ISU Department of Mathematics. Graduate Examination Policies and Procedures

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1 ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated knowledge of the techniques, terminology, and primary theorems, including being able to provide examples and counterexamples, apply theorems, and illustrate the strength of hypotheses? 2. Communication Has the student demonstrated the ability to communicate ideas in oral and/or written mathematics? Clean computations and clearly constructed arguments and proofs are essential. 3. Logic and Synthesis Has the student demonstrated sound mathematical logic in proofs and computations? Has the student demonstrated an ability to combine ideas? 4. Creative thinking Has the student demonstrated an ability to understand and interpret a problem correctly, try various approaches, persevere, and successfully solve the problem, especially by thinking creatively or using nontraditional approaches? Complete solutions to fewer problems are preferable to incomplete solutions to more problems. For example, if a written exam requires solution of ten problems, complete and well-written solutions to seven of those problems will result in a more favorable exam score than partial or poorly written solutions to all ten. Suggestions for Studying Students are encouraged to consult prior written exams that are available in the Mathematics Office, to practice communicating mathematics in an oral format prior to the oral exam, and to discuss study strategies with faculty members and more senior graduate students. If an attempt is unsuccessful, the student should seek specific guidance from committee members about how to improve performance in any future attempt. Master of Science Written Examinations Purpose. The written examination for the Master of Science degree in mathematics is intended to evaluate a student s competence in mathematics. Students seeking the M.S. degree must demonstrate competence in a single area of mathematics at the graduate level. Content and Length. Master of Science students must pass a written examination over material that is normally covered in one of the Department s 600-level course sequences ( , , , , , , , , , , ). The subject area is chosen by the student, in consultation with the Graduate Committee. The examination shall be designed so that a well-prepared student can complete the examination in no more than four hours. Committee Formation. The examination committee shall consist of three graduate faculty members in the Department. The student (or students, in the case that two or more students wish to take the same subject-area examination at the same time) may request a particular faculty member

2 to serve as chair or may request that the Graduate Committee assign a chair. Students are encouraged to select as the committee chair the faculty member from whom the student took the 600-level course sequence over which he/she will be examined. In either case, the chair of the committee shall have particular expertise in the chosen subject area and must be approved by the Graduate Committee. The chair shall appoint the remaining two committee members, who must be approved by the Graduate Committee. Committee Responsibilities. The examination committee, in consultation with the student s instructor in the course, shall provide the student with a domain of material (not necessarily limited to what was covered in the course) and at least two reference books for the material. The committee is responsible for writing the exam, administering the exam, and evaluating the student s performance on the exam. At a minimum, the examination committee must determine whether the student passed or failed the examination and communicate this result in writing to the Graduate Timing. A student may attempt the examination at any time after completing the appropriate course sequence, although the Graduate Committee recommends taking the exam as early as possible in the student s program. For students who are able to complete a 600-level sequence in their first year in the program, we recommend that the student take the exam during the academic year of the second year of the program. For students who do not take a 600-level sequence until their second year, we recommend that the student take the exam during the summer following their second year but early enough in the summer to meet deadlines for August graduation. Because the student is required to take an additional oral examination to complete the degree requirements and this examination must be taken at least two weeks before the end of the summer session, this necessitates completion of the written examination in the first half of July. The exact day and time of the exam will be determined by the examination committee, in consultation with the student(s) involved. Repeating the Exam. If a student fails the examination on the first attempt, the student can retake the examination at most two more times. Master of Science Oral Examination (Non-thesis Option) Purpose. The Master s oral examination is the final examination that the Graduate Catalog refers to in its general discussion of examinations. It is intended to evaluate the student s breadth of working knowledge of graduate level mathematics. The final step before graduation, it can take place only after the student has passed the Master s written exam and only if the student will have completed all coursework by the end of the semester in which the M.S. oral exam occurs. Content and Length. The Master s oral examination covers all the courses in the student s program of study, with emphasis on the 600-level sequences taken, and should normally last no more than two and a half hours. Committee Formation. The examining committee, which reports directly to the Graduate School, will consist of two Mathematics faculty members and a Graduate Faculty Representative (GFR)

3 from another department. To form a committee, the student should select, in consultation with the Graduate Director, a graduate advisor and a second member, both from the Mathematics Department, who are willing to serve. The student may propose a GFR from outside the department who has agreed to serve, but the Graduate School has the ultimate authority to assign that member. The committee becomes official when the Graduate School accepts the student s final program of study a step that must occur the semester before the student intends to graduate. Program-of-study forms are available in the department office. On the form, the student lists the courses that he or she wants counted toward the degree and the names of the Mathematics Department committee members. The student may also suggest a GFR. With signatures of the student, graduate advisor, and the department chair, the form then goes to the Graduate School. The chair s signature indicates departmental approval of the program of study and the exam committee. Committee Responsibilities. The examination committee is responsible for ensuring the examination covers appropriate breadth and depth and providing guidance to the student on how to prepare. The committee is responsible for working together to administer the exam and evaluate the student s performance on the exam. At a minimum, the examination committee together with the GFR must determine whether the student passed or failed the examination. This result is conveyed to the Graduate School by the GFR and should also be communicated to the Graduate Timing. The student and examining committee will work out a mutually agreeable time for the exam and inform the Graduate School, who will then schedule it. Repeating the Exam. If the examination is failed, Graduate School policies allow just one further attempt and require that it occur within three semesters of the first one. Doctor of Arts Written Examinations Purpose. The written examinations for the Doctor of Arts degree in mathematics evaluate competence in the core areas of algebra and analysis at the undergraduate level in preparation for teaching related material. Both must be passed before students seeking the D.A. degree may pursue Math 700, Teaching Internship. Content and Length. Doctor of Arts students must pass an examination in undergraduate mathematics. This examination shall consist of two separate components, one in analysis and one in algebra. The analysis examination covers material normally taught in Math , 326, 327, and 360, and some material from Math 423. The algebra examination covers material normally taught in Math 240 and , and some material from Math 406. Students are also expected to be competent in and able to apply basic material on logical structures, sets, proofs, and relations covered in Math 287. The examination shall be designed so that a well-prepared student can complete each component in no more than four hours. The two components may be administered on different days or on the same day, depending on the preferences of the students and faculty involved, but the former is recommended and has occurred most often in practice.

