**Department of Mathematical Sciences **

PO Box 172400

Wilson Hall 2-214

Office Number 406-994-3601

**Department Head**

Dr. Kenneth Bowers

**Professors **

M. Barge; dynamical systems.

J. Borkowski; experimental design, response surface methodology, sampling, quality control.

K. Bowers; applied mathematics, numerical analysis.

J. Dockery; applied mathematics.

W. Esty; stochastic processes, probability, mathematics education.

T. Gedeon; applied dynamical systems.

I. Klapper; applied mathematics.

J. Kwapisz; dynamical systems, ergodic theory.

J. R. Lund; numerical analysis.

**Associate Professors **

J. Banfield; statistical computation, pattern recognition.

E. Burroughs; mathematics education.

J. Cherry; spatial statistics, linear models, ecological and environmental statistics.

L. Davis; sensitivity analysis, optimal design, numerical analysis.

L. Geyer; dynamical systems, complex analysis.

M. Greenwood; functional data analysis, time series, model selection criteria.

J. Luebeck; mathematics education.

M. Pernarowski; applied mathematics.

J. Robison-Cox; statistical computing, graphics, mixed effects models.

D. Yopp; mathematics education.

**Assistant Professors **

M. Higgs; ecological and environmental statistics, bayesian hierarchial models, spatial statistics, computational statistics.

B. Lindaman; mathematics education.

T. Zhang; applied mathematics, numerical analysis.

**Degrees Offered **

M.S. in Mathematics

M.S. in Mathematics (Mathematics Education option)

M.S. in Statistics

Ph.D. in Mathematics

Ph.D. in Mathematics Education Emphasis

Ph.D. in Statistics

The Department offers graduate study leading to the degree of Master of Science in either Mathematics or Statistics. The M.S. in Mathematics degree is available with two options: Mathematics and Mathematics Education. The Doctor of Philosophy degree is offered in Mathematics and Statistics. The Ph.D. in Mathematics is available with two emphases: Mathematics and Mathematics Education. The Department also offers a Graduate Certificate in Statistics.

**Admission **

For regular admission to either the M.S. or the Ph.D. degree program, a student should have completed at least eighteen (18) credits of mathematics beyond calculus. For mathematics majors, this should include a year of advanced calculus. For statistics majors, it should include a year of statistical theory and a year of probability and statistical methods. Admission to the mathematics education program is determined on an individual basis. Successful applicants are accepted into both the Department and the Graduate School.

**Master of Science Requirements **

The Master of Science degrees are offered under Plan A (Thesis) and Plan B (Non-thesis). Of the required thirty (30) credit minimum, at least eighteen (18) credits of 500-level course work must be taken under either plan.

Available under Plan B is a comprehensive master's degree in either mathematics or statistics. Although no thesis is required in this plan, a sound knowledge of several areas of mathematics and/or statistics is expected. Also available under Plan B is a master's degree in mathematics with an option in mathematics education. This option is designed primarily for secondary teachers and is offered as a combination of on-line academic year course work and summer sessions. The mathematics education option requires completion of a program portfolio through a series of seminars.

For further information, refer to the For Master's Students section. Students are expected to be familiar with both the Department and the Graduate School degree requirements.

M.S. in Mathematics – Mathematics Option

M.S. in Mathematics Program Guidelines

The Master of Science degree in mathematics at Montana State University is designed to prepare students for further graduate work or for employment in academic, industrial, business, or government forums. Upon entrance, each student meets with the department's Graduate Program Committee to discuss career objectives and first year course work. During the second semester in the program each student forms a Graduate Committee and together, they outline the student's degree program. The prerequisites for the master's degree program in mathematics consist of the following courses or their equivalent: Linear Algebra (M 333) and Advanced Calculus (M 381-382). Students who have not completed these courses or their equivalent may still enter the master's program but it is suggested that these courses then be taken.

