This is a temporary static version of the TF wiki. Introduction 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 department. 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. Core 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. Tutorials 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. Compensation 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. Tutorials 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 involved. 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). Compensation 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. Apprenticeship 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 themselves. 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 apprentices. * 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 interesting. * 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 understood? * Keep your mind on the students. Concentrate on communicating with them. * 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 process. 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 http://bsc.harvard.edu/refer.html 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 Basics 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 sessions. 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. Homework 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 overcome. 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 altogether. * 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 students. * 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 card. 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 semester. 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. Expectations 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. Rapport 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 them.) * 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 integration: 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? Energy 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). Icebreakers 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 icebreaker. 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 regularly. * 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 critique. Some useful links Here are some more pages which discuss issues related to interaction in class. References: 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. 281-309. 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 problems. + 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 write. * 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, including * 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. http://bsc.harvard.edu/counseling.html * Peer tutors - Students may request a peer tutor from the BSC. For undergraduates, the cost is only per hour (and financial assistance is available). http://bsc.harvard.edu/tutor.html 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 recommendation * Advice for Those Writing Letters of Recommendation for Harvard Applicants to Medical School, from Harvard's Office of Career Services * Writing Student Recommendations, written by Heather Smith, a former English TF and resident tutor http://isites.harvard.edu/icb/icb.do?keyword=k1985&pageid=icb.page29706 http://isites.harvard.edu/fs/html/icb.topic58474/Verba-recs.html http://www.ocs.fas.harvard.edu/students/careers/medicine/applicationprocess/lettersrecwriters.htm http://www.ocs.fas.harvard.edu/ http://www.fas.harvard.edu/~english/teachingfellows/WRITING%20STUDENT%20RECOMMENDATIONS%20-%20heather%20smith.htm, 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 http://webdocs.registrar.fas.harvard.edu/ugrad_handbook/current/chapter1/index.html 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