Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
This is a temporary static version of the TF wiki.


   Welcome to the math department's teaching wiki! Much of the original
   information on this site was based on the TF handbook created by the
   East Asian Languages and Civilizations Department. Please feel free to
   add any information that might be useful to other members of the

Types of Teaching

Calculus Teaching Fellows

   Teaching fellows (also called section leaders) are responsible for
   teaching calculus (Math Xa, Xb, 1a, 1b, 21a, or 21b) to a class of
   about 25 undergraduates. They meet with their class 3 hours a week.
   They have a course assistant (usually an advanced undergraduate) to
   grade homework and to take a weekly problem session. Usually there are
   several classes following the same syllabus and with common exams.
   There is a course head (a member of our teaching staff) who will
   coordinate the various classes following the same syllabus and who is
   available to advise Teaching Fellows.

Calculus Coaches

   A calculus coach isn't responsible for preparing daily lessons but is
   responsible for looking after student learning, working intensively
   with students who are having trouble, and working at the Math
   Question Center. A coach and a section leader are paid the same salary.
   Both attend weekly course meetings, and both help write and grade
   exams. If you would like to learn more about the experience, talk with
   a current or former coach: Stefan Hornet, Jeechul Woo, Ruifang Song,
   Lin Han, or Chung-Jun Tsai.


   The math department runs a core course, QR28: The Magic of Numbers.
   (QR stands for Quantitative Reasoning.) Graduate students who have
   taught in this course include Cameron Freer, Rina Anno, Dawei Chen, and
   David Roe.

   One graduate student who has taught both calculus and QR28 says:

     "Teaching calculus requires more responsibility, since we are
     teaching the material in the first place to the students. While for
     QR28, we only have to lead weekly sections and grade homework, which
     is easy and less time consuming. In addition, students would need
     our help much more often in a calculus course than in QR28. So as a
     tradeoff, calculus takes us more time but has more fun to teach!"

Graduate Course Assistants

   Graduate course assistants (or GCAs) are course assistants for more
   advanced courses. A GCA is typically responsible for grading homework,
   holding a weekly problem session, and holding regular office hours.


   A tutorial is an undergraduate math seminar, for 6 to 10 students,
   usually at the junior undergraduate level, covering a topic of interest
   (of interest to both you and at least 6 to 10 undergraduates!) The
   tutorial leader chooses a topic and gives the beginning lectures, while
   coaching the undergraduates to give their own lectures in the final few
   weeks of the tutorial.

   Tutorials happen during the academic year, as well as during the
   summer. An academic-year tutorial is equivalent to teaching a section
   of calculus, both in terms of dollars and for those who are required to
   teach for their support. Summer tutorial leaders are paid the same, but
   a summer tutorial cannot substitute for term-time teaching.

Math Question Center

   The Math Question Center, or MQC, is a place for students in Math
   X, 1, and 21 to get help on their homework. The MQC is staffed mostly
   by CAs from Math X, 1, and 21, but there are a few positions for
   graduate students to work as MQC CAs. (Generally, positions are
   available only in the spring.) The responsibilities for MQC CAs are:

     * Working in the MQC 2 nights a week (2 hours each night).
     * Doing some administrative duties in the MQC, such as reporting how
       many students use the MQC.
     * Some additional work to be determined. (For instance, in Spring
       2007, MQC CAs also helped to grade exams in some courses.)

   The compensation for being an MQC CA is roughly the same as that for
   being a GCA.


  Teaching requirements and compensation for graduate students

Outside the Department

     * Summer school - There are jobs for graders and course assistants
       during the summer. Graduate students usually receive an email about
       these opportunities in the spring.

     * Extension school - The Extension School offers several math
       courses, including some designed for K-12 math teachers. Graduate
       students in the past have served as course assistants for some of
       these courses.


   A tutorial is an undergraduate math seminar, for 6 to 10 students,
   usually at the junior undergraduate level, covering a topic of interest
   (of interest to both you and at least 6 to 10 undergraduates!) This
   tutorial program gives you a chance to come up with a topic that you've
   always wanted to teach, but haven't had a chance to yet. The math
   department usually offers four tutorial courses (Math 99r) each
   academic year, two in the Fall semester and two in the Spring. In
   addition, when funding is available, we offer summer tutorials, which
   have been very popular. Tutorials provide a way for you to obtain
   teaching experience in a setting different from a calculus class, and
   are an opportunity for you to work closely with a small group of
   motivated undergraduates. It is a rewarding experience for those

What is a tutorial?

   The idea is for a tutorial leader (usually a grad student) to give the
   beginning lectures on a chosen topic, and at the same time to coach the
   undergraduates to prepare their own lectures for the final few weeks.
   The tutorial leader puts together the program of study, determines the
   reading list, assigns the topics and helps the students prepare. The
   students then write an exposition of their assigned topic, as well as
   lecture on it; they can submit their written paper to fulfill one of
   our undergraduate requirements. The upper limit for the enrollment of a
   tutorial is around eight.

   All math concentrators are encouraged to take a tutorial during their
   undergraduate career, so the level of the tutorials should not be
   pitched so as to exclude all but the best-prepared students. At the
   same time, the hope is to interest them in some mathematics which goes
   beyond their regular courses.

Academic year tutorial vs. summer tutorial

   There are some administrative differences between academic year and
   summer tutorials.

   Academic-year tutorials typically meet once per week for several hours,
   usually in the evening (along with an office hour during the week).
   Summer tutorials typically meet three times per week for six weeks
   starting from the beginning of July and ending in mid August. The
   meetings are usually held in the evenings so that the students can have
   real jobs to pay their bills. The precise meeting times and starting
   date can be arranged to the mutual benefit of you and your students.

