Department of Mathematics

The Pennsylvania State University

University Park, PA




Grateful acknowledgement is made to:  Former Department Head, Dr. George E. Andrews, who founded the Mathematics Mentoring Program; present Department Head, Dr. Gary L. Mullen, and Dr. W. Dale Brownawell for their support and valuable suggestions; Dr. John Roe, Director of the GTA Mentor Program, whose dedication to effective teaching continuously enlarges and deepens the scope of the Program; and to all those other teachers and graduate teaching assistants for providing essential, experienced-based perspectives contained in this Handbook.







                                                                             Jo Battaglia

                                                                             Mentor Coordinator

                                                                             August 2002





*GTA-Graduate Teaching Assistant




Introduction     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      3

Why a GTA Handbook?

What is the Mathematics Mentoring Program?

Pre-semester Preparation     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      4

Three essential preparations

Mentoring Meeting

Departmental GTA Training and Induction Program

Course Coordinator Meeting

Week 1     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       9

            How-to’s on making the best impression

                Addressing eight (8) student learning needs

                Week 1 Administrative check list

Week 2     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     12

            Week 1 review and reflection

                Developing classroom awareness – compiling student profiles

                Tips on giving quizzes

                Effective use of grading help

                Week 2 Administrative check list

Weeks 3 and 4     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     15

            Discussion of Classroom Observation #1

                Compiling a teaching profile

                Keying lesson plans

                Preparing for Exam I review

                Weeks 3 and 4 Administrative check list

Weeks 5 and 6     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     17

            Exam I analysis

                Keying lesson plans, updating profiles

                Weeks 5 and 6 Administrative check list

Weeks 7 through 9     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     19

            Administering and interpreting teaching evaluations

                Keying lesson plans

                Updating teaching profile

                Self-reflecting to heighten effective teaching

                Weeks 7 through 9 Administrative check list

Weeks 10 through 12     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     22

            Developing specific student learning skills

                Four specific exercises

                Developing student independent learning skills

                Weeks 10 through 12 Administrative check list

Weeks 13 through 15     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     26

            Discussion of Classroom Observation #2

                Updating teaching profile


                Final Exam preparation—maintaining the pace

                Compiling a teaching portfolio

                Weeks 13 through 15 Administrative check list

Appendix     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     28

            Resource List

                Mathematical Sequences Course Listings



Why a GTA Handbook?


It is often not until graduate teaching assistants (GTA) are well into their first teaching assignments that they come to realize that there is much more to the transmission of knowledge than originally meets the eye.  Consequently, common complaints such as the following begin to emerge:


  • The group of tuned-out students that sit in the back of class really annoy me.
  • I ask if there are questions but usually get little response, so in my class there is not much classroom interaction.  Students just aren’t interested in participating.
  • Attendance is dropping; I don’t understand why students don’t come to class.
  • My teaching evaluations were not good.  Students seem to write whatever they feel like at the moment.  Their remarks bear little resemblance to the actual class.
  • I am expected to write a syllabus or an exam, first-time things for me, yet there is practically no guidance on how to do so.


Experienced teachers understand that such complaints, some of which they themselves once had, are indicative of missing teaching skills, strategies, techniques, and course organization.  Consequently, the purpose of this Handbook is to guide the new GTA through his/her first teaching assignments.  It is designed to be used in conjunction with the Mathematics Mentoring Program.



What is the Mathematics Mentoring Program?


This Program provides for the pairing of a mentee, who is a new graduate teaching assistant or an older graduate assistant new to teaching, with a mentor who is teaching the same course or who has previously taught the course to which the mentee is assigned.  Mentors include tenured faculty, instructors, and experienced GTAs.


Mentees meet with their mentors on a regular basis.  During these meetings, mentees explore effective, course-specific teaching skills, techniques, and strategies.  Course policies, course organization, and additional technical/administrative information are also discussed.  In addition, GTAs are observed by their mentors twice throughout the semester, and mentees, in turn, are expected to observe other experienced teachers.


According to The Association of Graduate Schools in the American Association of Universities:


Since virtually all doctoral candidates, whether or not they

enter the academic sector, will be engaged in not only the

creation but dissemination of knowledge, the skills acquired

in learning how to teach will be fundamental to their future


And according to the magazine Penn State, Quality of Instruction, “teaching has always been recognized by the members of the Penn State community and the public as a major component of the University mission.  A recent statement of the strategic goals of the University underscores the important of the instructional function:  The quality of teaching and learning at Penn State ultimately determines the University’s impact.  Academic quality, therefore, is our highest priority.”


Consequently, the Mathematics Department believes its Mentoring Program to be a vehicle through which the University mission of excellence in teaching is supported and enriched.

GTA Pre-semester Preparation Checklist

(explanations of the listings follow)


§        Mentoring Meeting

Your mentor will arrange a meeting with you, usually via email, the week before classes begin.

  • Departmental GTA Training and Induction Program

This program, organized by Dr. John Roe, Mentoring Program Director, is for all new graduate teaching assistants.  It is a three-day orientation session, held the week before classes begin in August.

  • Course Coordinator Meeting

This meeting is conducted by the course coordinator for everyone teaching the course that you are teaching and is usually held a day or two before classes begin.  Be sure to receive the name of your course coordinator during the Training and Induction Program.



Mentoring Meeting


At this meeting with your mentor, you will be expected to discuss a number of issues related to teaching.  You will be asked such things as:


  • Do you have a teaching philosophy?  Historically, the relationship of institution to students was that of “in loco parentis”.  That is, teachers were viewed as and considered themselves to be substitute parents.  Today, however, that is not always the case.  While many teachers believe this role to be appropriate, others only do during the freshman and/or sophomore years.  Still others believe such a role is not appropriate under any circumstances.  What do you believe?  What do you expect in a teacher?  Is a student a near-adult needing guidance or an adult fully responsible for his/her learning?  What experiences have you had that will help you with such questions?  Because questions such as these evoke different responses among the faculty, you will find that teaching policies vary within our department.


