Guidelines for Teaching Math

GUIDELINES FOR TEACHING MATH

A Handbook for Adjunct Faculty Math Matters CCC

Teaching math can be very rewarding when one is able to make the subject useful & interesting, get the students excited & engaged in the learning. Effective math teaching is the result of many elements. The following thoughts, suggestions & recommendations for teaching math at CCC may help you be more effective by reminding you of tips & strategies you already know or, perhaps, providing some new ones. While some are quite obvious & intuitive, others are the result of experience & research.

 

Before you start

Make sure that:

· You attend the orientation session organized by your Division.

· You have obtained from your Department a starter packet with old syllabi, handouts, etc.

· You have obtained the detailed course description & a list of the chapters & sections that you must cover from the book.

· You have obtained the textbook & its supplements (e.g., instructor’s manual).

· You know about the course’s prerequisites, which majors do/do not require it, transferability to other institutions, etc.

· You have, if required, the appropriate graphing calculator (with instructions on its use).

· You know when & where your section meets.

· You know where to find chalk, do photocopying, find an overhead projector, etc.

· You have the College catalog, spelling out rules & policies.

· You know what to do in case of an emergency (inclement weather, sudden illness of instructor or student, locked classroom, safety problems, etc.)

· You have your class list.

· You know how, when & where to contact the chairperson & the secretary of the Math Dept., as well as college security.


Your syllabus

It should be ready for the first meeting with your class. It is a “contract” between you &

your students and, except for midstream adjustments, you should try to abide by it. It should

contain:

· Title & code for the course; number of credits

· Term, date, time & location of classes

· Instructor’s name, office phone & mailbox, e-mail address.

· Office hours

· Required textbook & technology.

The CCC Math Dept’s Web site

This site contains a series of links for online assistance to students & faculty as well as tips on how to study math.

 

Prerequisites for your course

· A list of topics & chapters in the book to be covered.

· Your attendance & class participation policies, clearly spelled out.

· Your homework, testing & grading policies, the relative weight of each test, the nature of the final exam. Specify how the course grade will be determined; indicate your grading scale.

· Your policy on cheating & missed examinations, clearly stated.

· The last day to drop the class without grade penalty, clearly noted.

· Some instructors make up a detailed calendar specifying dates for topics & exams.

· Some instructors include the homework assignments for the entire semester.


First day

This may be the most critical day of the semester - you send important messages to your class. First impressions endure.

· Remember: you may be nervous but they are terrified. Put your name on the board & tell them how you would like to be addressed. Break the ice. Smile at them. Tell them about your background, your family, even your vacation, whatever will show them you are a kind and caring human being.

· Call the roll. Handle discrepancies between class list & attendance.

· Hand out your syllabus - read it out loud & answer questions about it.

· Discuss how you intend to run the course, handle homework, tests, grading, attendance, latecomers, disruptions, missed tests, cheating, deadlines, etc.

· Tell your students what they can expect from you & what you, in turn, expect from them. Be clear & specific about this.

· Take a (written) survey of your class to know each student’s academic background & goals, professional aspirations, phone & e-mail, expectations about the course, concerns, anything about them they feel you ought to know (last math course taken & when, free time, job, hobbies, learning disabilities, etc.).

· Be clear & strict about prerequisites & their enforcement & about majors requiring the course.

· You could prepare (or borrow from a colleague) & have students take a short diagnostic test on essential material from the prerequisite course. It could be self-corrected. Tell students with a low score that they will have difficulties & try to convince them that it is to their advantage to either brush-up well or re-take the prerequisite course.

· Present an overview of the course, the main problems it tackles, why it matters, etc. A good way to create interest in the course is to state some problems they cannot do, but will be able to do by the end of the term.

· Show up on time & use the entire allotted time period. You could do a brief review of material they are supposed to have already mastered.

· Set a “serious business” pace & assign homework the first day.

Preparing a lesson

Preparation is essential if you wish to make the subject relevant & exciting.

· You may need extra preparation time if you have not taught the course before.

· Before the beginning of the semester, make an outline of the topics to be covered & how much time you should devote to each of them so that the entire course content is covered. Try to stick to this plan.

