use software and technology effectively in the teaching of mathematics and communicate mathematics effectively both orally and in writing.

### Admission

Undergraduates will apply after completion of the fall term of junior year. A minimum cumulative GPA of 3.3 in the mathematics classes Math 185, 186, 285, 243, 272, 361, and 377 is normally required.

**Degree Program**

(*150 credits)*

Students complete the required sequence of undergraduate courses during freshman, sophomore, junior and senior year. In the third year they enroll in one graduate class. In the fourth year they are enrolled in two 3 credit graduate courses, and complete the requirements for initial certification. In the fifth year, students complete an additional 23 graduate credits, for a total of 32 graduate credits. Graduates will demonstrate mastery of the tertiary mathematics curriculum, its connections to the secondary curriculum, the common core standards, and in particular, the foundations for Calculus, Algebra, Number Theory, Geometry and Statistical Inference. Students will take Masters Comprehensives in three content areas to be chosen by the student in consultation with the director.

### Required Graduate Courses:

MATG 632 Graduate Statistical Inference, MATG 622 Graduate Seminar for Mathematics Education , MATG 642 Graduate Number Theory, MATG 688 Graduate Analysis II, MATG 724 Contemporary Issues in Teching Mathematics,

and MATG 761 Classical and Modern Plane Euclidean Geometry and MATG 762 Modern Methods in Plane Euclidean Geometry.

One course chosen from MATG 671 Graduate Linear Algebra II or MATG 678 Graduate Algebra II , MATG 791 Graduate Colloquium or MATG 792 Graduate Colloquium, plus 3 additional elective credits of graduate mathematics (MATG 600-799).

### Courses

**MATG 622. Graduate Seminar for Mathematics Education . 3 Credits.**

This course is intended for prospective mathematics teachers. Topics in high school mathematics are examined from an advanced perspective. Topics include the real and complex numbers, functions, and trigonometry. The course requires a written project and an oral presentation. The use of appropriate technology will incorporated throughout the course. Prerequisites: MATH 243 and MATH 272.

**MATG 632. Graduate Statistical Inference. 3 Credits.**

Topics covered in this course include sampling distributions, point estimation,
interval estimation, testing statistical hypotheses, regression and correlation. Requires an historical analysis project in two parts: 1-2-page paper proposal, 10-page. Prerequisite: MATH 431.

**MATG 633. Mathematical Statistics. 3 Credits.**

This is a data intensive course on statistical inference. Topics covered in this course include regression analysis, hypothesis testing, analysis of variance, nonparametic modeling, and sequential tests of hypotheses. Students will utilize and evaluate statistical computer packages for use in teaching probability, statistics, and/or mathematics/economics courses. Not open to students with credit for MATH 433. Prerequisite: MATH 331.

**MATG 642. Graduate Number Theory. 3 Credits.**

An introduction to number theory with connections to the Middle and High school curriculum. Divisibility, prime numbers and their distribution, congruences, quadratic residues and nonresidues, Diophantine equations, elliptic curves, primality testing, applications to cryptology. Recent progress. The course requires a written project connecting the course content to the 6-12 curriculum. Prerequisite: MATH 272.

**MATG 648. Graduate Combinatorics and Graph Theory. 3 Credits.**

Fundamental concepts in combinatorics include binomial coefficients, inclusion-exclusion, and generating functions. Topics in graph theory include connectivity, planarity, colorings and chromatic polynomials, and max-flow-min-cut in networks. This course will require a written project and an oral presentation on some particular application of Graph Theory or Combinatorics. The project will consist of a case study that will require researching a particular area of application, and then formulating, solving, and analyzing an appropriate mathematical model. Findings will be presented at the end of the term. Not open to students with credit for MATH 448 or Cmpt. 335. (Cr. 3) Prerequisites: MATH 243, MATH 272.

**MATG 655. Graduate Operations Research. 3 Credits.**

Optimization, linear programming, simplex method, duality theory, transportation problems, scheduling problems, queuing theory. Students will be required to complete an independent project. The project will consist of a case study that will require researching a particular area of application, and then formulating, solving, and analyzing an appropriate mathematical model. Findings will be presented at the end of the term. Not open to students with credit for MATH 475 or 455. Prerequisites: MATH 272.

**MATG 664. Graduate Topology. 3 Credits.**

An Introduction to Topology, beginning with the concept of topological equivalence, and topological invariants. Knots and Links, colorings, knot polynomials, Euler characteristic, cut and paste techniques, classification of surfaces, 3-manifolds, and the fundamental group, the Poincare conjecture. This course will require a written project and an oral presentation on some particular application of, or historical development in Topology. Not open to students with credit for MATH 464. (Cr. 3) Prerequisite: MATH 243.

