The course requirements are 36 units of graduate credit in the major and 12 units in a supporting minor, which may be declared in Mathematics, although outside minors are encouraged. Students will normally either take the first year graduate core courses in Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Geometry–Topology (MATH 534A-B), or otherwise learn this material by the end of their first year of Ph.D. studies for the Qualifying Examinations.
Two year-long Mathematics course sequences that are not dual-numbered and are not part of the required core of algebra, real analysis, and geometry-topology are required. These two sequences must be in two of the three general areas: algebra and number theory, analysis and geometry, mathematical physics and applied mathematics. For most students one of the sequences will be Complex Analysis (MATH 520A-B). At least six units outside of Mathematics are also required.
Each student must present a coherent collection of courses in which the work outside of Mathematics is related to part of the studies in Mathematics. There are many such possibilities, including: algebra, and computer science or discrete methods in operations research; probability, and statistics or reliability/quality control; numerical mathematics, and computer science or computational science; mathematical foundations and history, and education; analysis, and physics or optics; etc.
The qualifying examination consists of three written parts, each with two segments, covering undergraduate and graduate level material. The undergraduate level material is linear algebra, rigorous advanced calculus, and complex analysis. The graduate material is at the level covered in the first year graduate core courses: Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Topology–Geometry (MATH 534A-B).
The first part is on linear algebra and algebra. The second part covers advanced calculus and real analysis. The third part treats complex analysis and geometry-topology. Copies of recent examinations are available in the Graduate Office.
Qualifying examinations are offered twice yearly, in August and in January, shortly before the beginning of Fall and Spring semesters. Students with prior preparation may attempt the examinations upon entrance to the program, or after one semester. Ideally a student will complete the qualifying examinations by the August following entrance to the PhD program. Students who do not complete the three examinations within five semesters and with a cumulative passing grade will not be continued in the PhD program unless the Graduate Committee, upon review of the case, finds extenuating reasons to justify such continuation.
The Written Comprehensive Examination consists of an outline of material in an area of concentration selected by the student with the guidance and approval of the student's committee. This is an area in which the student could write a dissertation.
The Oral Comprehensive Examination is primarily, but not exclusively, on material in the area of concentration. The examination covers background material for the general area together with advanced references in a more specific sub-specialty.
The Written and Oral Comprehensive Examinations together constitute the Comprehensive Examination. Students are encouraged to take the Comprehensive Examination by the end of their second year in the PhD program – in all events it must be finished within two years of successful completion of the Qualifying Examinations.
The language requirement is fulfilled by passing examinations in two of the following: mathematical Russian, French, or German; or a computer programming examination (usually in C).
The Ph.D. dissertation must be research of publishable quality. It is evaluated by an internal committee, and by one specialist in the area of the dissertation who is not a faculty member at the University of Arizona. Eighteen units of credit are awarded for an accepted dissertation.
The final oral examination is a presentation and defense of the student's dissertation; it is open to the public.
The course requirements are 36 units of graduate credit in the major and 12 graduate units in a minor in Education (or related field).
Students will normally either take the first year graduate core courses in Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Topology–Geometry (MATH 534A-B), or otherwise learn this material by the end of their first year of Ph.D. studies for the Qualifying Examinations. The remaining 18 units will be chosen in consultation with an advisor. These remaining units will include one year-long Mathematics course sequence that is not dual-numbered and is not part of the required core of algebra, real analysis, and topology-geometry. Some of the units will include relevant courses in Mathematics Education research (to be discussed with an advisor).
The 12 units in Education (or related) will be chosen in consultation with an advisor to ensure a coherent program of study. The courses will primarily be in Education. Courses in psychology, anthropology, sociology, women's studies, etc., may also be appropriate, depending on the student's research focus. Some suggested Education courses are listed below. EDUC 500, 501, 600, 601, 602; TTE 521, 524, 532, 545, 621, 640. Two courses in research design and methods (e.g., EDUC 600, 601, 602, or appropriate research methods courses in other fields such as sociology, anthropology, agriculture, ...) are required.
Two or more years of pre-college teaching experience are required. Students can fulfill this requirement through 9 units of practicum in local schools. Such students will take 3 units per semester to complete one practicum at the elementary school level, one at the middle school level, and one at the high school level. (Note: these 9 units do not apply toward the required 36 units of mathematics nor the 12 minor units.)
The same stipulations as given for the Ph.D. program in Mathematics.
Same as for the Ph.D. program in Mathematics.
Similar guidelines to those for the Ph.D. program in Mathematics, but the area of concentration will be in Mathematics Education.
The language requirement is fulfilled by passing examinations in two of the following: American Sign Language, French, German, Spanish, or Russian. Students may substitute a computer programming examination for one of the languages.
Same guidelines as for the Ph.D. program in Mathematics. The dissertation will be in Mathematics Education.
Same as for the Ph.D. program in Mathematics.