· Students are quick to detect the under-prepared instructor; lack of respect & poor attendance usually follow.

· Read ahead of time the appropriate chapter & sections from the textbook & try to prepare your lesson following similar approaches, terminology & notation. Read ahead in the text for the sake of perspective.

· Although you may know the material well, it is worthwhile to have the main points & examples of your lesson clearly written on paper. And yet, try not to read directly from your notes.

· You might start a class by presenting a brief summary of the previous class topics, for the sake of continuity. Explain how a new idea fits into the general scheme of the course.

· Find an interesting problem or application that will motivate students & stimulate curiosity in a new topic.

· When introducing a new concept, try not to start with an abstract definition. It is better to begin with concrete examples that students understand & that lead naturally, by the process of generalization, to the definition & concept you want them to know.

· Sometimes your students will understand a new concept better if you explain that it is similar to a more familiar concept – reasoning by analogy.

· Never introduce math terminology unnecessarily. Your audience may not understand your math vocabulary or technical jargon. Students speak a different language. This is one reason they tend to have trouble reading the textbook.

· By the same token, never introduce new symbolism unnecessarily. Students tend to be utterly confused by this. It has the effect of obscuring the idea being discussed & making it more difficult to learn.

· Avoid the temptation to tell your class everything you know about a subject. Concentrate on the core ideas & examples & return to them periodically.

· Try not to present the problems that are examples in the text; present similar ones, problems making the same points. Save the examples in the book for your students to study at home to reinforce their learning.

· Try to anticipate students’ questions.

· Decide what to emphasize, where to slow down & present additional examples.

· Be ready to explain concepts in multiple ways.

· Be ready to answer questions such as: “Why do I need to know this?”, “What is this good for?”, “What does this mean?”, “What are the possible real life applications of this?”, “Who uses this & why?”, “What are the historical origins of this?”, “Where does this lead?”.

· Intercalate historical tidbits & real world applications for context & added interest.

· Examples should progress from the simple to the complex.

· Emphasize the inner logic in the solution to problems. Hammer away the point that in math the clear display of this inner logic (showing all the steps) as well as the correct answer are equally crucial.

· Usually, students find an informal but intuitive & reasonable argument, example or picture to be more convincing than the complete proof of a math proposition. Professional mathematicians learn from proofs; students learn more easily from explanations.

· What is obvious to you may not be so to your audience & may require an elaborate explanation. Be ready for it. For example: How would you justify to an inquisitive student that the product of two negative numbers must be positive?

· We want our students to master the key ideas & techniques presented to them in a math course. However, let us not lose sight of what the ultimate goal of learning mathematics is: to be able to reason logically & rigorously, to develop higher order analytical & problem-solving skills, to adopt a mathematical outlook.


Your class performance

Your performance in front of your class should aim for total clarity in what you are trying to communicate.

· Your voice is one of your critical tools. Speak clearly & slowly in a relaxed voice; fill the room with it. Gain your class’s attention by changing your tone of voice as you make a point. Modulate your voice & enunciate well.

· Pause for effect before & after making an important point. State clearly that it is an important point. Lower your voice. Repeat the point. Write it on the board, underline it, do not erase it, return to it later. Give examples. Say that they will be tested on it. Repeat the important point. Ask for questions. Later, make sure they do get tested on it.

· Think of your lecture as “controlled conversation” between you & your audience. It should allow for communication & interaction between you & your students.

· Make eye contact with your audience, turn to them for individual attention, flash the emphatic smile, use your sense of humor, try to engage them all.

· Strive for as much informality in the classroom as your own personality and the circumstances will allow. Again, try to inject humor, smile occasionally. A smile helps to establish a congenial teacher-class relationship & a positive attitude among your students. By the same token, never give way to anger in the classroom.

· A well-organized use of the blackboard facilitates learning. Write neatly & in an orderly fashion. Use large characters. Write in complete sentences. Label. Make it intelligible. Write all the essential steps, the way you would like them to do it on a test. Do not clutter a blackboard with your writing. Underline the key ideas. Use different colors. Divide the blackboard into boxes. Write from left to right, top to bottom. Fill the boxes in succession. When all boxes are full, stand aside, read aloud what was written. Emphasize key points. Pause. Keep quiet. Ask for clarifications. Then, slowly start erasing.