   The undergraduate students get a small stipend for the summer tutorial
   as an inducement for attending, but there is no official Harvard course
   credit given for them. The academic term tutorials are counted for
   Harvard course credit (but there is no stipend for the undergraduates).


   An academic-year tutorial is equivalent to teaching a section of
   calculus, both in terms of dollars and for those who are required to
   teach for their support. The summer tutorial leaders are paid the same;
   but a summer tutorial cannot substitute for term-time teaching.

Applying to teach a tutorial

   Each spring, you should receive an email inviting you to submit a
   proposal for a tutorial. Generally, the proposal consists of a one to
   two page description of the tutorial topic with a rough syllabus,
   proposed reading material, and a list of a few final paper topics.

   As you are thinking about your proposal, keep in mind that a
   not-too-advanced topic is a good thing. Tutorials are advised to all
   math concentrators, and the department encourages all concentrators to
   take at least one tutorial during their time at Harvard (usually as a
   sophomore or junior). Remember that only a small proportion of math
   concentrators come from the Math 55 stream; topics need to be
   accessible to a larger group, if possible. (See Undergraduate math
   courses to find out what different paths math concentrators may take.)
   Successful recent topics include "Random graphs" and "Ramsey theory".

Recent tutorials

   If you're interested in teaching a tutorial, you are highly encouraged
   to talk with graduate students who have taught tutorials before. A list
   of tutorials taught in the past is at
   this page.


   Apprenticeships are designed to give graduate students who may not have
   taught very much before a chance to gain some experience and training
   before starting to teach on their own. Each participant is paired with
   a mentor who is presently leading a section of Math X, 1, or 21. The
   apprentice first attends a few of the mentor's classes, then conducts a
   practice lecture, and finally plans, delivers, and reviews three
   lectures of his or her own. All graduate students must successfully
   complete the program before they can be assigned a section to teach

   The following outline should serve as a guide to what happens at each
   stage of the Apprentice Program.The goal throughout is to help you get
   off to a strong start with your own section. Please note that it is
   your responsibility to schedule your visits, practice sessions,
   lectures, taping, and viewing.

   At the beginning of apprenticeship, you must work for one hour at the
   Math Question Center, or MQC. Please let the MQC coordinator know
   what date you're planning on attending. The MQC is staffed by Course
   Assistants and Graduate students and will provide you with one on one
   experience working with students while also letting you get a feel for
   what level the students are at.

Visit Your Mentor's Classes

   While you are in class:

     * Notice the level at which the course is taught. How rigorous is the
       presentation? What can you assume about the students' backgrounds?
     * Pay attention to the number and type of examples done. How
       difficult are they compared to homework and exam problems? Pay
       attention to the transitions from one topic to the next.
     * Notice the pace at which topics are covered. How does the mentor
       keep moving without leaving the class behind?
     * Think about motivation. Does the presentation make the material
       seem interesting and purposeful? How does the mentor connect new
       ideas to previously learned material?
     * Observe the blackboard work. Is it well organized and easy to
       follow? What do you think the students' notes look like?
     * How does your mentor promote interaction? Does he/she make eye
       contact with the students? What kind of and how many questions does
       the mentor ask?
     * Listen carefully to the students' questions. What do they tell you
       about the students' perspective? Do the students look like they
       feel free to speak up?

   After class:

     * Discuss your observations with your mentor and with other
     * Ask your mentor about why specific things were done as they were.
     * Although you may or may not want to conduct your class in the same
       way your mentor does, you should use him or her as a sounding board
       for your own ideas.

Conduct a Problem Session

   Before the problem session:

     * Discuss with your mentor and/or his course assistant what kinds of
       problems to do. Look over the recent homework assignments.
     * Plan a very short review lecture on a specific topic. You may
       present this for the first 5 or 10 minutes.
     * Choose and prepare problems which illustrate important techniques.

   During the problem session:

     * Do everything you can to involve the students.
     * One strategy: put a problem on the board; ask for ideas; write down
       only what they say; try not to make suggestions yourself until the
       class is completely stumped.
     * Another strategy: put problems on the board; have them work out
       solutions on paper; walk around; offer encouragement and
       suggestions; ask students to help one another.
     * Remember that the point is for them to do the problems, not you.

Give a Practice Version of your Lecture to your Mentor and Student Volunteers.

   Before the lecture:

     * Attend the class meeting before your lecture. Begin with a
       transition from where your mentor left off.
     * Prepare detailed lecture notes. Go over them with your mentor in
       advance checking the level, pace, motivation, etc.
     * Be sure to choose lots of examples the students will find helpful
       and interesting. These may not be the same ones you find
     * Have a few extra examples ready in case they are needed.
     * Plan your blackboard layout.
     * Think about how to get the students involved. How and where will
       you ask questions?

Give Three Lectures, the Second of Which is Videotaped.

   During each lecture:

     * Try to relax; a deep breath on occasion can help. Ignore the video
       camera as much as possible while taping.
     * Turn to face the class whenever you can. Make eye contact with
       students, including the front, back, and in-between rows.
     * Are you speaking loudly enough so that everyone can hear? Project
       your voice without shouting.
     * Can everyone see? Is your writing big enough? Don't erase or stand
       in front of what you just wrote.
     * Take all questions seriously. Be careful not to talk down to or
       belittle the class.
     * Are you checking in with your audience? Are you really being
     * Keep your mind on the students. Concentrate on communicating with
     * Try to look at the mathematics from their point of view.