Both non-thesis and thesis plans are offered for the M.S. in Mathematics – Mathematics Option degree:

*Non-thesis Plan *

This plan requires both completing the course work and passing the written comprehensive exam. At least 30 credits of course work are required. Of these, at least 18 credits must be numbered 500 or higher. Regardless, all of the following core courses must be completed:

- M 503 Advanced Linear Algebra (every Spring)
- M 504 Abstract Algebra (every Spring)
- M 505 Mathematical Analysis (every Fall)
- M 511 General Topology (every Fall)

Additionally students must fulfill a breadth requirement by completing at least two of the following courses:

- M 441 Numerical Linear Algebra & Optimization (every Fall)
- M 450 Applied Math 1 (Fall odd numbered years)
- M 454 Dynamical Systems I (Fall even numbered years)
- STAT 421 Probability (every Fall)

Either or both of these two required courses may be replaced by the corresponding semester of the appropriate 500 level course: M 581 (numerical analysis), M 560 (applied mathematics), M 595 (dynamical systems), or STAT 501 (probability), respectively. Any other exceptions to the course requirements must be approved by the student's graduate committee and adhere to the minimum policy requirements set forth in the Graduate Catalog (Plan B). Requirements for the written comprehensive exam are listed separately below.

*Thesis Plan *

This plan requires completing the course work, passing the written comprehensive exam, writing a thesis and an oral defense of the thesis. At least 30 credits must be completed of which 10 must be thesis credits. Students must also complete both the core and breadth course requirements described in the Non-Thesis Plan above. Any exceptions to the course requirements must be approved by the student's graduate committee and adhere to the minimum policy requirements set forth in the Graduate Catalog (Plan A). Thesis and oral defense requirements must be arranged with and approved by the student's graduate committee. Requirements for the written comprehensive exam are listed separately below.

**M.S. in Mathematics Comprehensive Exam **

The M.S. comprehensive exam is a written exam administered in disjoint 3-hour components. Each component is graded as pass or fail. To pass the comprehensive exam a student must pass four different components within two examination periods. At least two of these components must be from the following list:

- Linear Algebra (M 503)
- Abstract Algebra (M 504)
- Analysis (M 505)
- Topology (M 511)
- Numerical Analysis (M 441-442)
- Applied Mathematics (M 450-451)
- Dynamical Systems (M 454-455)
- Probability and Statistics (STAT 421-422)

The other two required components may be from the list above or from the following list:

The first examination period occurs in January with the specific dates and times for each component determined by the department.

Students must attempt at least four components the first examination period after 3 semesters of study.

Typically, these four, 3-hour components will be administered in a morning and afternoon of two different days. If the student fails one or more components in the first examination period, a failure will be reported to the Graduate School. The student must then pass the remaining required components in a second examination period administered either during spring semester (at least two months after the first examination) or the following January. No more than four components may be taken in the second examination period. If the student has not passed the remaining required components after the second examination period, a second failure of the comprehensive exam will be reported to the Graduate School.

M.S. in Mathematics - Mathematics Education Option (MSMME)

**MSMME Admission Requirements **

A typical MSMME applicant will have (1) a BS or BA with a major or minor in mathematics, (2) an undergraduate GPA of 3.0 or higher, (3) certification to teach mathematics, and (4) at least two years of successful mathematics teaching experience at the secondary level. Applicants who do not have the above qualifications (e.g., those teaching at a private school) will be reviewed on a case-by-case basis.

**MSMME Advising **

A designated MSMME faculty coordinator develops a program of study for each student and assigns the student a three-person faculty committee. The committee must include at least two faculty members from the Department of Mathematical Sciences. The MSMME coordinator is charged with facilitating the student's program while the committee chairperson oversees development and assessment of the student's program portfolio.

**MSMME Program Requirements **

The MSMME program requires 30 semester hours of coursework. The program of study includes required coursework in fundamental areas of high school mathematics content: algebra, analysis (calculus), geometry, and statistics. Elective courses offer further study in number theory, discrete mathematics, and mathematical modeling, as well as mathematics education courses in curriculum, assessment, standards, and learning theory.

- Core Content Courses (required):
- M518 Statistics for Teachers
- M524 Algebra for Teachers
- M525 Analysis for Teachers
- M527 Geometry for Teachers

- Pedagogy Courses (choose at least 2 of 4):
- M520 Standards-Based Mathematics for Teachers
- M521 Learning Theories in Mathematics for Teachers
- M528 Curriculum Design
- M529 Assessment Models and Issues

- Electives:
- M517 Modeling and Technology for Teachers
- M523 Number Structures for Teachers
- M526 Discrete Mathematics for Teachers

Courses are scheduled on a rotating basis that potentially allows completion of the program in two academic years and three summers. Adherence to this pace requires taking summer coursework in both online and face-to-face formats as well as online academic year courses.

