A Handbook for Mathematics Teaching Assistants
(Preliminary Edition)
Tom Rishel, Cornell University & The MAA
Contents
- Introduction
- Types of TA Assignments:
Recitation, Lecture, Grading
- Before You Teach: A Checklist
- Day One
- What Goes On in
Recitation?
- What Should be
on a Syllabus?
- Lesson Planning: Survivalist Tactics
- Grading Issues
- Cooperative Learning
- Technology
- Writing Assignments
- Making Up Exams
and Quizzes
- Using Cognitive Models to
Make Appropriate Problems(with Mary Ann Malinchak Rishel)
- The Active Classroom
- "What Was That Question Again?"
- Motivating Students
- How to Solve It
- Course Evaluations
- Get Along with
Colleagues
- What is a Professional?
- Teaching Methods for Various Types
of Classrooms
- Problems of and with Students
- Student Types: Who is the Audience?
- How to Get Fired
- Advice to International TAs
- Silly Stuff...
- ...And Not So Silly
Stuff
- The Semester in
Five Minutes
- Jobs, Jobs, Jobs
- Letters of Recommendation
- Mathematical Talks
- Becoming a Faculty
Member
- University and College Governance
- What Does an
Evaluator Evaluate?
- The Essence of
Good Teaching
- Case Studies
This is a text about teaching college
mathematics.
My view is personal, informed by over forty
years in higher education, over thirty of them teaching in some form or
another, and almost twenty of those involved with training and
evaluating teaching assistants and junior faculty. If I seem to
emphasize first-and second-person narrative in my writing, it is
because much of this text has come, literally, from discussion with
you, the TA or junior faculty member, about the real world situations
we are encountering daily in our classrooms.
At most every juncture in the text, I
emphasize nuts and bolts considerations over theory. This is not
because I believe that theory does not exist or is not important, but
because I think that good teaching starts with seeming
trivialities--"talk loudly, write large, prepare carefully, explain a
lot, be friendly." Only after we are familiar with such simplicities do
we begin to feel comfortable moving into theories of learning. This
last is not to say that such theories are never useful or
important--otherwise, Mary Ann Malinchak Rishel and I would not have
written the long section on how using cognitive methods can lead to
better examinations, for instance. However, I do think that you, as a
graduate TA or a young faculty member, will profit more and improve
faster from short, simple, clear suggestions that have immediate and
obvious impact in your day-to-day classroom. If this improvement leads
you to decide that you want to think more deeply about your current and
future teaching, so much the better. Send me e-mail so we can talk trishel@maa.org.
Finally, let me address a very common view about the discipline of
teaching; namely, as I was told again just last week, "Teaching can't
be taught." Well, maybe, just maybe, great teaching is lightning in a
bottle and can't be explained, but I claim emphatically that good
teaching can be taught. Of course, I am biased in my view, if only
because I have spent the last twenty years (roughly) trying to achieve
this aim. But, in fact, I believe not only that teaching can be taught,
but that if mathematics is to progress, it must be taught -- to the
bright young people who will carry it on after us. I hope that, by the
end of this volume, you will agree with me.
So, let's stop talking and get to work...
Return to Top
Most teaching assignments for
graduate students fall into one of three categories listed in the title
of this section.
Probably the most common TA
assignment in mathematics, and the one with which the majority of the
faculty began their careers, is that of recitation instructor. Those of
you who have received an undergraduate degree from a large university
will be familiar with the lecture-recitation format: a faculty member
lectures to a large class of students two or three times a week on an
assigned topic from a textbook, after which a graduate student answers
questions about the lecture and discusses assigned homework problems.
In this format, the lecturer decides which homework to assign, and
often determines the structure of the recitation. By this I mean he or
she may say: "Don't do all the problems; just the ones that are
designated not to be turned in for grading." Alternatively, the
lecturer may suggest that you begin each recitation with a couple of
"example problems." Generally, however, most instructors will give you
little or no advice, except to say something like: "Just do a standard
recitation." (For a sample "standard recitation," whatever that may be,
see the later section, What Goes On in Recitation.)
Another common assignment for TAs is to be asked to lecture. Schools
vary as to when in a graduate student's career this is to be done; at
some institutions you are handed an algebra and trigonometry text and
told, "Go teach this. Don't mess up!" Other schools wait for a year or
two until you have had some less demanding assignments before they ask
you to plan lessons, make up your own exams, determine grading policy,
and generally deal with the problems of teaching undermotivated
freshmen (or worse, undermotivated seniors!) the joys of precalculus.