To help you develop a general teaching philosophy, think about what your present beliefs are in terms of teaching and the learning process; of what you expect in a teacher; and of what you expect in a learner.  If you are not familiar with the American public school system, ask your mentor (or the Mentor Coordinator) for resource materials that will deepen your understanding.  The general teaching philosophy that you adopt will develop more fully as your knowledge of teaching and learning increases over the next fifteen weeks.

  • Do you know where your classroom is located?  It is a good idea to visit, beforehand, the classroom where you will be teaching.  This permits you to:


be certain of its location;


familiarize yourself with its size and layout;


assess the amount of blackboard space; how much of the

blackboard can be seen from the back of the room; and how

large you will need to write to ensure easy reading from the

back of the room;


survey visual aid capabilities and other features you may

consider important to your teaching.


  • What will be your personal classroom policies regarding quizzes, homework, participation, office hours?  Pick up a syllabus of the course you will be teaching at the GTA Training and Induction Program in order to familiarize yourself with it.  Although consistency among course classes is desired, a variation of personal classroom policies is encouraged.  Because you will be expected to discuss your personal classroom policies at your coordinator’s meeting, it is worthwhile to talk it over with your mentor beforehand.


Notice that the syllabus packet does not inform students of the number of quizzes that are to be given; how the quizzes are to be graded; and whether makeup quizzes are permitted and under what conditions make-ups may occur.  Consequently, you will need to provide such information.


Further, on the first page of the syllabus packet, under Grades, you will see that 50 or 100 quiz points are assigned.  Instructors often split these points in order to have a homework policy, an attendance policy and/or a participation-point policy.  If you would like to do this, the following information, together with your mentor’s suggestions, may help you develop specific policies and the accompanying point distribution.  Examples include:


            Homework policy

            homework is never assigned;


            homework is assigned but not turned in;


            homework is assigned and graded either weekly or per class:

                        usually 8 or 9 problems are assigned and graded,

                        more than 10 problems are assigned but only 8 or 9

                        are graded,

                        late homework may or may not be accepted;


            other suggestions by your mentor.



Attendance Policy

            an attendance sheet is posted for students to sign before or after



            attendance is taken visually once student’s names are learned;


other suggestions by your mentor.


Participation Point Policy

While rarely used, some department faculty greatly favor it.

Discussion with your mentor will provide you with the help

you need should you want to institute such a policy.


Office Hours

Office hours are required of all who are currently teaching—

at least two hours per week, which is the average.  Various

methods are used in the selection of these hours.  For example:

-office hours are set before classes begin;

-office hours are based on class discussion:

                        -students are given a choice—majority wins

                        -students submit their schedules so that the instructor

                          may choose the most appropriate times


You will need to provide information about your policies, in writing, to your class.


  • What is a lesson plan?  Sample class session plans are contained in your copy of The Penn State Teacher.  You will receive a copy from you mentor.  Your mentor will also help you with your daily lesson plans until you feel you are able to design them on your own.


  • Will you use a grader?  Each instructor is permitted 2 hours per week per course of grading help throughout the semester, given that class size exceeds 40 students.  A request form will be placed in your mailbox before classes begin.  You will be shown your mailbox location during the department tour segment of the orientation seminar discussed below.  You will be asked to complete and return the form as quickly as possible, if you choose grading help, so that you may be assigned a grader within the first week of classes.


Thinking about the issues listed above before your mentoring meeting will help you and your mentor make this the most productive session possible.


Departmental GTA Training and Induction Program


This three-day seminar, conducted by Dr. John Roe, is intended to provide you with crucial information necessary to perform effectively as a graduate assistant and teacher.  Here is a partial outline of the program:


§         Class Administration

§         Tour of Department

§         Technology Training

§         T.A. Responsibilities

§         Introduction to the Administration of a Class

syllabus, class lists, grade sheets, exams, classroom assignments, etc.

§         Student Information

composition, education backgrounds, etc.

§         Strategies for Teaching

case studies and panel discussions

§         Evaluation and Assessment

exams, homework, quizzes, etc.

§         PSU Policies and Issues


If you do not already know the names of your mentor and/or course coordinator, be sure you receive this information at the program.



Course Coordinator Meeting


This meeting with your course coordinator will provide you with:


§         Packets of syllabi and course policies to distribute to

students the first day of class

§         An estimate of current class enrollment

§         Notification of which exam(s) you are to construct.  Usually you

are paired with a fellow instructor for this purpose

§         An opportunity to ask any questions you may have about teaching

the course.  Experienced instructors will be invited to share any

course information they think will be helpful to you

§         Answers to questions you may have that have not been answered at

your mentoring meeting or the Training and Induction Seminar

Week 1:  The impression you make this week, particularly on the first day, is usually the one that most students will retain throughout the semester.  There are a number of questions uppermost in students’ minds at this time.  For example:


§         What is expected in terms of classroom behavior?

§         How many exams are there?

§         Will there be unannounced quizzes?

§         Will the class be a waste of my time?

§         Will I be able to learn from this instructor?

§         Will he/she be easy to talk to?

§         How much work will this involve both in and out of class?

§         Is tutoring help available?


The underlying concerns prompting these questions may be readily addresses by attending to such learning needs as:


  • A policy on classroom behavior

                        How will you react if a student reads a newspaper during class,

falls asleep, puts his/her feet up on a nearby chair?  What will you,

the instructor, choose to address and to ignore?  Your mentor will work

with you beforehand on the development of such a policy.


  • A copy of the course syllabus and course policies

                        As mentioned previously, these handouts will be given to each instructor

                        at the course coordinator meeting held just prior to the first day of



  • Additional policy information…

                        Regarding attendance, homework, missed quizzes and other related class-                                 time policies already formulated with your mentor.


  • An early assessment of the level of student prerequisite preparation

In these introductory classes, some students may need to be reassigned to a more appropriate level of learning.  There are a number of ways to ascertain this quickly:

-         Give a short quiz at the end of each class this week in which students recap key concepts or explain a specific concept in detail

-         Give a quiz after the third class

-         Call on as many students as possible during class in order to gauge their levels of understanding


You may then wish to talk privately, during your office hours, with students who score at either extreme of the scoring range, informing them of the opportunity to retest in order to enroll in the next higher sequence class, if that is the need, or to acquire tutoring help or to drop-down in the pre-calculus sequence.  In the latter case, this should be done under the guidance of your mentor as extreme care must be taken not to offend a student.


