The Ph.D. degree is a content-based degree program that has two concentrations: Concentration in Mathematics and Concentration in Collegiate Mathematics Education. Coursework is broadly distributed across the various areas of mathematics and mathematics education.
The Ph.D. degree is designed (a) to develop well-prepared teachers to teach college level mathematics and conduct independent research effectively in their areas of expertise or (b) to prepare students with advanced knowledge and applications in mathematics to work effectively in non-academic environments.
Program Requirements
Successful completion of the Ph.D., including coursework, internship, and dissertation, requires a minimum of 75 semester hours of graduate work beyond the bachelor's degree. Up to 30 hours of relevant graduate work may be transferred for students entering with a master’s degree. In order to obtain the Ph.D. degree the student must have a GPA of 3.0 (B) or better.
A student with a bachelor's degree must have earned at least 50 of the total 75 hours at the 600 level or above. Those entering with a master's degree must have earned at least 35 hours at the 600 level or above taken at CMU. At least 15 hours of the coursework must be earned at the 700 level or above, excluding the dissertation and the internship credits.
Coursework
The program requires a minimum of 57 hours of coursework excluding internship and the dissertation credit beyond the bachelor's degree or 27 hours of such coursework after the master's degree. At least 15 hours of the coursework must be earned at the 700 level or above, excluding the dissertation and the internship credits. These hours are distributed among core courses and concentration-specific courses. The minimum hours that are required in each category is specified in parentheses. An advisor will assist a student in the selection of the courses. Courses in which a student earns or has earned a grade below C (2.0) do not count toward meeting any graduate degree requirements.
Any student having satisfied any of the course requirements prior to entering the program may be excused from that course requirement. For this to be approved, the student must complete the Course Requirement Waiver Form supplied by the department. However, the required total credit hours will not be affected.
The three concentrations require a list of Common Core Courses, Internship, Dissertation, and concentration-specific coursework and qualifying exams. The list of coursework below is for students who have an undergraduate degree satisfying the admission requirements. For those who enter the program with a master's degree, up to 30 hours of relevant graduate work may be transferred depending on individual’s background and the chosen concentration in consultation with an academic advisor.
Common Core Courses (21 hours)
| MTH 525 | Advanced Linear Algebra | 3(3-0) |
| MTH 623 | Theory of Groups and Rings | 3(3-0) |
| MTH 625 | Theory of Modules and Fields | 3(3-0) |
| MTH 632 | Introduction to Real Analysis and Its Applications | 3(3-0) |
| MTH 636 | Introduction to Complex Variables | 3(3-0) |
| MTH 761 | Methods for Teaching College Mathematics | 3(3-0) |
| MTH 762 | A Survey of Research in Collegiate Mathematics Education | 3(3-0) |
Concentrations
Select one of the following concentrations:
Concentration in Mathematics (36 hours)
Mathematics Core Courses (9 hours)
Must select MTH 534 and MTH 645:
| MTH 534 | Applied Mathematics and Differential Equations | 3(3-0) |
| MTH 633 | Real Variables | 3(3-0) |
| MTH 637 | Complex Variable Theory | 3(3-0) |
| MTH 645 | Algebraic Topology | 3(3-0) |
Core Electives (15-18 hours)
Select 15-18 hours from Core Electives I and Core Electives II.
Core Electives I - Topics in Pure and Applied Mathematics (6-15 hours)
Select from the following:
| MTH 520 | Optimization Theory | 3(3-0) |
| MTH 578 | Combinatorics and Graph Theory | 3(3-0) |
| MTH 627 | Representation Theory of Lie Algebras | 3(3-0) |
| MTH 634 | Fourier Analysis | 3(3-0) |
| MTH 643 | Differentiable Manifolds | 3(3-0) |
| MTH 732 | Functional Analysis | 3(3-0) |
| MTH 734 | Partial Differential Equations and Applications | 3(3-0) |
Core Electives II - Advanced Topics in Mathematics (3-9 hours)
Select from the following:
| MTH 725 | Topics in Algebra | 3-9(Spec) |
| MTH 730 | Topics in Applied Mathematics | 3-9(Spec) |
| MTH 737 | Topics in Analysis | 3-9(Spec) |
| MTH 744 | Topics in Geometry | 3-9(Spec) |
| MTH 778 | Topics in Combinatorics | 3-9(Spec) |
Area of Specialization Courses (6 hours)
| MTH 796 | Special Topics in Mathematics | 1-6(Spec) |
| MTH 797 | Independent Study | 1-9(Spec) |
Other Electives (3-6 hours)
All courses at 500 level or higher, and approved by the advisor.