It is probably worth pointing out here that at some point in your
graduate career you should pursue a lecturing assignment, for two basic
reasons:
1. A graduate student who has lectured has a real advantage in the
job market (see the section, Jobs, Jobs, Jobs).
2. By lecturing before you take a first faculty position, you remove
some of the stress over teaching that goes into the tenure-pressure.
A third common TA assignment is that of grading, sometimes in an
elementary course, more often in an advanced undergraduate or even a
graduate course. Many TAs describe such assignments as "easy" or
"boring." While the assignments can be either or both, grading jobs,
however, can teach you how far you have come since the days when this
coursematerial was a real effort. These assignments can also show you
how hard it is to teach others to write clear, concise answers and
proofs. A third benefit to a grading job is that you can use it to
review the material that may be asked on a graduate comprehensive
examination. I will say more about the questions involved in grading
papers later on in the section titled Grading
Issues.
For now, think about:
Which type of TA assignment appeals to you most now? Is there one
that you might never want to do? Do you think that your opinions might
change later on in your career, or are they set in stone?
Return to Top
Return to Top
It is fitting that I begin writing this section
now, for today is the first day of the second semester. I have just
walked past a large lecture hall; the instructor is animated; students
are listening intently, wanting to know what is coming. For me, the
irony here is that I have passed this room during past semesters, often
observing several students sleeping or reading the campus news.
Day one of the semester is too important to throw
away. If all we do is call the roll and dismiss the class, what message
are we sending? Yet, many instructors do just that. "I didn't really
think about this class until now," maybe, or, "You don't need to be any
more serious about the material than I have been just now."
On the first day of class, students want to know
how the course will be run: what are the major topics, why is the
material relevant, and, of course, "How will we be graded?" In light of
these student interests, what can be done with day one? Here are some
suggestions:
Now let's discuss some aspects of each of the
categories above.
By the seemingly simple act of calling the roll,
you signal that you want to know the students. You will get to know
some names, and that will make the course more personalized. This can
lead to better attendance, fewer problems like cheating (since the
students feel more invested in the class, and since they know that you
know who they are), and better course evaluations for you at the end of
the course.
Handing out a syllabus is another common first day
activity. If you are new to teaching, you will have many questions as
to how to construct such a syllabus, some of which can be answered in a
later section, What
Should be on a Syllabus.
Many instructors assume that students will read what is handed to
them; I think this is incorrect. Every time I hand out a document,
whether it be a syllabus or a homework assignment, I read it to the
students. By reading through the syllabus, I allow students to ask
questions that I may not have answered clearly in my text, and I also
ensure that, within reason, students know what is required of them.
First-time graduate students are often teaching first-time
undergraduates. The undergraduates need to know how college operates:
"Should I bring my textbook to each class?" "Will you collect homework
every day?" "Do you answer questions during class, or do we wait until
later?" "Do you grade on attendance?"
More advanced students will have questions, too. Maybe they have
never had a mathematics course in college, or more likely, they just
want to know what the rules are: "I have lots of job interviews this
semester. Do you require attendance?" "Will you have answer sheets in
the library, the way they did last semester?" By the way, there is
nothing wrong with your answering, "I don't know; I'll check it out let
you know next class." Just make sure that you carry out your part of
this bargain and give them a definite answer at the next class.
As to more specific comments about how class is to be handled, we will
return to this topic in the section, Types of Assignments.
Students want to know whether and how often homework are going to be
collected. Will you grade each problem, or only some? How will they
know which? Do you have an idea of how you'll assign grades to the
homework? For instance, will you use a numerical system where each
problem is worth, say, from zero to five points? If you know what
system you or the course leader is using, now is a time you can tell
the students.
Similarly, you can describe when you will give exams, and whether
they will occur in class or in the evening. You can also describe where
the exams will be given, for instance, in a large lecture hall with 400
students, or in the classroom. You can also tell your class that "You
will have ninety minute exams, and I will show you some old exams for
review."
Then you can explain what you know of the final exam and grading
policies. Is the final cumulative? Does it have the same length as the
other exams? Does it count for more points than the earlier exams?
There are other bits of information you also should give: The names
of the texts for the course, your office hours, and any supplemental
texts or materials you will use.
Now that you have spent about twenty minutes on the nuts and bolts
of the course, it is time to turn your attention to content. What are
the topics your students will be learning? How do those topics relate
to other subjects they may be studying? In what ways will the material
be useful in the real world?