· Try to begin each class with a short discussion of material with which students feel relatively comfortable, rather than plunging directly into new territory.

· At the start of your presentation, write on the board the main goal(s) of the day’s lesson. Do not erase this. Explain why it matters. Once this goal has been attained, make sure your class gets the message by examining the goal statement on the board one more time. Ask them to summarize how it was achieved.

· When pertinent, provide (or have students derive) a strategy to find solutions to problems: an algorithmic, step-by-step approach is easier to apply & remember.

· Encourage conjectures & guesses; have your students discover certain patterns & ideas by themselves.

· Encourage total participation for difficult points or create “wake up” moments by telling everyone to solve a short problem, show it to a neighbor, raise hands if they have the same answer, etc.

· Call on all the students: male & female, young & old.

· Give students a fair amount of time to consider any questions that you raise. Otherwise they will think of the question as rhetorical.

· Admit mistakes when you make them. Thank the students who discover them.

· Manage your time efficiently. Use all of the available time. If you finish your lesson with some minutes left, use this time to sum up with your class the lesson’s main points & their applications, a sense of where you were, where you are & where you are going. Organize groups to get them working on the homework. Have them write their solutions on the board.

· Don’t rush out after class. Linger for a few minutes so that students may have an opportunity to ask questions. Erase the blackboard as a courtesy to the instructor of the following class; return chairs to their original position.

· Analyze & evaluate your performance & your students’ reaction after class. Jot notes to yourself to help you improve future strategies.


Homework assignments
One learns math by solving many problems. Students need to practice the techniques they have been taught.

· Tell students that prior to working on the homework they should review class notes & the examples in the book.

· Make sure that the assignment involves all the important topics covered in class.

· Make sure that exercises range from the simple to the more advanced.

· Emphasize that the solution method matters & that all the important steps should be shown.

· If pertinent, the homework should include some applications.

· Make sure that the homework prepares students for the upcoming test.

· Provide incentives or penalties that prod students to work on the homework: some of it could be graded, included on the next test, done & turned in by groups, etc. To encourage collaborative learning, you could grant bonus points to graded homework done in groups of two or three.

· Ask your students if they had trouble with the specific problems that contain core ideas of a topic. Do some of these problems in class.

· If you are collecting & grading some of the homework, late work is a persistent problem. Students can be deterred by downgrading late assignments, assigning extra work, etc.


Technology
You may want/have to use technology in one of the following forms: graphing calculators, computers, the Web, videos & multimedia.

· Calculator use has now become pervasive. Your students will have to be taught how to do it! They will not read the user’s manual. You will have to teach them the basics.

· Prepare or ask one of your colleagues for a handout with instructions on how to get the needed calculator results. Make copies for your class.

· Schedule a calculator lesson with your class, letting them know the date in advance.

· Make sure you have the needed equipment: overhead projector, cords, viewscreen, etc.

· Select & prepare a varied list of problems that are covered in the course & that lend themselves to a simple calculator approach (e.g., graphing, solving equations, generating a table of values, performing messy calculations, doing statistics, etc.). Make sure that your examples justify the use of technology.

· Practice your lesson in advance & have an alternate plan in case of system failure.

· Present the lesson to your class, making sure that your students practice the problems with you & are not left behind.

· Make sure that, as the semester progresses, your students also get tested on the use of this technology (if it is required in the course). Choose problems that require both math understanding & calculator use.

· Recall: teaching with technology is not the same as lecturing. Technology does not replace conceptual understanding - it should facilitate & reinforce it.


Questions
It is common practice to devote the first part of your class to answering questions on your lessons or previous homework. You could use this as a natural springboard for the topic of the day.

· Students should be encouraged to ask questions during your presentation. When a question is asked, repeat it out loud for the entire class to hear.