**MSMME Program Portfolio **

As a summative assessment experience under Plan B (non-thesis) MSMME students engage in an ongoing and embedded process of portfolio development. Teachers exiting the MSMME program must demonstrate a thorough understanding of the standards for content and practice that guide their profession; reflect on completed coursework as a coherent whole; and self-assess their acquired knowledge of mathematical content, pedagogical applications, and classroom research. The portfolio includes representative coursework, “living laboratory” research reports, and a series of reflections. Three credits of the program are designated for portfolio-related work.

A. Students participate in a 1-credit online portfolio seminar during spring semester of their first year in the program. They will:

- Read and discuss documents and articles related to standards of content and practice
- Complete a guided reflection for each course taken since admission to the program
- Synthesize knowledge from all courses completed up to this point
- Assemble reflections, work samples, and “living laboratory” reports into a portfolio

B. Students participate in a 1-credit online portfolio seminar during spring semester of their second year as well. They will:

- Read and discuss documents and articles related to standards of content and practice
- Complete a guided reflection for each course taken since the first seminar
- Synthesize knowledge from all courses completed up to this point
- Assemble reflections, work samples, and “living laboratory” reports into a portfolio

C. Finally, students attend a 1-credit, 1-week seminar their final summer. During this week all teachers near completion of the program will meet to reflect on program coursework and assignments, discuss current trends and issues in mathematics education, present “living laboratory” classroom research results, and share highlights from their portfolios.

**MSMME Program Completion **

Completion of the program requires 30 credits of approved coursework and satisfactory completion of the program portfolio. The portfolio will be reviewed by the faculty member supervising each of the three portfolio seminars and by the student's graduate committee chair.

The Graduate School mandates that each student must be enrolled in at least 3 credits of coursework during the semester they complete Plan B requirements and during the semester they intend to graduate. The MSMME program design ensures that a typical student will meet both of these requirements simultaneously by attending the third portfolio seminar and completing at least one 3-credit course during their final summer in the program.

M.S. in Statistics

**M.S. in Statistics Program Guidelines **

The Master of Science degree in statistics at Montana State University gives students a solid background in the applications as well as the theory of statistics. Students in this program prepare either for further graduate work or for academic, industrial, business, or government employment. Upon entrance, each student meets with the department's Graduate Program Committee to discuss career objectives and first year course work. During the second semester in the program each student forms a Graduate Committee and together, they outline the student's degree program. The prerequisites for the master's degree program in statistics consist of the following semester courses or their equivalent: Multivariable Calculus (M 273), Linear or Matrix Algebra (M 221), Probability (STAT 421), and Mathematical Statistics (STAT 422). Students who have not completed these courses may be accepted into the master's program with the understanding they should make up these courses.

Either Plan A (thesis and 20 credits of course work) or Plan B (30 credits of course work) can be chosen. In either case, all courses on a graduate program must be numbered 400 or higher and STAT courses must be numbered 408 or higher. The specific program of study depends on the student's previous training and experience. Regardless of the plan chosen, (i) at least half of the required non-thesis credits must be STAT courses, (ii) at least two-thirds of the required non-thesis credits must be numbered 500 or higher, and (iii) the following 15 semester core course credits are required:

**M.S. in Statistics Required Courses (15 semester credits) **

- STAT 501-502 Intermediate Prob and Math Stat - 6 credits
- STAT 505-506 Linear Stat Models, Adv Regression - 6 credits
- STAT 510 Statistical Consulting - 2 credits
- STAT 575 (Plan B below) 1 or 2 credits

**Additional requirements **

- The M.S. in Statistics degree requires completion of either a thesis or a writing project.
- Experience in data collection – either through a course such as Sampling or Design of Experiments, or a course taken in a former degree program, or real-life experience.
- For either Plan A or Plan B, the student must pass a comprehensive examination.

• Thesis (Plan A): The Plan A thesis typically requires 450-500 hours or work. The student must register for at least 10 thesis credits (STAT 590) in addition to the required 20 credits of course work. The student must give an oral defense or his/her thesis.