   Note: At the end of your lectures, it can be helpful to hand out and
   collect brief student questionnaires. You must do this after your
   second lecture. You may either use a form provided by the math
   department or create your own asking for student feedback. Whichever
   you choose, it is your responsibility to make copies and administer the

   After each lecture:

     * Talk with your mentor and, if possible, some students. Review the
       student questionnaires.
     * Decide what needs improvement and how to do it.

View the Videotape

   Between your second and third lectures:

     * Watch and discuss your tape with one of the Preceptors. It is
       usually a good idea to invite your mentor as well.
     * Have the completed student feedback forms with you to go over at
       the tape viewing. Copies of these forms should be kept to help with
       placement, etc.
     * Based on the tape, figure out what your strengths and weaknesses
       are (everyone has both). Develop a strategy for improving your
       third lecture.

Apprentice Review

   Before the end of the semester, mentors and apprentices should discuss
   the next steps with one another and with the Committee on Instructional
   Quality. In consultation with the Committee, the Department Chairman
   and the Director of Graduate Studies ultimately determine teaching
   assignments. Apprentices who receive teaching fellowships are expected
   to have:

     * shown responsibility,
     * prepared their classes carefully, and
     * demonstrated an ability to interact appropriately with students.

   Only the best teachers will be offered sections to teach. People whose
   support includes a teaching fellowship but who have not demonstrated
   readiness to teach a section of their own are typically assigned course
   assistantships, grading jobs, and other training activities to help
   them prepare to reapprentice during a subsequent semester.

Students with disabilities

   As a teacher, you have a legal and ethical responsibility to ensure
   that any student with disabilities in your classroom has equal access
   to learning. To learn more, please see the following resources

     * The section "Students With Disabilities" from the GSAS TF
       GSAS TF Handbook
     * The Accessible Education Office

Students with personal difficulties

   If you have a student who is experiencing personal difficulties, please
   do the following:

    1. If it seems appropriate, refer the student to the Bureau of
       Study Counsel, which offers counseling on personal and academic
       issues. (See for more
       information on referring students.)
    2. Let your course head know about the issue. If the student has a
       personal or family emergency, the course head may, for example,
       allow the student to skip some homework assignments. In addition,
       the course head may want to talk with the student's resident dean.
    3. If you feel the issue is an emergency and you can't find your
       course head, call the Bureau of Study Counsel for advice. You
       may also want to contact the student's resident dean directly. If
       the student is a freshman, call the Freshman Dean. If the
       student is an upperclassman, contact his/her resident dean. (You
       can find a list of resident deans and their phone numbers on pg. 61
       of the GSAS TF Handbook.

Also See

     * "What Can I Do If a Student Is In Difficulty?", starting on pg. 50
       of the GSAS TF Handbook

Logistical Information for Calculus Section Leaders


   First day of sectioned classes (Xa, Xb, 1a, 1b, 21a, 21b): The Monday
   after most classes start.

   Syllabus and Course-Specific Orientation: Provided to you by your
   Course Head. Your Course Head will schedule an orientation meeting.

   Texts: Any standard text will be provided by Nancy Miller, the math
   librarian (Rm. 337; phone: 5-2147; email: nancy). She is there in the
   mornings. For information about computer programs or calculators - see
   the Course Head.

   How do I find out when and where I teach and who is in my class? This
   will be determined the Thursday and Friday after classes start. There
   is no preregistration so we don't have information until the very last
   minute. Your Course Head will have that information first.

   Mailboxes: All CA mailboxes are outside SC 310.

What does a Course Assistant (CA) do?

   Your Course Assistant will

     * Attend class

     * Hold weekly problem sessions

     * Grade homework assignments, keep a record of the grades and provide
       a copy of the scores to the section leader each week.

     * Work at the Math Question Center one night every other week.

     * The Course Head may have your CA write up (and make web-ready)
       solutions to homework problems, but there is normally a designated
       "Head CA" to handle this chore.

     * At the Course Head's discretion the CA may be asked to grade weekly
       quizzes. If so, you would probably be asked to prepare the quizzes.

     * Give you a weekly report indicating problems students are having,
       who is not turning in homework, and what is happening at problem

   Since your Course Assistant will be grading homework and meeting weekly
   with the students at the review sessions, he or she should be a helpful
   source of feedback (e.g., in pointing out the areas in which students
   are experiencing difficulty, and in coming up with ways for each of you
   to assist the students who are having problems with the course). Talk
   regularly with your Course Assistant and ask to be provided with
   homework grade reports frequently so you stay informed as to how your
   students are doing.

   Your Course Assistant ought to be giving you weekly feedback forms (or
   equivalent communication). Please ask for this if it is not provided
   and communication is not optimal.

   Your CA will be assigned the weekend before calculus classes start. The
   CA coordinator(s) will send you an email indicating who your CA
   will be. You should try to contact your CA over the weekend so that you
   can meet briefly before the first class.

   If you are having problems with your CA, speak with the CA and try to
   solve the problem. If this does not work, contact the CA
   coordinator(s) as soon as possible.

Problem Sessions

   Your CA's weekly problem sessions are an important part of the course
   as they allow the students to work with the CA on problem solving and
   problem areas in more detail than allowed by the limited amount of
   class time.

   The CA is responsible for scheduling a time for the problem sessions.
   This should be done during the first or second class period. Give
   her/him a few minutes with the students to figure out a convenient time
   or give out forms on which the students give you information about
   their schedules. Your CA will then be responsible for getting a room
   allocated for the problem sessions.