Concentration in Collegiate Mathematics Education (36 hours)
Mathematics Education Core Courses (18 hours)
| EDU 614 | Advanced Educational Psychology | 3(3-0) |
| MTH 763 | Advanced Perspectives in Collegiate Mathematics Education | 3(3-0) |
| MTH 764 | Qualitative Research Methods in Mathematics Education | 3(3-0) |
| MTH 767 | Seminar in Collegiate Mathematics Education I | 3(3-0) |
| MTH 768 | Seminar in Collegiate Mathematics Education II | 3(3-0) |
| PSY 611 | Research Design | 3(3-0) |
Core Electives I (6 hours)
Select 6 hours from the following:
| MTH 520 | Optimization Theory | 3(3-0) |
| MTH 543 | Curves and Surfaces | 3(3-0) |
| MTH 578 | Combinatorics and Graph Theory | 3(3-0) |
| MTH 645 | Algebraic Topology | 3(3-0) |
| STA 580 | Applied Statistical Methods I | 3(3-0) |
| STA 584 | Mathematical Statistics I | 3(3-0) |
| STA 585 | Mathematical Statistics II | 3(3-0) |
| STA 590 | Applied Statistical Methods II | 3(3-0) |
| STA 684 | Theory of Statistical Inference | 3(3-0) |
Core Electives II (12 hours)
Select courses from either inside or outside of the department, in consultation with the student's advisor.
Qualifying Examination
In the Ph.D. qualifying examinations, students are expected to demonstrate a broad knowledge of the topics and be able to integrate concepts and explain them at an appropriate level. Qualifying Examinations will be offered in the following subjects, based on the material in the courses listed in the current Graduate Student Handbook of the Department of Mathematics.
- Algebra
- Analysis
- Mathematics Education
Each doctoral student must pass two examinations in two different subjects from those listed above.
- Students choosing the Concentration in Mathematics must pass examinations in algebra and analysis.
- Students choosing the Concentration in Collegiate Mathematics Education must pass examinations in mathematics education and at least one of algebra or analysis.
- If a student decides to switch to a different concentration after completing two qualifying exams, whether s/he will take additional qualifying exam(s) is determined by the Graduate Committee.
Full-time students who entered the Ph.D. program must pass at least one examination at the latest in the exam period just prior to the start of classes of their fifth semester and must have passed two qualifying exams at the latest in the exam period just prior to the start of classes of their eighth semester. Part-time students may request additional time from the department.
A maximum of three attempts in each subject are allowed. A third failure in one subject eliminates the student from the Ph.D. Program.
Internship (6 hours)
Students are required to take 6 hours of a teaching internship.
The teaching internship courses are at the 200 level or above offered in the Department of Mathematics. Students may teach a course lower than 200 level with the approval of the Graduate Committee. Prior to seeking a faculty teaching internship supervisor, the student is required to consult with his/her academic advisor.
The teaching internships must be in two different courses under the supervision of two different graduate faculty members in two different semesters in the Department of Mathematics. Courses with substantial overlap of contents cannot be used as two different internship courses.
The student must earn credits for internships by registering for MTH 766 for teaching internship.
| MTH 766 | Internship: College Teaching | 3-6(Spec) |
Dissertation (12 hours)
Upon successful completion of the qualifying examinations, the student will select a dissertation supervisor. A dissertation supervisor must be a graduate faculty member in the Mathematics Department. The student will form a dissertation committee of at least three graduate faculty members in consultation with the dissertation supervisor. This dissertation committee will be chaired by the supervisor(s) and must include at least two (2) other graduate faculty members. Two members of the dissertation committee must be from the Mathematics Department. A completed doctoral dissertation must be approved by the dissertation committee, and by the Office of Graduate Studies.
Students are required to register for 12 hours of MTH 898 (Dissertation). The dissertation must consist of original work and can combine scholarly, analytical, creative and expository skills. It could consist of research on a topic in mathematics, or research on a topic related to the teaching of collegiate mathematics. Before starting the dissertation work, the project to be undertaken must be approved by the dissertation committee, and by the Office of Graduate Studies.
Upon completion of coursework, qualifying examination, internship, and dissertation, the candidate for the Ph.D. degree must pass a final oral examination which is a dissertation defense in a colloquium format. The student's dissertation committee determines whether the student passes the oral examination.
The dissertation must be prepared according to the regulations prescribed in the Office of Graduate Studies most recent edition of the Preparation Guide to Doctoral Dissertations, Theses, Field Studies, and Plan B Papers and must be submitted to Dissertations Abstracts International.
Total: 75 semester hours