Let's be more specific about details; many of you will start
teaching with a first semester calculus course. You may want to say
something like this:
Calculus is usually split into two types: differential and integral.
Differential calculus deals with instantaneous rates of change: how
things change right now, not over six years or ten miles (those are
average rates of change), not over six seconds or six one-hundredth of
a second, but right now, this instant. We will be learning about this
instantaneous change this so-called derivative, how to find it, how to
manipulate it, and how to use it in problems from physics and chemistry
to business and economics. For instance, if the instantaneous change
takes place over time, then this derivative is the velocity of the
object that is moving, and this concept is of special interest to
physicists and engineers; it is one of their tools for explaining the
physical world. When Isaac Newton wrote F = ma, for instance, he was
saying that forces are related to acceleration, and acceleration is a
derivative, a rate of change.
Scientists are not the only people interested in calculus.
Economists and business people also use the subject; for instance, the
cost of doing business changes essentially instantaneously over time;
this change of cost is called marginal cost. Monitoring marginal cost
allows businesses to track their changes today, not over the last
twenty weeks or twenty months.
Then you might go on to explain how taking a derivative requires
having a function to work with; thus you will begin with a review of
some continuous and not-so-continuous functions. After that, you can
say that you will go on to talk about various methods of taking
derivatives of more and more involved functions, and then you will
discuss some applications of derivatives, such as how to maximize and
minimize profits, say, or maybe velocities, or areas of land.
At this point, I will leave as an exercise for you can decide what
you might want to say about integral and/or differential equations.
Meanwhile, let's shut the door on this first day calculus class, and
move down the hall to the precalculus class, where a more "activist"
discussion has begun:
Instructor (I):"... and we'll also talk about functions. Maybe some
of you have seen some functions, like, say, polynomials. Can you name
some functions that are polynomials?"
Two students together (S1and S2): S1: "Sure. y = axn + bxn-1
+..."
S2: "Unh maybe x2?"
I: "O.K. y = x2 works. It's a polynomial. Any others?"
S1: "x3?"
S2: "How about y = x2+ x + 1?"
I:"Yes." [Writing both polynomials on the board.] "Anything harder?"
S3: "How about the square root of x?"
I: [Writing y = x = x1/2 on the board.] "That one doesn't
work. Does anyone know why? "
[Silence. Then]
S1:" Cuz one-half is wrong."
I: "Good. One-half doesn't work as a power, right? I mean, y = (1/2)x2
is a polynomial, right? [Pause] So, this 1/2 points to the
power in x1/2 doesn't work--I mean, it's not 'legal' for
being a polynomial, although it is 'legal' for being some kind
of function, yes? (This [points] is called a power, by the way, and the
other is a coefficient of the polynomial. We'll define these terms
pretty carefully during the course..."
[A couple of minutes later.]
I: "How about some other kinds of functions? Have any of you heard
of trig functions? Can you name some?"
S1: "Sure. y= sin x."
I: "Yep, sine works. We'll study it, and the others, like cosine and
tangent and why they're all different from polynomials. 'Sine's'
picture, by the way,is, sin (x) right?
And, it comes up in spring and pulley mechanisms, and electrical stuff,
and things like that, and..."
Let's tiptoe away now, we get the idea.
This last instructor can teach us a lot about managing the
classroom. Notice how she accepted the answer she needed to her
first question, rather than going with the seemingly more complete
response from Student 1, who obviously knows a good deal of the
material she may be spending the semester teaching to the others in the
class. She also did a good job adapting to the incorrect answer y = x1/2
suggested by Student 3. She did so without emphasizing the student's
wrong answer; in fact, she turned a common mistake into a learning
experience for the entire class.
There are many good points to the classroom discussion we have just
witnessed, but in the interests of keeping the discussion short, let's
just say the following: Most people say that teaching precalculus is
boring, boring, boring, but this particular instructor doesn't make it
seem so.
Exercises:
Which of the two methodologies described above for a first-day
discussion of course material would you be more comfortable with?
Fill in the details of what you would say to a first semester calculus
class about the topics of integration and differentiation. (Your
answers may be nothing, of course, but you should then have an
explanation based on the syllabus.)
Return to Top
One typical format for a
recitation is this: The TA begins by asking if there are any questions
on the assigned homework problems. A student then asks to see "section
6.2, number 17." Other students chime in with "I couldn't do number
29," and "How about number 5?" Others ask for some problems from
section 6.3. One fairly quiet student says, "I wonder if you could do
an old problem from section 6.1?" Then, for good measure, another
student asks you to try one of the questions from section 6.4, the next
assignment, so we can see how they are done.
