Section 5 The second year of graduate study
The goal for all secondyear students is to advance to candidacy for the PhD by the end of spring term. Largely this is accomplished through the assembling of an Advancement Committee and passing an individuallytailored Advancement Exam.
Pure and applied math students have many requirements in common, but as students enter their second year — which begins in the summer term — some of the procedures to be followed begin to specialize. We begin by describing specialized procedures followed by ones common to all secondyear students.
Subsection 5.1 Advancement procedures — Applied Math Specific
In addition to course, TA, pedagogical training requirements noted later in this section, the following describes the outline of expectations for students wishing to work in applied math.
Subsubsection 5.1.1 Summer Research/Internship and Advancement Committee
Students will participate in a supervised summer research project or internship which was proposed and approved the previous spring. See section 4.3.5 for the planning and approval details.
By the beginning of the summer, the student should have at least partially formed their Advancement Committee, typically the two people who approved their summer research proposal. It is the student's responsibility to keep the department member(s) informed of the progress of the ongoing research.
Very soon after the summer research project is completed, the student should find a third member to fill out their Advancement Committee. That person may be someone who could act as a good resource for extending the summer research project, or who might be a source of research questions moving forward. On the other hand, if the research experience was not in a direction which the student wanted to continue, the third member could be the person with whom the student will ultimately work.
Satisfactory completion of the research project must include a written evaluation by the research supervisor, together with a report by the student approximately five pages long summarizing the research which was conducted. The report should detail the research performed (motivations, experiments performed, conclusions drawn) as well as citations to the relevant literature. This report must be acceptable to the Advancement Committee and will form a part of the student's Advancement Examination. The student should think of and frame the report as the beginning of a paper at least potentially intended for publication.
Subsubsection 5.1.2 Fall — Advancement Examination
The student together with their Advancement Committee details what will constitute the Advancement Exam for the student. Once approved by the Advancement Committee, the plan for the Advancement Exam must be submitted by the student to the GPC for approval. The plan may be submitted as early as the student wishes, but must be submitted before the end of the fall quarter of the second year and at least two weeks in advance of the scheduled exam.
As part of the Advancement Examination, the student will give an oral presentation on the summer research project. The first part of this presentation may be public, perhaps as part of the applied math seminar, though the Advancement Committee's presence is required.
Immediately following the public presentation, the general audience will be dismissed and the student's Advancement Committee will continue to ask questions stemming from the research project as well as more fundamental questions in the student's area of focus. Some of the questions asked by the committee will focus on how the student conducted the research, the tools that were used, how the student understands those tools mathematically, what the assumptions of the problem are, and so on, drawing on the concepts learned in the classroom.
If the student's research will no longer be along the lines of summer research, the Advancement Committee may use this time to ask questions that are perhaps less relevant to the research internship, but more relevant to research interests moving forward.
At the conclusion of the exam, each faculty member on the Advancement Committee will fill out a copy of the Advancement Exam Form (reproduced in the Figure 5.1 below). The outcome (pass or fail) will be by consensus, hence the same on all forms. Individual forms may offer different comments and perspectives. Signatures will confirm agreed upon roles of advisor and secondary advisor, a requirement for advancement to candidacy.
The student must attempt the Advancement Exam before the end of fall quarter of the second year. If a student does not pass, the committee may decide to allow the student a second attempt, which must take place before the end of the winter term of the second year.
Subsection 5.2 Advancement procedures — Pure Math Specific
In addition to course, TA, pedagogical training requirements noted later in this section, the following describes the outline of expectations for students wishing to work in pure math. A large part of the process of advancing to candidacy in pure mathematics requires students to pass an individuallytailored Advancement Examination, supervised by a threeperson Advancement Committee. The committee needs to be assembled and approved by the GPC by the end of fall term.
Subsubsection 5.2.1 Advancement Committee
Each student will assemble an Advancement Committee to aid and assess their preparation for writing a dissertation, including making up for any deficits revealed by the preliminary exam.
The student identifies two faculty members whose research encompasses the student's desired research area(s) who will work with the student to determine the format, content, and requirements of the Advancement Examination. A third faculty member from a different (though perhaps allied) research area is added to complete the Advancement Committee.
While researchactive emeriti and postdoctoral fellows are eligible for Advancement Committees, tenured and tenuretrack faculty are preferred given their expected longterm status within the department.
Subsubsection 5.2.2 Advancement Examination
The student's Advancement Committee is responsible for administering an individuallytailored Advancement Examination. Once approved by the Advancement Committee, the plan for the Advancement Exam must be submitted by the student to the GPC for approval. The plan may be submitted as early as the student wishes, but must be submitted before the end of the fall quarter of the second year and at least two weeks in advance of the scheduled exam.
At the conclusion of the exam, each faculty member on the Advancement Committee will fill out a copy of the Advancement Exam Form (reproduced in the figure below). The outcome (pass or fail) will be by consensus, hence the same on all forms. Individual forms may offer different comments and perspectives. Signatures will confirm agreed upon roles of advisor and secondary advisor, a requirement for advancement to candidacy.
The student must attempt the Advancement Exam before the end of winter quarter of the second year. If a student does not pass, the committee may decide to allow the student a second attempt, which must take place before the end of the spring term of the second year.
Subsubsection 5.2.3 Practical considerations
The student starts to assemble their Advancement Committee with a choice of two potential advisors. This process should probably begin by the summer, but students are encouraged to build relationships with faculty working in areas of potential interest throughout their first year. If the student does not yet have a idea of with whom they might want to work, the summer seminar, Math 117, may provide some insight into the research interests of faculty the student has not yet met. In any case, this is another prime opportunity to talk to your firstyear advisor who can help sort out your interests and options, and help begin to formulate plans.