  • A GTA who is well prepared and presents the material with clarity

                        To help you achieve this level of preparedness, practice each lesson

beforehand.  This will help assure that:

-         You know exactly what you will be covering.  In so doing, you will be certain of the information you wish to emphasize, and that you are keeping pace with the syllabus.

-         You have worked each example and/or problem you will be using in class.  (Other non-text examples and analogies also aid understanding.)  This will help keep blackboard calculation errors to a minimum and to assess the clearness of your handwriting.  In addition, you will be better able to anticipate the amount of student-instructor interaction necessary to student understanding.

-         If you anticipate language problems, you have practiced using the blackboard to emphasize terms you have difficulty pronouncing.

-         You have developed an organized approach pertaining to administrative matters, such as writing announcements, class agenda and homework answers on the blackboard before instruction begins (arriving early helps); collecting and/or returning homework; and taking attendance.  In addition, you will have assured that the amount of class time needed for such activities is held to a minimum.


Your mentor will be of great help in this area as he/she has taught the course a number of times.  During this discussion you will also be urged to contribute your own ideas.  Please remember that teaching is a learning situation—for all of us.


  • Student participation in the lesson

                        This may be achieved in a number of ways.  For example, by

-         Frequent questions by you, the instructor, during the lesson.  In turn, the type of interaction, whether sparse, slow, uncertain, or frequent, will indicate to you the degree of student comprehension.

-         Participation in a short quiz at the end of class.  This, in turn, permits not only a daily assessment of student understanding, as previously mentioned, but, also, an extension of that participation into the next class period, i.e., by using the quiz results during your next lesson discussion with the students.

-         Submitting homework whether on a daily or weekly basis.  Collected, graded and then returned by next class period, homework assignments require students to take responsibility for their learning.  In addition, the instructor receives information useful in building student profiles.  (Your records will immediately indicate absences, lack of performance, etc.)

-         Being addressed by name.  Students appreciate this!  Consequently, they become comfortable with you not only during the lesson but also before and after class.  They are also more likely to visit you during office hours, to submit homework and to attend class regularly.  Consequently, they are more likely to perform at a higher level than might otherwise be the case.  In turn, you will have created a classroom climate conducive to learning.


Some of our math professors believe assigned seating during the first

            week is an excellent method of learning student’s names quickly.  Others

            take the class outside on the first day for a Polaroid picture of the entire

class.  Still others rely on constant interaction with students as their tried and true method—as they return quizzes and homework before and after each class.


  • Clearly defined student responsibilities outside the classroom

                        Students need to realize that most of their learning will occur outside the

                        classroom.  You may want to alert them to the two-hour rule, that for

every 50 minutes of class time, two (2) hours of study outside of class is

often required.  Students often assume incorrectly that they may easily get

by with less.  However, it is because they are at such introductory levels of

understanding that the two-hour rule particularly applies to them.


  • Information about tutoring help

                        In addition to office hours:

-         The Math Center drop-in tutoring help is available and is explained in the flyer included in each hand-out of the course syllabus and course policies packet.

-         Some instructors also conduct weekly review sessions.  (Your mentor will provide you with the necessary details concerning room reservations.)

-         You may want to encourage the formation of study groups as students become acquainted with each other.


Further, according to Biology Professor R. B. Mitchell, “If you are able to pay more attention to what the class knows than to what you do or do not know, and if you are able to improvise and respond to the class while making clear progress towards defined goals—then you are prepared enough.”


Administrative checklist:       Room reservation procedures

                                                Exam I construction responsibilities

                                                Grader request form submission

Week 2:  Quality teaching involves the use of the learning acquired during mentoring discussions; from other resources such as the literature and your associates; and during your daily classroom experiences.  Consequently, quality teaching involves a constant integration of the old with the new, just as does our own personal learning, no matter our field of major study.  During this week, then, as you prepare your lesson plans, it may benefit you to:


§        Review and reflect on last week’s activities

Are you keeping pace with the syllabus?

What questions do you have regarding the construction and/or results

of your first quiz?

How do this week’s lesson plans reflect last week’s mentoring discussions and teaching experiences?

On a scale of 1 to 5, what is the frequency of student participation, of overall student comprehension?

What other questions come to mind?

§        Develop classroom awareness

Which students are not turning in homework?

Which students participate the least and which the most?

How many absences have occurred?  Has a student been absent more than once or never come to class?

Which students scored poorly on the first quiz, which scored well?  Is there a correlation of quiz grade with homework performance, class participation, and/or attendance?


Review and Reflection


Keeping pace with the syllabus

Sometimes there is a tendency to move ahead of the syllabus believing its pace is too slow, and the elementary concepts are self evident.  However, your students have tested into these classes and, subsequently, lack such basic understandings.  The syllabus is paced to ensure that such understanding occurs.


This is not to say, however, that the syllabus is inflexible in its use.  For example, being slightly ahead of the syllabus, perhaps a half a class, is often beneficial because it permits greater review opportunities.  However, being two classes ahead is usually inviting trouble.  If you strongly believe that your class would benefit by this faster pace, then you may want to weave constant review work into each class period so that you are able to measure the degree of student retention and thus, be certain that your belief is correct.


Constructing the first quiz and/or interpreting quiz results

How many problems were there?  What kind of problems were given?  How long was the quiz?  Was it given at the beginning or at the end of class?  How many quizzes will you give during the semester?  How often will you give a quiz?  Will quizzes be given on the same day each week?


Be sure to discuss questions such as these with your mentor.  You will find that:

-         three to four problems is typical for a 10-12 minute testing period.  Have your mentor check their degree of complexity in order to assure appropriateness.

-         using old exams as examples of problem type is helpful.  This also serves to familiarize students with exam format.

-         quizzes are usually given at the end of class.  Students may then take advantage of extra after-class minutes.

-         quizzes are usually given once a week, often on Fridays.  However, both frequency and day do vary among faculty.


Constructing lesson plans that reflect gained learning and experiences

For example, as you discuss lesson plans with your mentor, point out some ways you are using what you have learned—in presenting a specific concept, in encouraging student participation, and in gauging the degree of student learning taking place.