You, as the person in charge,
can field questions in the order in which they occur, taking section
6.2, number 17 before number 5 from the same section, say. Or, you can
ask for a list of all the problems at the start of class, collect them
on the board, and do them in the order in which they occur in the
textbook. The advantage of the first method is that you answer
questions in the order in which they arrive. The disadvantage is that
the student who couldn't do one of the easy problems may be totally at
a loss as to what you are talking about when you start off with the
hardest problem in the section. The second method solves the latter
problem, but only at the risk of "falling behind in the material." This
is a point you may not consider too important, but students always do.
A via media for making the best
of both methodologies is to collect the questions as above. Then tell
the students you will do the section 6.4 question "if there is time at
the end of class." Starting with current material, do two or three of
the problems from section 6.2, one or two from section 6.3, and then go
back to the one from 6.1. Finally, if there is time, you can "suggest a
hint to get people started" on the 6.4 exercise, which, after all,
essentially no one has looked at but the one student who asked. In this
way, you emphasize current material of most interest to the majority of
the class, while at the same time showing that you are willing to deal
with "old and new business" as time permits. And, by giving just a hint
as to how to do the new problem, you allow the entire class the
opportunity to puzzle out the secrets of that particular problem.
It should be clear by now that,
since recitation consists mainly of discussing homework problems, you
should show up on time and be prepared to discuss past and current
assigned problems. A shocking number of TAs and instructors try to
"wing it" often with unpleasant consequences for themselves, their
students, and for their end of term evaluations. So I will say this
again, with emphasis:
A recitation instructor will
show up on time prepared to discuss past and current homework problems.
No excuses are acceptable; this is part of your job.
This means that you will read
through all the problems the night before recitation, you will perform
the required computations (Yes, the chain rule is dull, and you have
used it so often before, but, just when you don't prepare a set of
problems because they're too easy, that's when you'll get stuck in
front of your class on the day before the exam.), and you will get "the
answer in the back of the book," because that's the one the students
prize so highly.
Why do you want to prepare
meticulously when you know this stuff so well? Because:
People never learn course
material as well as when they have to explain it to others.
Even though you took and passed this course some years ago, that
doesn't mean you can't learn from a refresher. After all, it was six
years ago in high school that you took AP calculus, right?
Textbook authors love to put little tricks into the exercises to keep
students on their toes; these tricks can trip up unsuspecting
instructors, too.
You are getting paid to do these exercises.
Even a TA who has done this course three times already needs to recall
where the pitfalls are placed.
You can probably add one or two
more well placed reasons to this list. Remember those reasons when you
decide to take a day off from preparing.
One final thought on this
topic: Course evaluations bear out the importance of instructor
preparation in students estimations of teaching. Even those faculty who
are described as "boring" and "unmotivating" usually receive overall
evaluations in the B-minus to B-natural range if students say that they
"can do the coursework" as shown by their being well prepared.
In this section, I have
emphasized the importance of being prepared in teaching recitations.
Preparation is important, but it isn't the only thing. For more
advanced advice, see the sections on The Active Classroom and Motivating Students.
Exercise:
Name some of the topics you think I have slighted or ignored in the
above discussion. How essential do you think they are to good
recitations?
Return to Top
Some departments keep syllabus
files, which provide a major impetus for institutional, not to mention
personal, memory. Even if such a file is not readily available, you can
still find out who taught your course last time, what books they used,
which chapters they covered, how pleased they were with the outcome,
and what they thought of the students. In the unlikely possibility that
the previous instructors all have retired or left town, you can get
some hints and advice from members of the curriculum committee, or
maybe even from the staff member who deals with the campus bookstore.
If all else fails, you can look at the appendix of this text for some
sample syllabi of randomly chosen undergraduate courses.
Enough said about how to find
old syllabi; now, what should yours describe?
First, give the name, number
and section of the course. Especially if multiple sections are taught,
you want to identify yours as specifically as possible. Also write
days, times and room numbers on the syllabus; e.g., MWF 10-10:50, 304
White Hall. Put your name on the syllabus (some prefer Professor A. B.
C. Jones others like D. Smith), your office number and hours if you
know them. If you haven't decided your office hours yet, promise to
write them on the board as soon as you do know them, and do so
often over the next few weeks.