· Some instructors find it effective to have a supply of index cards in the back of the classroom for students to write questions about the lecture or homework. Then you can choose to answer the best or more frequently asked questions at the start of the next meeting.

· Students tend to be shy & hesitant about asking questions in class - they may have to be prodded. Instead of asking, “Are there any questions?” , you may get a better response by being more specific: “What are the main features of a parabola?”, “Do you all know how to find them?”, “Let’s examine problem 37 on page 186. Did you have any trouble with it?”, “What are the main ideas contained in the Central Limit Theorem?”.

· You may hear a question that makes no sense. Avoid sarcasm. Never ridicule or put the student down. You could respond with: “Maybe what you are asking is…?”.

· When asked a silly question: respond quickly as if it is an honest one & move on.

· When asked an unintelligible question: you could ask in return “Could you rephrase that?” or “Are you asking about (something sensible)?”.

· When asked a question that is off the subject, tell the student: “Let’s talk about it after class”.

· If somebody asks a good question, say so clearly: “That’s a good question & I’m glad you’re asking it”.

· When asked a question you cannot answer - just say: “I don’t know the answer right now, off the top of my head. But I’ll investigate that point & I’ll let you know the next time”. And by all means, do.

· When asked: “Is this going to be on the test?”, your answer could be something along the lines of: “It certainly will, otherwise we wouldn’t be doing it”.

· Again, be ready to answer questions such as: “What is this good for?”, “Why do I need to know this?”, “What does this mean?”, “What are the possible real life applications of this?”, “Who uses this & why?”, “What are the historical origins of this?”, “Where does this lead?”. If nobody asks any of these questions, you could.

· After asking for questions, wait, while looking at the class, to give the students time to formulate their ideas. Ditto, when posing a question to the class.

· Propose that each student & a partner come up with one important question.

· Keep lines of communication open with your class. Invite them to visit you during office hours. Some may prefer to e-mail you. Try to be responsive to this. A mid-semester evaluation survey may allow your students to provide you with valuable feedback - both for improving your teaching as well as allowing students to self-assess the effectiveness of their own study habits.


Composing & giving exams
You really have your students’ attention during an exam cycle - from the moment they start getting ready for it & until the time they get it back & graded. Take advantage of this opportunity by making it a productive learning experience.

· Drive home the important ideas; give your students a list of the topics & key problems on which they will be tested. If some of these topics are too involved & time-consuming for an in-class test, they could be part of a graded take-home project.

· Some instructors feel that a fair distribution of problems is as follows: 40% on really basic material, 40% on less routine problems that require some thought & a multi-step approach & 20% on more challenging problems / applications & that will allow you to discover the more proficient students. More challenging problems could be assigned bonus points.

· Students deeply resent being tested on new types of problems they have never seen before. In this regard, & to encourage good attendance & effective study habits, you can tell them in advance that you will use three sources of problems: the examples presented in class, the corresponding examples in the text &, mostly, the assigned homework problems. This gives you plenty of leeway & they usually feel it is fair. They want to be reassured that they will not be unpleasantly surprised on a test.

· You could ask your students to give you a list of exercises from the homework they feel it would be fair to be tested on, write the list up on the board & let them know that you will choose some of those as problems for the next test.

· A practice test, for which you could provide a solution key, as well as a review session before the exam, are effective tools that tell students what the central ideas are & what kinds of problems to focus on for the exam.

· There are many arguments against tests with a multiple-choice format & in favor of the handwritten, show-the-full-solution exam. One is the issue of granting partial credit. Also, the handwritten exam is a form of communication between student & teacher. It allows the teacher to gauge the extent of the students’ ability to give a complete, well-justified math argument & to display problem-solving skills.

· A well-crafted test should have a reasonable number of problems of reasonable length, which should not be inter-connected. Making test questions significantly interdependent is a form of double jeopardy & should be avoided. The test could be written in such a way that a problem could be broken down into several parts - guiding the students through a sequence of steps that culminate in something of value. Be sure that the questions you ask elicit the basic information that you seek.

· Some instructors give the easy questions first to allow the less confident students to get off to a good start.

· When writing instructions for the exam, clearly indicate that students must show all the steps & justify their answers.