This is a larger undertaking than it appears to be as you may need to step beyond what you feel comfortable in doing in order to increase your teaching effectiveness.  For example, you may feel the need to establish more eye contact, more student questioning, and/or more movement about the classroom—adjustments that will feel strange at first.


Developing Classroom Awareness


Reread the previous questions that appear under this topic heading.  Do you notice the common thread tying together the answers?  That is, you, the GTA, need to be engaged in some form of record keeping in order to answer each of these questions.  Maintaining a record of each student in terms of his/her homework (see the tips below), class participation, quizzes, exams and attendance performance, provides data that enhances your awareness of each student’s general performance at any point in time; presents contrasts among students; and permits analysis of each student’s learning situation.  This is called developing student profiles.


The data that student profiles provide will carry you into a deeper analysis of your teaching.  For example, if a student is doing poorly, you will be able to judge quite accurately the condition(s) responsible for such performance and the action needed to be taken.


If the student seldom turns in homework, has been absent, and so forth, then, clearly, the student is not acting responsibly.  How would you use this information?  What action would you take?


If, on the other hand, that student is working hard, is never absent, and so forth, then the student is taking responsibility, but…how would you use this information?  What action would you take?

At this point in time you may believe that paying such a degree of attention to your teaching will take up too much of your time.  That, of course, is a good point, however, it is not true.  As is true in your own academic learning, it is just a matter of organization.

Here are some tips that may help you:  (Excel and other similar spreadsheet software programs are useful in organizing such data.)


-         Have your grader keep a record of homework performance, enclosing a weekly update with the latest homework return.

-         Provide manila collection envelopes for homework pick up by your students/grader, and for homework returns, handouts, etc., not picked up in class.  Some GTAs and graders prefer to use GTA/grader mailboxes for pick ups and returns.  Some GTAs also prefer to keep a binder containing all lesson plans and handouts for easy referral and access to students.

-         Photocopy your class list without student ID numbers and use as monthly attendance sheets that students fill in based on either passing around or posting during class.

-         Photocopy your class list without student ID numbers and use as a record of quiz performance.  (You will receive a running record of individual exam performance after each exam.)

-         Learn student names as quickly as possible so that you may identify students readily.  This together with student profile information will enable you to handle various matters concerning individual students quickly and efficiently as they arise.

-         Ask your mentor, peers, and other associates for other tips.


Your mentor will also discuss with you at this time the details regarding your first classroom observation.  Sometimes GTAs wish to observe an experienced teacher before their own first observations.  Your mentor will be able to provide you with suggestions.


According to Prof. W. J. McKeachie, University of Michigan, “Good teaching involves more than communicating the content of one’s discipline; a good teacher needs to both motivate students to continue learning and to teach them the skills and strategies needed for continued learning.”


Administrative check list:      Student reminders of date/time/location of Exam I/conflict exam/make-up exam

                                                Date/time/location/procedure for grading of Exam I

                                                Classroom observation procedure

                                                Drop/add protocols

Weeks 3 and 4: By this time you will have had your first classroom observation.  Your mentor will discuss the results with you.  The assessment form you receive will be of great help in providing you with a profile of your current teaching capabilities.  Just as student profiles enhance your awareness of student learning needs, compiling a teaching profile of strengths and weaknesses will serve as a guide for enhancing your teaching abilities in the coming weeks.


Many GTAs find that it is good practice to use the teaching assessment as a check list when preparing lesson plans, and as a daily reminder of particular areas of performance that are less than desirable.  Consequently, it is often helpful to mentors that GTAs bring their teaching assessments with them to future discussions.  GTAs can then more clearly show how lesson plans are keyed to those results.


This is also the time to be reminding students of the upcoming exam and to provide them with the dates, times and locations of the conflict and make-up exams as well.  During Week 4, you will be providing students with practice exam packets.  The course instructors assigned to write the exam will notify you when they are ready for distribution to your students.  The intention is to provide students with review opportunities concurrent with the material that is yet to be learned and that is included in the exam.


Course instructors use these packets in a variety of ways:


-         Students work on the packets outside of class.  Questions are answered before and after class, during office hours, and/or during class.  (Instructors who conduct weekly reviews use the packets at that time as their primary review source.)

-         Specific parts of the packets are assigned as homework.  Questions are answered as suggested above.

-         Instructor and students work together on selected problems during part of each subsequent class.  Students are also urged to work on the packets outside of class.

-         Towards the end of each class, students break into groups to work on selected problems.  The instructor helps answer group questions.  Students are also urged to work on the packets outside of class.

-         Some combination of the above is used.  The instructor also helps students organize study groups.

-         Other suggestions by your mentor.


Review is important to both students and instructor.  For example:


-         Students become acquainted with the exam format and the types of questions that will be asked.

-         Students are provided the opportunity to firm up areas of weak comprehension, and of which they are often unaware.


-         Instructors are provided the opportunity to further assist those students who will need extra help in order to perform well.

-         Review often lessens the tendency for instructors to overestimate student comprehension.

-         Review often encourages student-student interactions which, in turn, encourage study group formation.



During this time frame it is important that you remember to continue improving upon

those activities that you believe aid the teaching-learning process.  For example:


-         Continue to work on learning students’ names so that teaching may become comfortable.  You may notice that as students come to know you, they will ask and answer more questions during class even though they may not be sure of the correctness of their responses.  And you, the instructor, will begin to sense an appropriate reaction to that student whose response is incorrect.

-         Continue to develop student profiles.  Doing so enables you to face each class more fully aware of each student’s current capability.  This is particularly beneficial during this review period.  In turn, you will be better able to tailor your review more specifically to students’ learning needs.


At this time, you should also verify with the course coordinator the time and location of Exam 1 as well as the grading procedure you are to follow.


According to Dr. L. S. Shulman, President, Carnegie Foundation for the Advancement of Teaching, “The taxonomy of learning impediment consists of three basic taxa:  amnesia – learning is not remembered; fantasia – students think they know what they do not; inertia – students subsequently cannot use what they think they have learned.”