• Writing Project (Plan B): The Plan B writing project typically requires at least 90 hours of work, for which the student earns 2 credits of STAT 575. With permission from the student's committee, additional credits of STAT 575 (no more than 4 total) may be earned. Students should enroll in Stat 575 in their final Spring semester, and must give a seminar on the writing project before graduating.

**M.S. in Statistics Comprehensive Exam **

The M.S. comprehensive exam consists of a written exam over material from STAT 501, 502, 505, and 506.

The exam is given each August with the specific date determined by the department. The exam is graded as PhD pass, M.S. pass, or fail. Examinees will be informed of the results within five working days of taking the exam. The M.S. comprehensive exam may be repeated once.

** ****Ph.D. Requirements **

Students in mathematics are expected to develop competence in real and complex analysis and at least two areas chosen from applied mathematics, dynamical systems, functional analysis, numerical analysis, partial differential equations, probability, topology or other topics the student's committee may approve.

Students in statistics must demonstrate proficiency in the Ph.D. core (linear models, probability, and mathematical statistics) by passing the M.S. comprehensive exam at a Ph.D. Level.

Students in mathematics education must demonstrate competence in three areas: (1) at least one Ph.D. level mathematics topic (see above) to be determined by the student's graduate committee (2) current theory in mathematics curriculum, assessment, and instruction; and (3) educational statistics and research methods.

The student's graduate committee determines additional requirements. Degree candidates are expected to be familiar with both departmental and Graduate School degree requirements.

Ph.D. in Mathematics

Described below are the Department of Mathematical Sciences requirements for the Ph.D. in Mathematics. These departmental requirements supplement those set out by the Graduate School in the Graduate Catalog for Ph.D. Students.

There are no foreign language requirements or qualifying exam for a Ph.D. in Mathematics.

**Ph.D. In Mathematics Committee **

- The Ph.D. committee must include a minimum of five members excluding the Graduate School-assigned Graduate Representative.
- A committee must be formed before the end of the student's second semester of study.
- The Committee Chairperson (Advisor) must be a faculty member within the Department of Mathematical Sciences.
- The first three committee members listed on a candidate's Program of Study read and assess the dissertation.

**Ph.D. In Mathematics Course Requirements **

- A minimum of 30 credit hours are required (see the Graduate Catalog for Ph.D. Students for details).
- A minimum of 18 credit hours must be dissertation credits (M690)
- The Ph.D. student's Program of Study listing their intended coursework must be approved by all committee members.
- The student must take a minimum of 4 credits of the M 594 seminar series.

Typically, a Ph.D. student takes 18 credits of mathematics in courses numbered 500 or higher to prepare for their comprehensive examination. Students are encouraged to begin some form of doctoral reading or research (either informally or in the form of M 689 credits) with a committee member by their second year of study.

**Ph.D. In Mathematics Comprehensive Exam **

The Ph.D. Comprehensive examination consists of both a written and an oral comprehensive examination. The candidate must pass the written comprehensive exam before taking the oral comprehensive examination.

*Written Comprehensive Exam *

How a student may choose and retake exam components is determined by (a)-(f):

- The written comprehensive exam consists of 4-hour exam components graded as Pass or Fail.
- The candidate must pass three components to pass the written comprehensive examination though they may attempt more.
- If a candidate fails a component it may be attempted at most one more time.
- The candidate must pass the following "required" components:
- M 547, M 551 Measure Theory and Complex Analysis

- Normally the remaining components are from the following list of "standard" components:
- M 511-512 Topology
- M 595-596 Dynamical Systems
- M 584-585 Functional Analysis
- M 581-582 Numerical Analysis
- M 544-545 Partial Differential Equations
- M 560-561 Applied Mathematics
- M 547, 586 Measure Theory and Probability

- At most one "nonstandard" component not from (i)-(viii) may be taken. To take such a component a petition form must be completed.

*Oral Comprehensive Exam *

After passing the written comprehensive exam the candidate must pass an oral comprehensive exam at a date agreed upon by the candidate's committee. Normally the oral comprehensive exam is a thesis topic proposal where the candidate's ability to conduct research on the proposal is assessed. When this is not the case, the candidate will be informed of the nature of the oral comprehensive exam by their committee. The candidate has at most two attempts to pass the oral comprehensive examination.