· Encourage questions during the examination about the interpretation of the problems, but reserve the right not to answer some.

· As a concession to the math-panicked students, some instructors give students the opportunity to rework the exam at home once it is returned to them. Both the in-class & the home component are counted as part of the grade.

· To complete the learning cycle, you could write constructive comments on the students’ test & prepare for them a complete solution key that you discuss with them in class. As an incentive to have them learn from their mistakes & consolidate previous learning, you could tell your class that you will pick one question verbatim from the last exam & put it on the next one.

· Try to grade & return tests to your class within a week or so - while your students still remember the test material.

· If there is a technology component in your course, you may find it useful to give your students a graded take-home project that involves heavy use of technology. Interesting real life application problems could also be part of this project.

· You should give a cumulative final exam. It gives the student the opportunity to have a comprehensive, non-fragmented understanding of the central ideas of the course & a better sense of their multiple inter-connections. Some instructors tell their classes that to study for the final they can study from their previous exams - the problems will not be too different.

· On the subject of make-up tests, some instructors tell their students that when a regular test is missed for a well-justified reason, they must take the cumulative final. That score becomes also that of the missed test. Others tell students that the lowest test score will be dropped & that the missed test will then count as the lowest.


Grading
You can think of grading as a learning opportunity for all involved – the instructor gauges how well the material was taught & the test designed, while students determine how well they know the material & studied for the exam.

· Put your grading policy on your syllabus & make it clear to your class at the very first meeting. It then becomes a matter of public record. You must be firm & consistent when applying this policy.

· Students may have to be reminded from time to time of this policy & its implications for them as the semester progresses. Before the deadline to drop a course, tell your class that by then they should have a good sense of the level of the course & their performance so far & that some may be better off withdrawing from the course rather than risking failure. You may want to meet individually with those students to drive home this point. The fatally optimistic should be reminded that acing the final exam is a rather unlikely occurrence.

· Do not discuss a student’s grade / performance in public. Do not compare students to each other in public.

· The more frequently students are tested, the better they seem to learn. Additionally, the course grade will tend to provide a more accurate reflection of their level of understanding & performance. However, there are time constraints. As a rule of thumb, you should get from them a set of at least 3 scores (including tests, sets of quizzes, graded homework, projects, etc.) plus a comprehensive final exam. Some could be dropped. Assign appropriate weights to each. Some instructors drop the lowest such score. Others apply a changing scale where the best score gets the highest weight. Whatever grading policy you decide to implement, it should be fair & defensible (with your students, colleagues, dean).

· There are times when students will insist that they know the material, but for various reasons their records do not show it. At such times you should remind them that the grade is an indication of their actual performance.

· The course grade should reflect a student’s performance during the entire semester. The grade a student gets should be the grade he or she has earned. It is a done deal, not a matter to be negotiated.

· We want our students to write correct, clear & complete solutions to problems. When this is not the case & when merited, giving partial credit is justified. The process of giving partial credit is not an exact science but a matter of good judgment, which we must use fairly & uniformly when grading a test. However, some students will try to contest it. You should be able to explain & defend your decision, respectfully but firmly.

· Some instructors also take into consideration neatness, clarity, intelligibility & organization in their students’ presentation of a solution to a problem.

· For the sake of consistency, some instructors grade one problem at a time.

· If the solution to a problem you are grading is incorrect, identify the error & indicate the correct step. You could also write encouraging remarks where the work is especially good.

· When you return the exams, list the grade ranges & the median grade on the board.

· It is more appropriate to tell those questioning your grading criteria to see you after class once the test is returned. Make sure they have examined the answer sheet first. Don’t rush into a decision: tell the student that you will take the test home to re-examine the given solution as well as the grading.

· Occasionally, you may realize you made a mistake or were too harsh when grading a problem. You should admit so to the student & change the grade.

· Never penalize the honest student. If a student tells you that her score was added incorrectly & that she deserves fewer points, thank her for the honesty but do not change the score.

· Some instructors adopt a rule that a student will always receive a letter grade which is no more than one grade below the one he or she has before taking the final exam. Thus, for example, a student who has an A coming into the final, will get at least a B in the course.