**MATG 671. Graduate Linear Algebra II. 3 Credits.**

A continuation of the topics introduced in Linear Algebra, (MATH 272), with emphasis on orthogonality, inner product spaces, eigenvalues and eigenvectors, diagonalization, quadratic forms and numerical linear algebra. The course requires a written project connecting the course content to the 6-12 curriculum. Not open to students with credit for Math 325 or 471. (Cr. 3) Prerequisite: MATH 272.

**MATG 678. Graduate Algebra II . 3 Credits.**

This is the second part of a two-semester sequence. We undertake further study of algebraic structures, such as rings, fields and integral domains. Significant results include the Fundamental Homomorphism Theorem and Unique Factorization. The course requires a written project connecting the course content to the 6-12 curriculum. Not open to students with credit for Math 316 or 478. Prerequisite: MATH 377.

**MATG 688. Graduate Analysis II. 3 Credits.**

This course is a successor to Analysis I. The approach followed here is a rigorous treatment of the material found in Calculus I and II leading to an introduction to measure theory and the modern definition of the integral. The first part of the course covers the Riemann Integral, infinite series, sequences and series of functions with an emphasis on uniform convergence and its consequences. This leads to the need to extend the definition of the integral to allow for the treatment of more complicated functions. The approach of Lebesgue leads to a new integral with vastly improved convergence properties. Not open to students with credit for MATH 488. (Cr. 3) Prerequisite: MATH 387 (formerly 313).

**MATG 690. Graduate Complex Analysis . 3 Credits.**

This course focuses on the complex plane, complex functions, limits and continuity, as well as analytic functions, the Cauchy-Riemann equations, the Cauchy Integral Theorem, and consequences. Additional topics may include: power series, Taylor and Laurent Series, classification of singularities, the Residue Theorem and its applications, conformal mapping, and selected applications. This course will require a written project and an oral presentation on some particular application of, or historical development in complex analysis. Not open to students with credit for MATH 490 or 407. Prerequisites: MATH 387 (formerly 313).

**MATG 699. Research in Mathematics. 3 Credits.**

Investigation of a research topic in mathematics culminating in a written paper and oral presentation. Prerequisite: Permission of the Graduate Director. (Cr. 3).

**MATG 724. Contemporary Issues in Teching Mathematics. 4 Credits.**

Discussion of issues related to mathematics instruction at the secondary and early college level: how to develop student competence in effective communication, cooperative learning, use of technology, quantitative literacy, knowledge of content, professional responsibilities. Students will gain experience running review sessions and assist in labs, and evaluate lesson plans incorporating the use of technology as appropriate. Selected readings and evaluation of lesson plan portfolios generated in Algebra and Number Theory. (Cr. 4).

**MATG 761. Classical and Modern Plane Euclidean Geometry. 4 Credits.**

This is a the first part of a two-semester introduction to classical and modern plane Euclidean geometry. The sequence fills a critical gap for prospective secondary school mathematics teachers, who may have to teach courses in Euclidean geometry with an undergraduate background that may have no geometry or may include geometry courses whose nature is not well-adapted to the demands of most secondary geometry curricula. The first part of the course includes a recapitulation of the classical content of high-school Euclidean geometry, based on the SMSG axioms, with additional topics and most exercises well beyond the high-school curriculum. For example, the classical results about the concurrence of various special lines in a triangle are obtained as consequences of a single overarching result, Ceva's theorem. Ptolemy's theorem for cyclic quadrilaterals is proved, and then is used in the second semester to obtain the formula for sin(x). Heron's formula for the area of a triangle in terms of the sides and semi-perimeter is derived, and the equidecomposability of plane polygons of equal area is proved. Finally, a geometric treatment of the conic sections and their reection properties is presented. The course will use Geogebra, Geometers SketchPad or an equivalent software product. (Cr. 4) Prerequisite: MATH 243.

**MATG 762. Modern Methods in Plane Euclidean Geometry. 4 Credits.**

This is a the second part of a two-semester introduction to classical and modern plane Euclidean geometry. The course continues with the introduction of modern methods. Topics include trigonometry, coordinate methods and the algebra associated with the conic sections, complex numbers, vector methods, transformations, and inversion with respect to a circle. Many of the results of the first semester are revisited from new perspectives (for example Heron's formula is found by complex number methods), and a host of more modern results are obtained. The course will use Geogebra, Geometers SketchPad or an equivalent software product. (Cr. 4) Prerequisite: MATG 761.

**MATG 791. Graduate Colloquium. 1 Credit.**

Weekly participation in Departmental seminar. Written expository summary of talks given by other speaker. Readings and discussions of good presentation practice. Graduate students will give several presentations during the semester. 1cr.

**MATG 792. Graduate Colloquium. 1 Credit.**

Weekly participation in Departmental seminar. Written expository summary of talks given by other speaker. Readings and discussions of good presentation practice. Graduate students will give several presentations during the semester. 1cr.