**Ph.D. In Mathematics Dissertation Requirements **

Once the Ph.D. candidate has passed the comprehensive exam (both written and oral parts) the student has at most five years to submit an acceptable dissertation and pass their final defense. The first three committee members listed on a candidate's Program of Study must be given a dissertation draft at least two weeks prior to the Final Defense. Regardless, all committee members must have access to a dissertation draft at least one week prior to the Final Defense. The dissertation should embody the results of extended research by ·the candidate, be an original contribution to knowledge, and include new material worthy of publication. The dissertation must be submitted as an electronic dissertation, in final form to the Graduate School not later than 14 working days before the end of the term in which graduate work is completed.

**Ph.D. In Mathematics Final Defense **

Department policies on the final defense and all other administrative procedures regarding the degree completion are exactly those as set out by the Graduate School.

Ph.D. in Mathematics Education

The Department of Mathematical Sciences offers a Ph.D. in Mathematics specializing in Mathematics Education. This program blends the study of advanced mathematics with coursework covering current trends in K-12 curriculum, assessment, and instruction, current research on mathematics teaching and learning and educational research design. As an emphasis within the Ph.D. in Mathematics, this program follows similar requirements regarding the Ph.D. committee. Credit requirements are also similar.

This pathway is designed for candidates who plan a future of teaching, research, and service focused on mathematics education in K-12 or collegiate settings. The program carries a strong emphasis on the pedagogy, content, and issues that characterize K-12 school mathematics, and graduates typically go on to faculty positions that involve teacher preparation and research in K-12 mathematics education. Doctoral students conduct research in areas that align with the faculty's ongoing research interests and currently funded projects.

The education components of the program are coupled with a strong preparation in graduate-level mathematics . Ph.D. candidates possess or earn the equivalent of a master's degree in mathematics and must complete a doctoral-level comprehensive examination in one area of mathematics, equipping graduates to seek employment in university mathematics departments as well as schools of education. A separate program leading to an Ed.D. in Curriculum & Instruction, with a 15-credit concentration in mathematics, is offered by the Department of Education in the College of Education, Health & Human Development. Information about this program is available through the MSU Education Department.

**Ph.D. in Mathematics Education Admission **

Applicants should possess a solid background in mathematics content, most often indicated by an earned master's degree in mathematics, statistics, or mathematics education. Applicants with a strong undergraduate degree in mathematics or mathematics teaching will also be considered for the program. Undergraduate applicants will need to take necessary coursework to ensure both masters level competency in mathematics and roughly the equivalent of a secondary teaching credential in mathematics. Candidates who enter the Ph.D. program from baccalaureate programs can expect to add two to three years to their program of study while they complete these prerequisites.

Ideally, applicants will have teacher certification with a mathematics endorsement and/or two years of K-12 teaching experience. Applicants who lack K-12 teaching experience will be expected to acquire the following equivalencies by the time they complete the coursework:

- Doctoral candidates are expected to possess a minimum level of competency in secondary mathematics instruction. This is represented by an undergraduate degree in mathematics along with at least six credit hours of education coursework, earned by:
- Completing a secondary mathematics methods course (EDU497)
- Completing a pedagogy course from M520, M521, M528, M529
- Students who lack sufficient exposure to instruction at the elementary or secondary level will be required to complete school-based internships prior to beginning dissertation research. Each internship calls for extensive field experience as well as participation in a Mathematics Education seminar. Internships may include:
- Elementary level – teach, tutor, and observe students in a K-8 classroom
- Secondary level – teach one or more courses in a high school setting

**Ph.D. in Mathematics Education Comprehensive Examination **

The written comprehensive examination for the Ph.D. in Mathematics Education consists of three separate components. Each component is completed in a 4-hour session.

• One component in mathematics chosen from the examination areas accepted for the Ph.D. in mathematics. This includes Real and Complex Analysis (M547-M551) or another sequence approved by the student's committee.

• One component in mathematics education including M528, M529 and any other mathematics education courses required by the student's committee.

• One component in educational statistics and research methods.

The mathematics and education/research components are typically completed in different years. Upon successful completion of all three written comprehensive examinations, the student must pass an oral defense of the comprehensive examinations. The student's full committee attends the oral defense, which is often coupled with presentation of the dissertation proposal.