   Students who are unable to attend the problem sessions for your section
   can feel free to attend the problem sessions for one of the other
   sections. (Since all of the sections will be following the syllabus at
   about the same rate, the problem sessions should be relevant for any
   student.) Problem session times and places will be posted as soon as
   the CAs sign up for rooms on the Math Bulletin Board outside SC 310 on
   the third floor of the Science Center. The available rooms list is put
   up Tuesday afternoon during the first week of sectioned calculus
   courses for CA signups. If an appropriate room is not available, the CA
   should contact the Science Center Scheduler.


   Typically, homework will be assigned at the end of each class period
   and will be due at the beginning of the following class period,
   although this can vary from course to course. The CA is responsible for
   arranging a convenient method of collecting and distributing the
   homework. Remind your students frequently (during the first couple of
   weeks of the course, at least) that there will be solutions to the
   homework problems available on the web. Check with your Course Head
   about a late homework policy.

Office Hours

   Hold two distinct office hours per week. It is best to pick the times
   early on and leave them unchanged for the duration of the course.
   Students are typically hesitant about coming to office hours. Assure
   them that there is nothing to fear! Encourage them to come to your
   office hours if they are experiencing any difficulty. You will need to
   publicize your office hours frequently. Many high school students are
   not used to the idea of office hours.

   Note: After exams, it is a good idea to meet with the students who did
   poorly in order to determine the causes of the difficulty. Freshmen in
   this situation can often use some advice concerning their approach to
   the material. Try to establish what the difficulty is and how it can be

   The Bureau of Study Counsel (5 Linden Street) is available for
   students in need of intensive counseling. They also run a tutoring
   service for students who want/need their own personal tutor.

   Tutors are generally paid for by the student.

Weekly Meetings

   There will be weekly course meetings to discuss the course and the
   goals for the coming week and ensure high quality of communication
   throughout the course. Keep your Course Head posted on any special
   cases or problems concerning your students - e. g. academic trouble,
   disappearing acts, strange behavior, plagiarism and cheating issues,
   time conflicts with exams, etc.

Record Keeping

   You are in charge of keeping track of the examination scores for all
   your students. A copy of students' scores should go to the Course Head.
   You have responsibility for your students' grade records.

Class Lists and Drop-Adds

   At the first or second class, you should pass around a sign up sheet,
   later you can compare it to the sectioned list. Don't expect this to be
   your final class list, many students drop and add classes during the
   first couple of weeks. Your class list probably won't "settle down"
   until about the third week. In the meantime, by looking at the homework
   grade reports your CA gives you, you should be able to determine, at
   least to some extent, who's in your class.

   Important: If a student wants to "add" into your class or wants to
   switch sections, you must refer him or her to SC 308. You cannot give
   permission to add or switch classes.

Teaching Observations

   As part of ongoing attention to teaching, you will be asked to get one
   of your classes videotaped. A preceptor will watch the tape with you.
   We hope this service will be helpful.

Student Evaluations of Instructors

   Midway through the semester, the Math Department will ask you to set
   aside ten minutes of a class period in order to let the students fill
   out a midterm evaluation form. The purpose of this midterm evaluation
   is to give the students a chance to provide some feedback while there
   is still a chance for you to do something about it. The forms include
   questions about your CA. The forms will be kept in the Calculus Office,
   SC 308. Stop by the office to see the responses. If there are problems,
   please seek help.

   Near the end of the semester, you will ask students to fill out a
   University course evaluation questionnaire, (CUE), the results of which
   you will not be able to see until after the course is over and the
   final grades have been submitted. The university takes these
   evaluations very seriously. Keep in mind that when you ask students why
   they rate teachers high or low, they often say that they are rating in
   part on whether or not they feel that the instructor cares whether the
   students learn.

Answering questions

   As you teach, students will naturally have questions. Here are some
   tips for handling student questions.

     * Before you make any attempt to answer a student's question, make
       sure you understand what the student is asking. Students don't
       always formulate their questions clearly on the first try, so you
       may need to do some probing! One way to do this is to say something
       like, "I think what you're asking is blah blah blah; is that
       right?" If you're still having trouble figuring out what the
       student's asking, try to isolate when the confusion arose. ("Did
       you follow everything we were doing up to this line on the board?")
     * Once you know the question, make sure all of the other students
       know it, too. Often, a student will mumble a question and the
       teacher will answer, while the rest of the class has no idea what
       question is being answered. So, before you answer, repeat the
       question for the whole class to hear.
     * As always when you teach, be sure to explain things in language the
       students understand.
     * After you think you've answered the question, check back with the
       original student to make sure that you've really answered it. ("Did
       that answer your question?")
     * Be supportive of all student questions; if students feel you are
       dismissive of their questions, they will stop asking them
     * If a student asks a question that you aren't able to answer on the
       spot, it's perfectly acceptable to say something like, "That's a
       really good question. I'm not sure how to answer it right now, but
       I'll think about it and let you know the answer next time."
       Similarly, if a student asks a question that would take you too far
       off-topic, you can offer to show them the answer outside of class.

Feedback and evaluation

   All teachers are evaluated at the end of the semester, but it's a good
   idea to start getting feedback from your students much earlier than
   that! This way, you have time to adjust your teaching if necessary.

Midterm Evaluations

   All calculus courses (X, 1, and 21) do midterm evaluations. These are
   anonymous evaluations filled out by students in class (while you are
   not in the room). The questions on the form are determined by the
   course head. Normally, your course head will also discuss the feedback
   with you.

Early Evaluations

   You are encouraged to do informal evaluations early in the semester
   (perhaps around the 3rd or 4th week). Generally, you make up this
   evaluation form yourself. The Bok Center has several example forms.