Administrative check list:      Practice exam distribution

                                                Conflict exam/make up exam sign-up form submissions

Exam I date/time/place – classroom posting, including conflict/make-up exam information

                                                Review session room reservation

                                                Extra office hours/review session – postings

                                                Study group posting

                                                Exam grading process

                                                Exam return protocol

Weeks 5 and 6:  Exam I will have been graded and returned to students by this time.  You will have received a print out of the exam results from your course coordinator.  Your mentor will help you analyze this information.  For example, look through the listing to see if, in general, students performed as you expected.  If not, ask your mentor to help you with a further analysis in order to more clearly understand why. 


You may need to adjust teaching strategies as you move into your second segment of teaching.  This will mean re-examining current student profiles and current teaching practices.  Using additional information from your mentor, and from the rest of this section, will also be useful.


Calculate the exam average for your class and compare it to the course average.  If your class average is below the course average by more than a point or two, then further analysis, like that suggested above, will help you discover whether you simply have a lower-performing class, what other causes are responsible, or whether it is some combination of the two.


In addition to the tabulation of individual test scores, it is a good idea to also review the Response Table (Figure 1) tabulation that is also return with the results of a multiple choice exam.  (If you do not receive one, ask your course coordinator for a copy.)  This tabulation lists the percentage of students selecting each of the problem options.  It further delineates the degree of difficulty of the test and provides insight into the types of errors that are occurring. 

For example:


-         Study the A through D frequencies listed for each problem in order to determine the specific problems students found easy, moderately difficult, difficult and extremely difficult.  Then with the help of your mentor, examine the specific types of errors that were made.  Do they imply a lack of attention, retention or comprehension?

-         Notice that the resulting implications will determine the need for future lesson plans keyed to the particular types of errors that are occurring.

-         Use the additional plus and minus information contained in the Response Table to help you further analyze student responses.  The + symbols, for example, will indicate to what degree higher performing students were overconfident; and the types of errors such students make given this circumstance.  The – symbol will indicate ineffectual test items; and the quality of responses that make them so.  Your mentor will help you with these implications.


GTAs are often not sure of what further use to make of examinations.  Practices vary among faculty.  These include:


-         Posting the solution key either on the bulletin boards located on the first floor of McAllister Building or on office doors or close by.  Students are expected to refer to the key if they have questions.

-         Requiring corrections as a homework assignment.  Time in class may or may not be used answering questions--usually, a study of the solution key is sufficient.


The approach you choose often evolves from the analysis of the examination.


Your mentor will also discuss with you information pertaining to the teaching

evaluations to be conducted in the coming weeks.


According to Neil Johnson, Penn State CELT Program Director, “Classroom research is

the teacher’s systematic application of her (his) own investigative skills in the learning

environment for the express purpose of documenting what and how students are



Administrative check list:      Exam I analysis

                                                University Testing Services (exam analysis help)

                                                Evaluation protocol

                                                Exam II construction responsibilities

                                                Date/time/location for grading of Exam II

Weeks 7 through 9:  Students will be evaluating your teaching during Week 7.  Although you have the choice of whether or not to share the results of your evaluations with your mentor, it is a good idea to do so in order to get the most benefit out of the evaluations.  Your mentor will help you to interpret the results in terms of keying the review of each item to your current teaching practices.  For example:


§         Quality of the instructor (or of the teaching):

Tabulate the average student response.


-         If the response has been positive

Think about the good things that you are doing that cause such a rating.  You will want to reinforce such methods in the future lesson plans.


-         If the response has been negative

Discuss with your mentor ways to improve that incorporate additional students’ evaluations of the clarity of your examples, of your explanations, and of your responses to student questions.


§         Effectiveness of homework:    

Tabulate the average student response.  (This item is not included in Math 110 evaluations)



-         If the response has been positive

Recount the strategies that you use so that you will not forget them in future lesson plans.


-         If the response has been negative

Discuss with your mentor ways to improve—do you return such feedbacks as homework, quizzes and examinations promptly (during the next class); and how do you provide feedback during class time?


Students need to know you are making use of their evaluations.  After all, that is the point of mid-term evaluations—to be evaluated in terms of student thinking so that you may make adjustments that are more in keeping with their learning needs throughout the rest of the semester.  Consequently,


-         thank the students the next class period for their evaluations.

-         in future lesson plans, always try to think of appropriate comments that refer to their evaluations whenever you are using a teaching method or strategy that has been influenced by their evaluations.  Or if you need to return feedback more promptly—as you make that adjustment during the next return, mention their appreciation of early returns.


The self-reflection in which you are presently engaged and that has occurred over the previous weeks will serve to sharpen your teaching growth curve to the point that may now enable you to:


-         use your student profiles to single out those students in need of extra help before the second exam; and to prepare students for the coming exam with a review that, based on observed rather than assumed, strengths and weaknesses.

-         Use test feedback to address attention, retention and comprehension issues.  For example, you are providing learning opportunities that sharpen students’ abilities to differentiate between slightly contrasting meanings or details; to use pertinent algorithms readily and efficiently; and to explain concepts clearly and concisely.

-         Use the classroom observation and your teaching evaluations as teaching aids for each class preparation in order to

·        remind you of those areas in which you may need to improve, such as, pace or presentation. 

·        Continue those practices on which you were complimented, such as, approach or availability. 

·        Motivate yourself to make other adjustments, such as, in teaching style or in teaching strategies.

-         Use your mentor’s additional teachings not only to heighten your technical diagnostic expertise but to deepen personal analysis.  For example, at this point in time try to think about

·        Whether or not teaching is comfortable.

·        How easily you communicate with students.

·        Which concepts are most troubling to students.

·        Those students who were absent and the frequency of such absences.


Consequently, you may also find that as the review day approaches for Exam II, that you have already done much in the way of preparation.  For example, you


-         regularly include problems from previous material in homework assignments in order to boost retention.

-         solve a review problem or two at the start of each class.

-         use the practice exams as soon as they are received for additional review.

-         are working with poorer performing students.

-         provide opportunities for students to practice troublesome concepts.

-         have encouraged student interactions so that the formation of study groups is more likely.

Review day should be a natural extension of previous class activities.  If it is not, but is more like being asked to figure out an intricate schematic in 50 minutes, then we instructors have not done our homework.