Along with passing the comprehensive examination, a Ph.D. candidate must have the support of a research advisor in order to transition from Ph.D. coursework to dissertation research. This advisor is not necessarily the committee chairperson of the student's original committee. While still completing their coursework, Ph.D. candidates should seek out faculty to discuss potential research opportunities and eventually establish a formal advisor relationship.

Ph.D. in Statistics

**Ph.D. in Statistics Program Requirements **

The Ph.D. program in statistics at Montana State University prepares students for academic, industrial, business, or government employment. To earn a Ph.D. in statistics, a student must pass the M.S. comprehensive exam at the Ph.D. level, pass the Ph.D. comprehensive exam, and write and defend a Ph.D. dissertation. The exams are described below. The dissertation must be an original contribution to statistical science and must include new material worthy of publication. There is no departmental foreign language requirement for the Ph.D.

A Ph.D. student typically takes at least 24 credits of statistics in courses numbered 500 and higher. Additional course work in statistics and/or mathematics may be necessary, depending on the candidate's chosen area of specialization and background. For instance, a Ph.D. student is expected to have completed all courses required for the M.S. degree in statistics and may need to make up one or more of these courses. Also, it is expected that a Ph.D. student will take directed study courses (STAT 689) in his/her area of specialty. STAT 690, dissertation credit requirements, are listed in the Graduate Catalog. Two credits of STAT 510, Statistics Consulting Seminar are required.

**Ph.D. in Statistics Qualifying Exam **

The Ph.D. qualifying exam is identical to the statistics M.S. comprehensive exam except that the exam must be passed at the Ph.D. level (i.e., Ph.D. pass). A student who earned an M.S. in Statistics from MSU need not take the PhD qualifying exam if the M.S. comprehensive exam was passed at the Ph.D. level. Other students are expected to take the Ph.D. qualifying exam as soon as course work in the M.S. core has been completed. Two attempts to pass the qualifying exam are allowed.

**Ph.D. in Statistics Comprehensive Exam **

The topics and format of the written comprehensive exam for the Ph.D. in Statistics will be determined by the student's committee.

Each student must devise areas of concentrated study. The concentration areas must be approved by the student's committee and must include, in total, an amount of material equivalent to at least 6 graduate level courses. An area could involve course material from outside the department.

Each component of the written comprehensive examination is graded separately as pass or fail. A failed component may be repeated once. Once the written comprehensive examination has been passed, the student must pass the oral comprehensive examination. The student's committee will inform the student of the structure and timing of the oral exam.

Graduate Certificate in Statistics

** **Training in statistical methods is a required part of the education of graduate students in engineering and the sciences. The Graduate Certificate in Statistics is designed to provide additional education in statistical thinking and methodology over and above the basic coursework taken by the typical graduate student. This transcriptable certificate will provide a clear record of additional training in statistics for future graduate programs or employers. The Graduate Certificate will also be of interest to those currently employed in technical businesses in the and to post-baccalaureate students.

Course Requirements

All students must take

STAT 511/512 - Methods for Data Analysis I/II

Students choose 2 courses from the following list, at least one of which must be either STAT 446 or STAT 526

- STAT 446 - Sampling
- STAT 526 - Experimental Design
- STAT 431 - Nonparametric Statistics
- STAT 436/536 - Introduction to Time Series Analysis
- STAT 437 - Introduction to Applied Multivariate Analysis
- STAT 439 - Introduction to Categorical Data Analysis
- STAT 448 - Mixed Effects Models
- STAT 524 - Biostatistics
- STAT 528 - Statistical Quality Control

Current graduate students must:

- Obtain the approval of the department head/graduate coordinator of the student's major department and the Department of Mathematical Sciences.
- Obtain a grade of B or better in all coursework counted toward the certificate.

Other students must:

- Obtain the approval of the department head/graduate coordinator of the Department of Mathematical Sciences and the Graduate School.
- Obtain a grade of B or better in all coursework.

**Financial Assistance **

Graduate assistantships are available to qualified graduate students in mathematics, mathematics education, or statistics. Graduate Teaching Assistantships (GTA) usually require teaching one course each semester. Graduate Research Assistantships (GRA) are also available to qualified students. Time requirements are similar to those for teaching assistantships. See the Graduate Assistantships section for detailed information on appointment criteria.