Other ways of gathering feedback

   You may want to have more regular ways of collecting feedback from your

     * Some professors give "one-minute papers" at the end of each class,
       in which students write anonymously what they still find
       confusingly about the class material. This article describes
       how one professor used minute papers.
     * One math TF says:

     I use brightly-coloured (actually, neon) index cards as an informal,
     "quick-response" system. Sometimes I'll distribute these in class
     and ask students to write down anything they want to say -
     questions, comments, requests, whatever. I don't set time aside for
     this - I just ask students to jot their thoughts down as they occur
     to them during the class, and give me the card at the end.
     Occasionally I'll have a specific question (eg. "did you understand
     the Chain Rule today?") and I'll ask them to answer that on an index
     Also, in some classes I've asked students to keep three or four
     index cards in their folders, so that if something suddenly occurs
     to them during class, they can jot it down and give me the card at
     the end of class.
     Oh, the most important thing is that these are anonymous, unless the
     student wants to write their name on it. Also, about collecting the
     cards at the end of class: if there's a CA, I ask students to give
     the card to him/her so that I have no idea who wrote and who didn't.
     If there's no CA, I keep a box on my table for students to drop the
     card into as they leave. My idea is that I want it as anonymous,
     casual and impersonal as possible.

Interpreting Evaluations

   The Bok Center has a useful set of articles on

     * Interpreting Evaluations
     * Interpretive challenges
     * Profiles of problem classes - four common problems and how they
       show up in feedback

   If you need help interpreting your evaluations, talk to your course
   head or a preceptor.

Responding to Feedback

   You should talk with students after you've received their feedback. Let
   them know what sorts of feedback you received. Mention things that are
   going well, things that you might not have the power to change, and
   things that you plan to try to change. This is an opportunity to let
   students know that you care about them and want to improve their
   experience in the class. Here is an example email showing how one
   professor discussed feedback with his students.

First day of class

   The first class sets the tone for the entire semester. Students' first
   impressions can be lasting, so make them good! Experienced teachers
   often talk about the first day "setting the contract" for the semester:
   students will leave your first class with a firm idea of what your
   class is like, and it is difficult to change this later in the

   There is no magic formula for the perfect first class, and what you do
   will depend very much on your personal style. However, try to establish
   the following during your first class:

     * The expectations of the course.
     * A rapport with the students.
     * An engagement of the students with the material.
     * An energy in the classroom.


   If you are a CA or calculus TF, your students will learn the course
   requirements from the course head before you meet them. However, there
   are other expectations that you will set on your first day:
     * Expectations about class time: What's the format of the class? Is
       it discussion-based or a lecture? What is the role of the students?
       Are they passive listeners or active participants? Are they welcome
       to ask questions?
     * Expectations outside of class time: For freshmen, the transition
       from a high school math class to a college math class can be
       startling; try to let them know what to expect. (Talk to your
       course head to find out typical issues.) If you're teaching a
       tutorial, let students know how the work is different from that of
       other math courses they have taken.
     * What students can expect from you: How can students contact you for
       extra help? Will you be open to answering their questions? If they
       send you email, how soon will you respond?

   You may set some expectations by stating them explicitly; for instance,
   you might tell your students directly, "We will not have time to go
   over homework in class, so you're responsible for looking over your
   graded homework. Homework solutions will be posted on the course
   webpage, and you are welcome to come to office hours to discuss with
   me." However, you can also set some expectations through your actions:
   if you plan a very interactive first class, students will expect
   interaction throughout the semester.


   Course evaluation research has found a positive correlation between
   students' ratings of rapport with their instructor and their actual
   achievement. Anecdotally, many math TFs simple find class more
   fun when they have a rapport with students. Some math TFs who have had
   difficult semesters point to lack of rapport as a major stumbling
   block. So, it's certainly important to establish a good rapport with
   your students.

   How can you start establishing rapport in the first class?
     * Introduce yourself to the class. Undergraduates at a Bok Center
       panel emphasized that they want to know their TFs are people, not
       just "talking heads".

     * Learn your students' names.

     * Find out something about your students. You can accomplish this
       with an icebreaker, and you can also ask students to fill out
       surveys on the first day. (In calculus sections, CAs already
       distribute surveys on the first day, so you can just copies from

     * Remember that establishing rapport is not just a first day
       activity; many TFs spend some time in the first few weeks of the
       semester talking with each student individually for 10 or 15
       minutes to find out more about them. In the first class, you can
       have students sign up for these informal meetings.

Engaging students with material

   Besides just convincing students that you're a great person, you also
   want to convince them that the course material is interesting. You
   might tell them some interesting applications of the course material,
   or you might give them a cool problem to do. Good problems are rooted
   in something students already know and lead in to new course material.
   Here are some problems that TFs have used on the first day of class to
   get students thinking.

     * Bottle calibration problem - frequently used to start Xa and 1a
     * For a Math 1b course in which the first topic was applications of

     The definite integral is defined to be:

     (a) a limit of a Riemann sum
     (b) the difference in the evaluation of an antiderivative at the interval endpoints
     (c) an area
     (d) all of the above

     * For Math 21b (linear algebra):

     Old McDonald has a farm. On this farm he has some pigs and some
     chickens. Together, his animals have 15 heads and 52 feet. How many
     pigs and how many chickens does Old McDonald have?


   If you've succeeded in creating a rapport with students and engaging
   them with the material, chances are you've already created a good
   energy in the classroom. Here are some specific ways to add energy:

     * Be enthusiastic!
     * Get students involved. They will add energy for you. 
     * Bring physical props (if they are relevant).