According to Emerita Prof. Mary McCammon, Dept. of Mathematics, Penn State, “…just because you explain something logically and clearly does not mean that the students will understand everything the teacher dictated, and the students must understand what you are trying to teach them.”


Administrative check list:      Submission of conflict exam/make up sign-up forms

Student reminders of date/time/location of Exam II/conflict exam/make up exam

                                                Exam III construction responsibilities

                                                Teaching evaluation discussion with your mentor


Weeks 10 through 12:  During this time period mentors will urge you to take

advantage of the many insights you have gained to increase the learning success of your

students.  This task of increasing students’ ability to learn may be broken down into

elements too numerous to discuss at this point in time.  For the purposes of this

Handbook, the task is divided into two general areas—developing specific learning skills,

and developing independent learning skills.


Before beginning, be sure to conduct an analysis of Exam II similar to that of Exam I in

order to have a current performance profile of each student and of the class as a whole. 

Also, your mentor will have additional suggestions and be able to provide you with

further reading resources that will help you with this task.


Developing specific learning skills

The performance profiles of students at the introductory mathematics level usually

indicate weakness in attention, retention and comprehension.  The following exercises—

devised by Professors of Engineering Charles Yokomoto and Roger Ware, University of

Indiana-Purdue University—provide students with practice in strengthening such

weaknesses.  The exercises are provided in detail so that they may best provide ideas of

how to incorporate such practices into your own instruction.


Exercise 1 – Knowing Things Cold


Symptom:         “The exam was too long.”

                        “I knew the material but couldn’t finish.”


Cause:              Not enough mental rehearsal to put                   Experts such as

                        information into short/long term             accomplished musicians,

                        memory, thus student needs extra                      chess players and athletes

                        time for recall.  Erroneously                               execute basic skills without

                        thinking that you should not                               thinking.

                        memorize details translates to not

                        needing to remember anything.


Cure:                Use mental rehearsal and repetition                   Experts call on thinking

                        to know “cold” what can be known                   only when thinking is

                        cold.  Save valuable time for the                        required.

                        difficult thinking required in

                        problem solving.

                        Practice!  Practice!  Practice!


Exercise:           Write everything you know about a                   When information is known

                        specific task; write step-by-step rules                well, it can be linked with

                        for a procedure.                                                other information to solve


Exercise 2 – Learning a Procedure


Symptom:         “I couldn’t remember how to do it.”

                        “It takes me a long time to solve a problem.”


Cause:              Knowledge and information is thought   Math exams at the

                        of in general terms, not as procedures.   introductory levels usually

                        All solutions are derived from basic                   contain straight-forward

                        principles.                                                         problems which call for

execution of basic



Cure:                Select the basic procedure that the                    Students who ignore learning

                        problem calls for.                                              basic procedures must use

                                                                                                valuable time on

examinations to re-derive



Exercise:           Execute a basic procedure that is

                        well-known; write down the steps

                        in a frequently-used procedure.


Exercise 3 – Selecting an Appropriate Strategy


Symptom:         “I don’t know where to begin.”

                        “What’s the best way to solve this?”


Cause:              Knowledge and information are                         Successful problem solvers

                        viewed as an amorphous whole             are armed with a collection

                        rather than in terms of procedures,                    of useful strategies.

                        routines and strategies.  Satisfaction

                        with learning one way to solve a

                        problem prevents the investigation

                        of others.


Cure:                Know and be able to apply the             We need to teach students to

                        appropriate procedure or routine                       be mindful of strategies and

                        via the selection of an appropriate                     thus to provide practice in

                        strategy.                                                           the development of strategy

                                                                                                selection skills.


Exercise:           List different ways to approach a

                        particular problem; design a strategy

                        to solve the problem.

Exercise 4 – The Inverted Problem


Symptom:         “This is a trick question!”

                        “When knowns and unknowns are switched, I get confused.”


Cause:              Problem solutions have been                             To invert a problem is to

converted to templates.  There is                       re-word the problem so that

a lack of awareness that learning                       unknowns are the knowns  

must take place above the level             and vice versa.

of computational routines.


Cure:                Note that knowns and unknowns                      We invert problems so that

                        may be switched, giving the                               students may gain a better

                        original problem the appearance                        understanding of the inner

                        of a totally different situation.                             workings of the material.

                        Recognize that learning involves

                        knowing how to work problems

                        backward as well as forward.


Exercise:           Solve a homework problem that

                        has been rewritten so that what

                        was the unknown is now the given

                        and vice versa; articulate the

                        relationship between the two;

                        select a solved homework problem

                        and design an inverted problem.


Developing independent learning skills

Most of our students are usually in need of such development.  It is often evidenced by such remarks as:  “I couldn’t do some of the homework.”; “How can I do the homework when I don’t understand what’s going on in class?”; or “I think the page number was wrong, so I couldn’t do the homework.”  Such complaints indicate that these students are not taking enough responsibility for their learning.  They are not taking advantage of the tutoring help that is available; are not taking enough time to figure things out for themselves; or are engaged in some combination of the two.


The first cause may be easily dispensed with by discussing with such needy students the tutoring help that is available.  You may want to remind them also of your own office hours.


Treatment of the second cause requires more work on the part of both the GTA and the student.  However, it is time to develop, or if you are presently doing so, to reinforce, teaching strategies designed to provide independent learning opportunities.  Your students will soon be moving into courses within their majors where, as you know, being able to learn independently is essential to success.

Several indirect approaches might include:


-         limiting the number of problems that you rework from a quiz, examination, or homework assignment to one or two.  This will subtly force students into the habit of attempting to figure out problems for themselves and/or seek outside help.

-         Establishing administrative policies that require missed homework assignments be obtained from a fellow student; or that missed handouts and returns be picked up during office hours or from an envelope on your door or nearby.  Such policies place responsibility on the shoulders of your students rather than on yours.