4 Committee Formation. The examination committee shall consist of three graduate faculty members in the Department. The chair of the committee shall be a member of the Graduate Committee. The chair shall appoint the two remaining committee members, who must be approved by the Graduate Committee. Committee Responsibilities. The examination committee is responsible for writing the exam, administering the exam, and evaluating the student s performance on the exam. Evaluation is to be done separately for each component. At a minimum, the examination committee must determine whether the student passed or failed each component and communicate these results to the Graduate Timing. The D.A. examination will be offered twice each academic year: during the first two weeks of fall semester and during the first two weeks of spring semester. The exact day(s) and time(s) of the exam will be determined by the Graduate Committee, in consultation with the students. The examination will not be offered at any other time during the year. All D.A. students must attempt the examination at the beginning of their second full year of residence in the D.A. program. Repeating the Exam. If a student fails a component of the examination, the student can retake that component at most two more times. The student is not required to retake both components of the examination if only one was failed. Detailed Content of the Exam. The D.A. Examination in Undergraduate Mathematics is intended to evaluate a student s competence in the core areas of undergraduate mathematics, defined to be the following specific subjects, in addition to the general material on sets, logic, proofs, and relations covered in Math 287. A. Analysis 1. Calculus (Math 170, 175, 275, 327) This includes all areas of single and multivariable calculus from limits through Stokes theorem, including the Implicit Function Theorem. Questions can involve computations (e.g., implicit differentiation, partial fraction decomposition, line and surface integrals), explanations (e.g., why and how definitions and theorems are applied in a given situation), and proofs (e.g., limits, convergence of series). 2. Differential Equations (Math 360) This subject includes methods for solving general first-order equations, higher-order linear equations, and linear systems. Also included are the geometry of solutions (direction fields and phase portraits) and the existence and uniqueness of solutions. 3. Elementary Real Analysis (Math 326 and some material from Math 423) The real analysis subject includes the theorems of calculus (limits, convergence, continuity, mean value theorems, Taylor s theorem), basic concepts of topology (compactness, completeness, connectedness, metric spaces), properties of real numbers (sup, inf), and ideas of convergence of sequences (uniform, pointwise) and of series (absolute, conditional). B. Algebra

5 1. Linear Algebra (Math 240 and some material from Math 406) This is the subject of vector spaces, matrices, and linear transformations. Questions can involve computations (e.g., solutions to linear systems, bases, eigenvectors/eigenvalues, orthonormalization, determinants), terminology (e.g., dimension, rank, null space, linear independence, inner product, adjoint matrix, positive definiteness), and proofs (involving vector spaces, subspaces, linear transformations, etc.) 2. Abstract Algebra (Math 407, 408) The subject of abstract algebra covers groups, rings, and fields. Topics involving groups include the definition, examples, symmetry groups, subgroups, homomorphisms, product and quotient groups, cosets, the counting formula and the group isomorphism theorems. Topics involving rings include the definition, examples, polynomial rings, homomorphisms, ideals, quotient rings, maximal and prime ideals, and integral domains. Topics involving fields include the definition, examples, field extensions (algebraic and transcendental), and the degree of an extension. Advanced results and material from graduate courses are not covered on the D.A. Examination in Undergraduate Mathematics. Doctor of Arts Oral Examination Purpose. The D.A. oral examination is intended to evaluate a student s competence in four core areas of graduate level mathematics in order to begin thesis work. Content and Length. The D.A. oral examination covers the material in the required sequences Math , , , and and should normally last no more than two and a half hours. Committee Formation. The committee conducting the examination will consist of at least three Mathematics faculty members. If the student has already formed a thesis committee by submitting a program of study to the Graduate School, then departmental members from that committee could conduct the exam. (If a GFR has been named, he or she is not required to attend.) An exam committee not formed from a thesis committee must be approved by the Graduate Committee. Committee Responsibilities. The examination committee is responsible for ensuring the examination covers appropriate breadth and depth and providing guidance to the student on how to prepare. The committee is responsible for working together to administer the exam and evaluate the student s performance on the exam. The committee may decide that the student has passed the exam, failed it, or passed just some of the four parts. The committee will report its decision to the graduate committee and to the Graduate School. Timing. Students should take the oral exam as soon as possible after completing the four core sequences and before they have proceeded far in thesis work. Scheduling is by mutual agreement between the student and committee members. Repeating the Exam. If a student fails the examination on the first attempt, the student can retake the examination at most two more times.