   Many teachers use icebreakers to start off their first class.
   Icebreakers can serve multiple purposes. They can establish energy and
   help students establish a rapport with you and with each other. They
   can also be used to help set the expectations of the course; many
   icebreakers send the message that you want the students to talk.

   Here are some icebreakers that have been used successfully by TFs:

     * Name Bingo: Each student gets a grid filled with descriptions like
       "has been to more than 40 states". Students walk around and try to
       find someone who matches the descriptions. Hre is a sample
       board (from Bret Benesh).
     * Have students pair up and meet each other. Then have each student
       introduce his/her partner to the class.
     * "The Name Tag Activity": Students write some information on a
       nametag and then walk around and meet other students. Here are
       instructions given to Math Xa workshop leaders using this

General advice for getting started

   Below is some general advice to help your class go smoothly.

     * Be extremely well-prepared for your classes. Clarity is essential.
       How will you make the material both clear and engaging? What
       questions will you ask students? Think about your presentation from
       the students' perspectives. What will they find confusing? Write
       out lesson notes in advance. Keep track of what homework you have
       assigned and of where you leave off at the end of each class.

     * Clarify for yourself the goals and purpose of the class. Then pass
       this information along to your students. Point out to your class
       how the topic or question at hand fits into the bigger picture, a
       general approach to problem solving, how it connects with things
       they have discussed, things they will discuss, the course, the
       world at large... It has been shown that students who know why
       they're doing what they're doing -- even proofs or theory -- are
       both more interested and more satisfied.

     * Schedule office hours during the first or second lecture and
       encourage students to make use of them.

     * (For calculus TFs) Have your CA schedule the problem sessions
       during the second lecture and encourage students to attend them

     * Try to learn the names of your students as soon as possible. Do
       your best to establish good communication between you and your
       students and amongst the students.

     * Encourage student participation during lectures. In particular,
       encourage them to ask and answer questions. They can be quite
       reticent about this, so try to be unintimidating and respectful.
       Listen carefully to what they say in order to figure out what they
       are thinking. Try to promote an atmosphere in your classroom in
       which students feel comfortable asking questions, giving wrong
       answers, and taking risks. Based on our experience, we think that
       mathematics is best taught in an environment where the student is
       an active participant; this is why we teach calculus in small
       sections as opposed to a large lecture.

     * Finally, show your enthusiasm for the class and the subject matter.
       Your level of enthusiasm is very obvious to students and will
       affect their own levels of engagement.

Making your class (more) interactive

Why teach interactively?

   Here are some of the many reasons graduate students have come up with:

     * Students learn better.
     * Students have more at stake because they're contributing.
     * Student misunderstanding is more likely to be corrected.
     * It forces students to organize their thoughts to speak.
     * The teacher gets feedback about whether students know what's going on.
     * Being able to contribute is an important skill for students to have.
     * It's just more fun, for both students and teachers.
     * It keeps students awake.

   Course evaluation research has also found a positive correlation
   between interaction in class and student achievement.

How can I teach interactively?

   Here are many suggestions, gathered from graduate students:

     * Ask questions, of all sorts:

          + Computational
          + Conceptual
          + Yes/no or multiple choice
          + What is the definition of ...?
          + What's the next step?
          + Level playing field questions -- questions for which there is
            no right or wrong answer

     * Do in class problems: have students work by themselves, in groups,
       or both.

     * Have students vote on the right answer to a question, and then
       follow up by:

          + Having a class discussion
          + Asking students to explain
          + Having a public "debate"
          + Asking students to discuss in small groups

     * Target particular students, for different reasons:

          + Make sure they've corrected previous misunderstanding
          + To get them involved
          + To give them confidence (if you know they have the right
            answer to something)

     * Give a group test (a test where students can work in groups)

     * Have students put things up on the board; have other students

Some useful links

   Here are some more pages which discuss issues related to interaction in


    1. Cohen, Peter A. Student ratings of instruction and student
       achievement: A meta-analysis of multi-section validity research.
       Review of Educational Research, Vol. 51, No. 3. (Autumn, 1981), pp.

Office hours

   We asked undergraduates in 1a and 1b what they like their TFs to do in
   office hours. They offered lots of advice, which is applicable whenever
   you're working with a student. In fact, most of their advice applies
   whenever you're teaching, whether you're in front of a whole class or
   working with a single student.

   One thing students emphasized repeatedly is that, when they come for
   help with a problem, they do not want you to do the problem for them;
   instead, they want you to guide them through the process of solving the
   problem. In the end, they want not only to be able to do this
   particular problem, but to be able to solve new problems (like the ones
   on their exams!).

   Here is some of the specific advice students offered:

     * Make sure you know what the student's question is.
     * Establish what the student already knows.
     * Lead rather than tell. Don't do the problem for the student;
       instead, ask directed and leading questions to guide the student to
       the solution.
     * Remember that the goal is not just for the student to complete this
       problem, but also for them to be able to do other problems on the
       same material or using similar strategies. So:
          + Get at the thinking process or strategy.
          + Make sure the student can generalize his/her thinking to other
          + Ask the student to recap the entire problem at the end to make
            sure he/she understands the whole process.
     * Write things down as you say them. Even better, have the student
     * Explain things in terms that the student can understand. Be careful
       about notation.
     * Drawing pictures can help.
     * If a problem is about a general situation, use specific examples to
       explain (for instance, if a problem is about "a continuous function
       f(x)", show some specific examples of possible functions).