Several direct approaches might include:


-         introducing a particular section of new material by asking the students for the information.  Break students into groups of three or four and have them work together on the examples.  Then ask them to derive the algorithm(s) involved.  You may need to plan this with the help of your mentor.  Your mentor will help you choose an appropriate section.  If you are teaching Math 004, the division of polynomials may be an appropriate choice; for Math 021, solving quadratic formulas; for Math 022, calculating asymptotes using a handout to provide clear, concise examples; for Math 026, simplifying expressions using sine or cosine addition formulas; for Math 017, patterns involved in Pascal’s triangle; and for Math 018, relating Gauss-Jordan elimination to solving systems by addition-subtraction.

-         Having students work independently on a similar exercise, or on a stepwise solution to an assigned problem immediately after introducing a new concept.


Use such approaches occasionally throughout the remaining weeks.  As a final note, you now know those students who are failing or are at risk of doing so.  Your mentor will work with you on how to best approach these students.  Based on their profiles, it may be best to encourage such a student to work harder so that he/she may pass; or to take advantage of the late drop and repeat the course without penalty.


Administrative check list:      Submission of conflict exam/make up sign-up forms

Exam III date/time/place – classroom posting to include             conflict exam/make-up exam information

Student drop deadline

Final Exam date/time/place – classroom posting

Final Exam conflict procedure

Exam III analysis

Weeks 13 through 15:  During this time period you will be observed by your mentor for the second time.  Your mentor will then discuss with you your current teaching strengths and weaknesses; to note those areas of greatest and least growth; and to help you to more fully recognize and appreciate the complex causal relationships between quality teaching and quality learning.  Thus, if your mentor were to ask you the following, how would you answer?


-         Name and explain the importance of a teaching practice you would like to improve.

-         What are the teaching and learning ramifications when both students and teacher explain the whys of proofs and applications?


At the same time, however, students have an equally active role to play.  If a student is consistently absent, fails to turn in homework, or exhibits other half-hearted learning behavior, then your greatest teaching efforts will be of no benefit to the student because he/she has chosen not to take advantage of them.  The teaching-learning process can only operate at full throttle when both teacher and learner are doing so.


Caution:  During these final weeks you will be teaching the last of the new material which is often the most demanding of students.  At the same time, attendance often falls as students become taxed with end-of-semester demands.  Consequently, to ensure that you are reinforcing students’ resolve to do well,


-         depend upon the teaching strategies, methods and techniques that you have developed

-         engage in further discussions with your mentor

-         reread this booklet for additional ideas


When all new material has been presented, the final homework assignment and quiz given, quiz grades submitted, administrative details finalized—when everything is on target—it is also a good time to pull together all that you have learned during this semester into binder form so that you have easy access to such information during your next teaching assignment.


This is call compiling a teaching portfolio, and it is no more than what you would do after completing a semester of academic study.  Do you save class notes with pertinent diagrams and algorithms added for future referral—and past quizzes and examinations with corrections in the margins?  To compile a portfolio you would do similarly, saving notes on your mentoring discussions; samples of effective teaching strategies, methods, techniques, and skills that you used; accompanying analyses and syntheses—and classroom and student observations together with the critical self-reflections they generated.  Your mentor has access to sample portfolios by teachers from a variety of disciplines across campus and will be glad to share them with you.


Consequently, just as past academic learning is the building block for future knowledge, so, too, a teaching portfolio provides the building block for growth in your future teaching assignments.


In addition, if you intend to pursue a career in academia, compiling a teaching portfolio is an effective way to represent your scholarship in teaching just as compiling a portfolio of your research efforts will represent your scholarship in research.  Developing the portfolio habit now will significantly ease future work in this area of professional preparation.


According to Professor M. Secor, Department of English, “Real (teaching) spontaneity, enthusiasm, and energy in the classroom are not at all opposed to conscious reflection…it’s like playing the piano.  Thus music sounds energetic and spirited and spontaneous only after enough practicing and self-reflection have occurred.  In music, it is like the beautiful, free rubato of Chopin, released within the bounds of strict form and tonality.”


Administrative check list:      Final Exam construction responsibilities

                                                Extra office hours – posting

                                                Final Exam review session room reservation

                                                Date/time/place of Final Exam review session – posting

Date/time/place of Final Exam – posting to include conflict exam information

                                                Date/time/place of Final Exam grading

                                                Course quiz grade submission procedure

                                                Course grade assignment procedure

Resource List



See your mentor or mentor coordinator for copies of:


The Penn State Teacher II:  Learning to Teach; Teaching to Learn

Enerson, Johnson, Milner, and Plank

The Center for Excellence in Learning and Teaching

July 1997


Penn State:  Quality of Instruction; Surveys of Students and Teachers at University Park

Willitz, Moore, and Enerson

The Center for Excellence in Learning and Teaching



The TA Coordinators Handbook:  A Summary Report of Best Practices at Penn State

The Center for Excellence in Learning and Teaching



Teaching Tips:  Strategies, Research, and Theory for College and University Teachers

McKeachie, Chism, Menges, Svinicki, and Weinstein



Teaching First:  A Guide for New Mathematicians

Thomas Rishel

Mathematical Association of America, 2000


How to Teach Mathematics

Steve G. Krantz

American Mathematical Society, 1999



Additional Instructor Help


University Testing Services

(for help in exam analysis)

23 Willard Building, 863-2802


The Center for Excellence in Learning and Teaching

(offering teaching workshops)

236 Grange Building, 863-2599



Campus Organizations


Graduate Student Association

ITA (International Teaching Assistant) Support Group

See Mentor Coordinator for current updates

Mathematical Sequences


Algebra/Geometry Sequence (pre-calculus)


MATH 004, 021 GQ, and 022 GQ form a three-semester sequence in intermediate algebra, college algebra, and analytic geometry. MATH 026 GQ is a one-semester course in plane trigonometry.  Students are placed in the appropriate course depending on the results of the mathematics proficiency examinations.


004. INTERMEDIATE ALGEBRA (3) Algebraic expressions; linear, absolute value equations and inequalities; lines; systems of linear equations; integral exponents; polynomials; factoring.  This course may not be used to satisfy the basic minimum requirements for graduation in any baccalaureate degree program.

Prerequisite:  MATH 003 or satisfactory performance on the mathematics proficiency examination.


021. (GQ) COLLEGE ALGEBRA I (3) Quadratic equations; equations in quadratic form; word problems; graphing; algebraic fractions; negative and rational exponents; radicals.