· You will often hear sad & convincing stories of why a low grade in your course will have dramatic consequences (e.g., financial aid, college or job prospects, etc.). You can try to be accommodating & yet the issue of fairness with the entire class should not be ignored.

· Keep thorough & well-organized records of students’ performances.

· Keep your students’ final exams for at least one semester beyond the end of the term. They provide valuable evidence to defend your grading in case somebody contests it.

· Some instructors feel that only well-justified & extreme circumstances deserve the grade of incomplete. It certainly should not be the fall back option for the entire class. Before the temporary grade of “I” is granted, you should make specific arrangements (in writing) with the student regarding when & how it will be removed.


Discipline, honesty & civility
The best approach to foster an atmosphere of civility & counter potential discipline problems in the classroom is for you to set the tone by presenting the image, from day one, of a well-prepared, respectful & serious professional.

· From the very first interaction, you should strive to make clear to your class what your expectations are, what will not be tolerated (e.g., cheating, student conversations, reading the newspaper, eating, sleeping, etc.), how it can be avoided & how you plan to handle it.

· You are an authority figure in the classroom (you are older than most, you have a college degree, you give out the grades, etc.) & there are times when you will be faced with situations that require exercising this authority, swiftly & effectively.

· Nip behavioral problems in the bud by courteously but firmly tackling them as soon as you feel the problems are real. Ignoring the problems may create a widespread, uncontrolled & disruptive class dynamic for the rest of the term.

· In general, faced with distracting or disruptive situations, you should focus attention on the behavior, not on the person. Instead of saying “John, stop talking!”, you could comment on the fact that background chatter makes it difficult for everyone to pay attention. Or, without putting anybody on the spot, simply tell the whole class: “OK, let’s quiet down”. Of course, there may be times when the only effective remedy is to directly tell John to stop talking.

· Avoid debating in class with a student who is misbehaving. Better to talk with the student after class, in a friendly & respectful manner. Ask the student what triggers this kind of behavior & ask/tell him what can be done to avoid a recurrence of it. You could offer him some choices, allow him to save face.

· On vary rare occasions you may have to face disrespectful or aggressive behavior from a student. Don’t be confrontational - handle the situation with care. The student may have serious personal or psychological problems. Courteously ask the student to see you after class. If the situation seems to be getting out of hand, you can suspend the class or call security. You can also tell the troublemaker that you have run out of options & that he should talk to the chairperson of your department. Make sure you promptly brief your chair/dean on the situation.

· You must have a policy, laid out clearly in your syllabus & discussed with your class, regarding instances of academic dishonesty. The student found cheating should be penalized as soon as you have reliable evidence. If cheating occurs during a test, you have every right to give the student a grade of “F” on the test or even in the course (unless the student decides to drop the class). But make the policy clear before the problem happens. And keep the evidence that prompted you to take such drastic action.


Favorable attributes (of a good math teacher):

· Knowledgeable
· Friendly, caring & helpful
· Enthusiastic about the discipline & its teaching
· Fair
· Clear
· Responsive
· Responsible
· Well organized
· Punctual
· Patient


References
1. How to Teach Mathematics (2nd Ed.), by Steven G. Krantz, American Mathematical Society, Providence, RI, 1999.

2. A Handbook for Mathematics Teaching Assistants (Prelim. Ed.), by Thomas W. Rishel, The Mathematical Association of America, Washington, DC, 1999.

3. You’re the Professor, What Next?: Ideas and Resources for Preparing College Teachers, Bettye Anne Case, Editor, The Mathematical Association of America, Washington, DC, 1994.

4. College Mathematics: Suggestions on How to Teach It, by the Committee on the Teaching of Undergraduate Mathematics, The Mathematical Association of America, Washington, DC, 1979

5. Can We Make Mathematics Intelligible?, by R. P. Boas, The American Mathematical Monthly, Volume 88, Number 10, December 1981

6. The Craft of Teaching (2nd Ed.), by Kenneth E. Ebling, Jossey-Bass Publishers, San Francisco, CA, 1988.