   One of the keys to accomplishing all of this is to let your students
   talk. Otherwise, there's no way for you to establish what they know or
   speak their language.

Preparing for class

   Preparing well for class is the most important part of teaching. From
   the syllabus and the weekly course meetings, you should be able to
   formulate precisely in your own mind the goals of each class. Outline
   on paper the definitions, general explanations, and specific examples
   to be covered in class. Work out your examples beforehand to ensure
   that they are good illustrations of the ideas being covered, and that
   unsuspected difficulties will not arise during class, robbing you of
   precious class time. Use your class time effectively because there is
   seldom enough of it. Two or three well-thought-out examples are worth
   much more than five or six off-target examples.

   You may want to set aside a few minutes, at the beginning or end of
   class time, to write down (on the blackboard) the reading and homework
   assignments for that day, upcoming office hours, and any other
   information likely to be helpful to the students.

Presentation skills

   If you are interested in improving your presentation skills, here are
   some options:

     * Watch the video The Act of Teaching, in which Nancy Houfek, Head
       of Voice and Speech for the American Repertory Theatre at Harvard,
       introduces some acting techniques that are useful for teachers. You
       may watch the video online or borrow a copy from the Bok Center
       (Science Center 318).

     * The Bok Center provides individual coaching on presentation
       skills. If you are interested in coaching, contact Rebekah
       Maggor (rmaggor), who directs the Bok Center's Program in
       Speaking and Learning.

Using worksheets in class

   Your course head may sometimes give you worksheets for your students
   (or you might make up your own worksheets). Here are some different
   ways to effectively use worksheets in class:

     * Work through a problem as a class, asking students to suggest steps
       to take.

     * Give students time to do the problem individually or in small
       groups. As they work, you and your CA can walk around and see how
       they're doing. Afterwards, it's important to make sure that all
       students understand how to do the problem. Here are some ways you
       might accomplish that:

          + Do the problem yourself on the board (perhaps asking for input
            from the students).
          + Ask a student to put the problem on the board and explain it
            to the class.
          + If students have differing solutions, ask multiple students to
            put their solutions on their board, and let the class discuss.
          + Ask students to vote on the correct answer. This can be
            followed up in many different ways:
               o Discuss as a class to arrive at the right answer.
               o Have students discuss in small groups, and then hear from
                 a student who changed his/her mind.

     * You might want to combine some of these methods. For instance, you
       might have students first try the problem individually and then
       compare answers with their neighbors before you do the problem on
       the board.

How do I decide which method to use?

   There are no hard and fast rules about how to use a worksheet in class.
   However, here are some things to keep in mind when you decide what you
   want to do:

     * Doing math vs. watching math: Most mathematicians feel that the
       only way to learn math is to do it. If you do a problem as a class,
       there may be some students who just watch without really doing the
       problem themselves. On the other hand, if you ask students to try
       the problem individually or in small groups, you can get everybody
       to attempt the problem.
     * Feedback for the teacher: Often, a teacher will give a problem to
       check how well students understand the material. If you do the
       problem as a class, you usually hear from at most a handful of
       students, and you may not find out whether other students
       understand. On the other hand, if you have students work on the
       problem individually and walk around to see how they're doing, you
       can get a glimpse of each student's work.
     * Knowing math well enough to do a problem vs. well enough to explain
       it: Sometimes, students will be able to do a problem but are not
       able to explain their reasoning clearly. Encouraging students to
       talk (either to the class or to fellow students) encourages
       students to organize their thoughts.
     * Time: Having students do things almost always takes more time than
       doing them yourself. Obviously, doing things like having students
       debate will take more time than simply telling them the right
       answer. Try to balance your time constraints with what you want
       students to get out of the problem.
     * Seeing multiple strategies: Often, there are multiple ways to solve
       a problem, and it can be useful for students to compare different
       strategies. (In fact, Jon Star of Harvard's Graduate School of
       Education has done research showing that having students compare
       different strategies can lead to better conceptual understanding as
       well as problem solving flexibility. Star recommends having
       multiple strategies on the board side-by-side and having students
       discuss the advantages and disadvantages of the strategies.)
       Students are more likely to come up with multiple strategies if you
       give them some time to try the problem. Then you can figure out
       ways to get multiple strategies on the board to compare.

   Keep in mind that you don't have to use the same method for each
   problem. Figure out what you want students to get out of each problem
   and how to choreograph to accomplish that.

Some Examples

   Here's an example from a former Math 1a TF:

     I was teaching the definition of the derivative, and the first
     worksheet problem was:

     Let f(x) = x^2. Find f'(a).

     I asked the students try it by themselves, talking to a neighbor if
     they wished. As they worked, my CA and I walked around to see how
     they were doing. Some students had no trouble with the problem, but
     many (who had taken calculus before) told me, "I know the answer is
     2x, but I don't know how to use the definition to get that."

     So, as a class, we discussed how to interpret the definition and
     apply it to this specific problem. Then I had them complete the
     problem (by themselves or with a neighbor) before we finished it as
     a class.

   In this instance, the TF probably spent a lot more time on the example
   than she would have if she had let the class discuss. On the other
   hand, she gained some important information, such as the fact that many
   students were not comfortable applying a general definition to a
   specific example. This allowed her to tailor the discussion to focus on
   the students' confusion. In addition, all students tried the problem
   and had to figure out what they didn't understand.