Prerequisite:  MATH 004 or satisfactory performance on the mathematics proficiency examination.


022. (GQ) COLLEGE ALGEBRA II AND ANALYTIC GEOMETRY (3) Relations, functions, graphs; polynomial, rational functions, graphs; word problems; nonlinear inequalities; inverse functions; exponential, logarithmic functions; conic sections; simultaneous equations.

Prerequisite:  MATH 021 or satisfactory performance on the mathematics proficiency examination.


026. (GQ) PLANE TRIGONOMETRY (3) Trigonometric functions; solutions of triangles; trigonometric equations; identities.

Prerequisites:  MATH 021 or satisfactory performance on the mathematics proficiency examination; 1 unit of geometry.


Calculus Sequences


MATH 110 GQ, 111 GQ and MATH 140 GQ, 141 GQ are two sequences that discuss differential and integral calculus.  They differ in the areas where calculus is applied.  The MATH 110 GQ, 111 GQ sequence includes applications from business; the MATH 140 GQ, 141 GQ sequence includes applications from the physical and engineering sciences. A student who wants to change from one sequence to another should consult with the chair of the mathematics department.


110. (GQ) TECHNIQUES OF CALCULUS I (4) Functions, graphs, derivatives, integrals, techniques of differentiation and integration, exponentials, improper integrals, applications.  Students may take only one course for credit from MATH 110, 140, 140A, 140B, and 140L

Prerequisite:  MATH 022 or satisfactory performance on the mathematics proficiency examination.

111. (GQ) TECHNIQUES OF CALCULUS II (2) Analytic geometry, partial differentiation, maxima and minima, differential equations.

Prerequisite:  MATH 110.


140. (GQ) CALCULUS WITH ANALYTIC GEOMETRY I (4) Functions; limits; analytic geometry; derivatives, differentials, applications; integrals, applications. Students may take only one course for credit from MATH 110, 140, 140A, and 140B.

Prerequisites:  MATH 022, 026; or MATH 040 or 041; or satisfactory performance on the mathematics proficiency examination.


141. (GQ) CALCULUS WITH ANALYTIC GEOMETRY II (4) Derivatives, integrals, applications; sequences and series; analytic geometry; polar coordinates. Students many take only one course for credit from MATH 141 and 141B.

Prerequisite:  MATH 140, 140A, or 140B.


Other Math Courses


017. (GQ) FINITE MATHEMATICS (3) Introduction to logic, sets, probability. Prerequisite:  2 units of high school mathematics.


018. (GQ) ELEMENTARY LINEAR ALGEBRA (3) Matrices and vectors; transformations; systems of linear equations; convex sets and linear programming. Prerequisite:  2 units of high school mathematics.


030. (GQ) PROBLEM SOLVING (3) Concepts in problem solving; reducing new problems to old ones; techniques for attacking problems; building mathematical models.


035. (GQ) GENERAL VIEW OF MATHEMATICS (3) Survey of mathematical thought in logic, geometry, combinatorics, and chance.


036. (GQ) INSIGHTS INTO MATHEMATICS (3) Examples of mathematical thought in number theory, topology, theory of symmetry, and chance.

Prerequisite:  1 unit of algebra or MATH 004.


041. (GQ) TRIGONOMETRY AND ANALYTIC GEOMETRY (3) Straight lines; circles; functions and graphs; graphs of polynomial and rational functions; exponential and logarithmic functions; trigonometry; conic sections.

Prerequisite:  MATH 021 or satisfactory performance on the mathematics proficiency



140A. (GQ) CALCULUS, ANALYTIC GEOMETRY, ALGEBRA, AND TRIGONOMETRY (6) Review of algebra and trigonometry; analytic geometry; functions; limits; derivatives, differentials, applications; integrals, applications.  Students may take only one course for credit from MATH 110, 140, 140A and 140B .

Prerequisite:  satisfactory performance on the mathematics proficiency examination.


140B. (GQ) CALCULUS AND BIOLOGY I (4) Functions, limits, analytic geometry; derivatives, differentials, applications from biology; integrals, applications from biology. Students may take only one course for credit from MATH 110, 140, 140A, and 140B.

Prerequisites:  MATH 022, 026; MATH 040 or 041 or satisfactory performance on the mathematics proficiency examination.


141B. (GQ) CALCULUS AND BIOLOGY II (4) Derivatives, integrals, applications from biology; sequences and series; analytic geometry; polar coordinates.  Students may take only one course for credit from MATH 141 and 141B.

Prerequisite:  MATH 140B.


200. (GQ) PROBLEM SOLVING IN MATHEMATICS (3) Mathematical ways of thinking, number sequences, numeracy, symmetry, regular polygons, plane curves, methods of counting, probability and data analysis.  For elementary education students only.


220. (GQ) MATRICES (2) Systems of linear equations; matrix algebra; eigenvalues and eigenvectors; linear systems of differential equations.

Prerequisite:  MATH 110 or 140.


230. CALCULUS AND VECTOR ANALYSIS (4) Three-dimensional analytic geometry; vectors in space; partial differentiation; double and triple integrals; integral vector calculus.  Students who have passed either MATH 231 or 232 may not schedule MATH 230 for credit.

Prerequisite:  MATH 141.


231. CALCULUS OF SEVERAL VARIABLES (2) Analytic geometry in space; partial differentiation and application.  Students who have passed MATH 230 may not schedule this course.

Prerequisite:  MATH 141.


232. INTEGRAL VECTOR CALCULUS (2) Multidimensional analytic geometry, double and triple integrals; potential fields; flux; Green's divergence and Stokes' theorems.  Students who have passed MATH 230 may not schedule this course. Prerequisite:  MATH 231.


250. ORDINARY DIFFERENTIAL EQUATIONS (3) First- and second-order equations; numerical methods; special functions; Laplace transform solutions; higher order equations.  Students who have passed MATH 251 may not schedule this course for


Prerequisite:  MATH 141.


251. ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (4) First- and second-order equations; special functions; Laplace transform solutions; higher order equations; Fourier series; partial differential equations.

Prerequisite: MATH 141.