If plans change and a new potential advisor is still one of the other members of the chosen Advancement Committee, no formal changes are needed as the change will be registered when faculty sign onto their roles in the Advancement Examination Form. If the new potential advisor is not currently a member of the committee, a new plan will be called for, and submitted to the GPC for approval.
 Other minor revisions to the plan for the Advancement Exam (e.g., changes to the syllabus that do not substantially change the content) do not need further GPC approval. The goal is to have a reasonably faithful written record of the student's plan for advancement. In case of doubt, just ask the GPC.
Subsection 5.3 Advancement Procedures — All students
Subsubsection 5.3.1 Courses
All fulltime students must enroll in three courses each term.

For secondyear students, there is a classroom course requirement of four courses. This means the other courses can (if desired) be chosen from among reading courses (127), independent reading (137) or independent project (148) as appropriate.
If you are taking for only one reading course in a term, register for Math 127. If two of your courses are reading courses, register for Math 127 and 137. Note: the courses (127, 137, 148) require a faculty supervisor and syllabus; grades are reported by the faculty supervisor.
Remember that students following an applied track need to take 106 and 116 for a second time since the content alternates between even and odd years. Similarly the content of Math 111, 113, and 114 typically alternates and will add considerably to the student's breadth. Students following the pure track need to take the summer seminar, Math 117.
Finally, in the term a student takes DCAL's Teaching Science Seminar, they should register for Math 147, which will designate that they have completed the theory end of the teaching seminar. Math 147 does not count as one of the four required classroom courses.
Subsubsection 5.3.2 TA and grading responsibilities
These responsibilities are exactly the same as in the first year, remembering to register for Math 107 (tutoring) in the second term in which you TA. Second year students will also serve as natural mentors to the firstyear graduate TAs.
Please also see Reference Sheet for TAs which includes a great deal of information about resources and regulations.
Subsubsection 5.3.3 Pedagogical Training and Requirements
Students must successfully complete both theoretical and practical components of pedagogical training as part of their graduation requirements and to be eligible to teach their own course in the later years of their time at Dartmouth.
Pedagogical Training — theory
For the theoretical portion, students will attend one of the offerings of Dartmouth’s Center for the Advancement of Learning’s (DCAL) Future Faculty Teaching Seminar, offered at least twice a year. This seminar covers the basic pedagogical literature with an emphasis on application within STEM fields and incorporates brief but rigorous opportunities for lesson design, practice, and feedback. Students should register for Math 147 in a term in which they take the seminar.
Pedagogical Training — Math specific training
To anchor these ideas within the mathematical curriculum, students will participate in a practicum experience mentored by a regular faculty member. Students should register for Math 148 in the term in which they do this training, and it may take several forms. For example, our traditional summer math camp will run whenever faculty are available to mentor and supervise. Other instances fall under the umbrella of a supervised teaching assistant assignment, which has three components.
 First, the student needs to autonomously design (some) curricular elements.
 Second, the student needs to teach these elements to a subset of the students enrolled in the course, typically during xhours.

Third, the student must receive feedback from a faculty member and respond to it via revision of the material.
Resources: We note that the department already has in place teaching mentors, which have been used in varying capacity over the years. The teaching mentors should coordinate with the student and course instructor for establishing expectations and goals for the practicum, and should regularly communicate with the student during the term.
Logistics: Students should discuss with their advisor how they might go about fulfilling this requirement. Student teaching assistant assignments must coordinate with course scheduling, so some flexibility is needed. To ensure objective feedback, a representative from DCAL will be asked to observe graduate student taught classes. A discussion between the graduate student, course instructor, student teaching mentor (if appropriate), and DCAL representative will follow.
Benefits: The plan provides flexibility while also ensuring that our PhD students have plenty of opportunities to interface with students. Moreover, the plan plays to our students' strengths and will allow them to stand out on job applications. The students may also provide a great resource for undergraduates in courses that have not traditionally had TAs, for example, by teaching MATLAB in some of the applied mathematics courses, or running problem solving sessions in some midlevel courses.
Subsubsection 5.3.4 Certificate in Mathematical Pedagogy
There is a higher level of pedagogical training — beyond the training required by the graduate program — for those who aspire to secure positions with a significant teaching component.
To facilitate this, we have created a certificate program within the Guarini School of Graduate and Advanced Studies. Upon completion of the requirements, a student receives a certification and notation on their transcript. Students with an interest in this certification should discuss the matter with their advisor.
The requirements for the certificate are included in the form below. To download a copy, use the download link.
A typical path through the course of training for a student interested in the certification might be:
 Year 1: Two terms serving as a Teaching Assistant, typically in calculus courses and (if possible) paired with a more senior student. Attendance of the TA Training Seminar in the Fall term.
 Year 2: Two terms serving as a Teaching Assistant, one of which will incorporate a more intensive collaboration with the instructional team. Attendance of the TA Training Seminar in the Fall term. Students will attend the DCAL course for one offering.
 Year 3: The student is eligible to teach a section of their own course, likely under the supervision of the faculty member who is the course supervisor.
 Year 4: Depending on the student's performance in their first independent teaching experience, the student may have a similar experience in year 4 or be trusted with more autonomy (e.g. with a standalone course).
 Year 5: Depending on the student's performance in their first two independent teaching experiences, the student may have a similar experience in year 5 or be trusted with more autonomy (e.g., with a standalone course).
 Interspersed throughout: Students will pursue the other components of the certification.