   Contrast that with this anecdote provided by a former Math 21b TF:

     We were talking about matrix multiplication and inverses, and the
     problem was to simplify (AB)^ - 1. I started by asking students to
     suggest answers; I wrote these all on the board, and then I asked
     students to vote on the right answer. There was no clear consensus,
     so I asked them to discuss with their neighbors for a minute. Then,
     I had them vote again, and nearly everyone agreed that the correct
     answer was B^ - 1A^ - 1. I asked a student to explain to the class,
     and then I gave an alternate explanation.

   This TF chose completely different choreography from the 1a TF but
   still got students to do the problem themselves. The students also had
   to explain their reasoning to others, and the TF was able to get
   feedback about what they understood.

Improving your teaching

General Resources

     * Your colleagues.
     * Your course head.
     * The preceptors are always available to talk about teaching.
     * Visiting someone else's class is a great way to get new ideas. Just
       check with the teacher in advance to make sure it's okay to sit in
       on the class.
     * The Bok Center for Teaching and Learning provides many useful
       services, including:

          + Videotaping your class (You may watch the video by
            yourself, with a preceptor, or with a Bok Center staff member.
            The tape is completely confidential and cannot be watched by
            anybody without your permission.)

          + Individual consultations on anything related to teaching,
            such as speaking skills.

Especially for International TFs

   The Bok Center offers a great deal of support for international TFs,

     * individual and small group coaching to improve speaking
     * workshops on a variety of topics

   For more information on these services, visit the Bok Center
   international TF page.

Links to books, videos, etc.

     * "Teachers from Other Countries", a short section from 
       the Torch or the Firehose: A Guide to Section Teaching
     * Teaching American Students, a Bok Center guide explaining
       American students' expectations and offering advice. You may
       request a free copy here.
     * Teaching in America: A Guide for International Faculty, a Bok
       Center video. You may borrow this video from the Bok Center by
       visiting Science Center 318.

Student resources

     * Bureau of Study Counsel - The BSC runs workshops on topics like
       test-taking and time management. They also provide [4]private
       counseling on all sorts of issues, both personal and academic.

     * Peer tutors - Students may request a peer tutor from the BSC.
       For undergraduates, the cost is only  per hour (and financial
       assistance is available).

Writing letters of recommendation

   At some point, your former students may start asking you for letters of
   recommendation. Here are some links with useful advice:

     * Bok Center tip sheet
     * GSAS guide for teaching fellows on writing letters of
     * Advice for Those Writing Letters of Recommendation for Harvard
       Applicants to Medical School, from Harvard's Office of Career
     * Writing Student Recommendations, written by Heather Smith, a
       former English TF and resident tutor,

Preparing for the job market

   If you are applying for an academic job, you will probably be asked to
   submit a teaching statement as well as a teaching reference letter. In
   addition, you may want to create a teaching portfolio. See the [3]Bok
   Center's guide to teaching portfolios, which describes components of a
   teaching portfolio and offers lots of tips on creating one.

   If you've had a good semester of teaching, it's a good idea to ask the
   course head for a letter of recommendation at the end of the semester,
   even if you're not applying for jobs yet. This will allow the course
   head to recall details that he/she might not be able to remember at a
   later date.

Academic calendar

   The calendar for the current academic year can always be found at
   This calendar includes dates such as the last day
   undergraduates may drop a class or decide to take a course pass/fail.

Undergraduate math courses

From TeachingWiki

   Jump to: [1]navigation, [2]search

   Courses are usually described by their number, and the easiest way to
   learn what these mean is to see the [3]course catalog.

   The standard undergraduate courses that graduate students teach in are:
     * Calculus:
          + Math Xa and Xb - pre-calculus plus the material from 1a
          + Math 1a - first-semester single variable calculus
            (differentiation and integration, up to the Fundamental
            Theorem of Calculus)
          + Math 1b - second-semester single variable calculus
            (integration, series, and differential equations)
          + Math 21a - multivariable calculus
          + Math 21b - linear algebra and some differential equations

          Most students in the calculus courses are not math
          concentrators. Many concentrate in fields like economics or life
          sciences which require some math. Some students take calculus
          primarily so they can apply to medical school. Others simply
          take it out of interest.

     * QR28 - The [4]Core is Harvard's breadth requirement (it is
       currently being phased out, to be replaced by a "General Education"
       requirement) The math department offers one course, [5]QR28: The
       Magic of Numbers, that students may take to fulfill part of their
       breadth requirement. Students who take this generally do not need
       math for their concentrations, and no math background beyond high
       school algebra is necessary.

   One TF who has taught both calculus and QR28 describes the difference
   from the students' perspective:

     "Calculus is harder for them. The study involves more technical
     calculations and a much heavier homework load. Some of them could do
     really bad in exams;

     For QR28, we teach the material in a more down to earth approach,
     since the students usually don't have any mathematical background.
     Most homework problems are very basic and could be solved based on
     intuition. Most students can perform well in exams."

Undergraduate math concentrators

   There are many different options for students concentrating in math.
   Some start in Math 1 or Math 21. Others take Math 23, 25, or 55 in
   their first year. This guide describes these courses a little more.

   The following documents for undergraduates will give you an idea of
   what math concentrators take in later years:
     * What concentrators are advised to take freshman and sophomore year
     * Sample 4-year course schedules for math concentrators

How students decide what to take

   All entering freshmen take a placement exam before the fall semester
   starts. The placement exam is primarily useful for students who want to
   take some level of calculus but are not sure where to start. The
   placement exam gives students a recommendation of X, 1a, 1b, or "beyond
   1b" (meaning they can take any course after 1b, such as 21a, 21b, 23,
   25, or 55). The placement exam is not binding; it simply gives students
   a recommendation.

   In the fall, there is always advising for students who are wondering
   what math course to take. More information is available [11]here.

External links