It is worth saying here that
I am always amazed but probably should not be at how little verbal
information is processed, and I am reminded of this fact over and over
again throughout my career. ("I told the students that that topic would
be covered on the exam, but they didn't listen.").
Name the required and
recommended texts and readings, including edition numbers, specifying
which texts are required and which are recommended. Then explain which
chapters will be covered (Thomas-Finney, Chapters 1-7). If you are new
to teaching, you may not be sure as to which chapters are required. It
is very important that you find this information, for if students go on
to the next course without having seen some of the requirements, a lot
of people will be annoyed and upset, and people don't often forget what
and who caused their problems.
Also on the syllabus, discuss
homework, exams and grading in general; if you try to get too
specific about requirements, students will come back to tell you how
you have "changed your syllabus"--"unfairly," of course. (For
more details about grading schemes themselves, see the section entitled
Grading Issues. For now, we will
stick with what goes into the syllabus.) Will you be assigning homework
by the class? By the week? The month? The entire semester? Will you
collect and grade all the homework, or just some? If just some, will
you be announcing in advance which ones you will grade? When will you
collect these problems, e.g., "Right at the start of class each
Monday"? Do you want the homework written out in any particular format?
Then there are exams. Do you know when they are to occur? If so, put
that information into the syllabus, along with any other details you
may have, such as how long the exams will be and where they will take
place.
As far as grading is concerned, offer a general statement like,
"...three equal exams, along with a comprehensive final exam counting
double [alternatively, one-and-one-half?] the value of the exams.
Homework and class participation will also count about ten percent of
the total grade." In this way, you offer the students a framework,
while at the same time allowing yourself some leeway-- "what is class
participation" for instance, and how do you propose to measure it? Yet,
it's logical to suggest that such participation is worth something, and
you do want to have a mechanism for rewarding students who make an
extra effort.
At this point, syllabi often diverge, depending on course, material
and style. For instance, I have seen a few-- very few, actually
-- faculty put a short description of their academic credentials in the
syllabus. Others, especially those who are teaching in a fairly
nontraditional way, will feel the need to describe the classroom
situation as they see it happening. For instance, they might describe
how their "project-oriented" calculus sections will work, what kinds of
writing assignments they will offer in their geometry class, or how
they will handle group work in their precalculus class.
Regardless of what you put into your syllabus, it would be well to
remember that this document takes on the character of a contract with
the students; you are telling them what you plan to do, and in turn
what you expect them from them. Thus it behooves you to take a little
care with what you write. You might consider passing it by an older,
wiser faculty member for approval.
Courses often require unplanned or unexpected changes in midstream.
Most of these are acceptable to students. On occasion, however, some
adjustments you understand to be minimal or benign will elicit an
unexpected outburst "Why are you canceling exam three? I was counting
on that one to boost my grade! You can't do this unless the entire
class agrees, you know!" And then, heaven forbid, if you decide to
"take a vote" on the question, those students with higher grades plus
those who just don't want to take an exam along with those who just
want to get the course over with will simply outvote the three really
angry ones who want the exam. The ultimate outcome is that you end up
giving an exam you hoped not to give, while you have lost the respect
of, and authority over, your students.
No one can avoid all difficulties or see all the possible problems
about to appear. However, you need to think carefully about your
syllabus before you start the semester. That and getting input from
colleagues is a strategy that will make for a more coordinated course.
The outcome of such planning will then be better for you and for the
students, and will make your course less work in the long run.
Exercises:
What goes into your course? What would you add to the above
syllabus? Is there anything that you think should be subtracted from
the syllabus, and if so, why?
How would you resolve the problem discussed in this section of the
student who wants to take the third exam? Is he being unfair? Are you
wrong for suggesting that the exam be dropped?
Suppose you came to town on Thursday; it was
your first time at the college. Suppose further that classes begin on
Monday, that you have to move into your new apartment, register for
classes, wait for the cable to be connected, and oh yes you have to
start teaching your very first class on Monday morning. For what to do
on Day One, of course, you can look back to that section in these
notes. Of course, that doesn't let you off the hook that easily,
because you are stuck trying to build a syllabus. And then, once you
have given the students a general introduction to the course, you will
have to start making up lesson plans. Further, you won't just have
lesson plans for Tuesday (or, if you're lucky, Wednesday) you have to
plan an entire semester's worth.
In this section, let's consider the most
basic aspects of lecturing. Later, in sections called The Active Classroom, and Motivating Students, we will look at
more refined aspects of making such plans.
Once, some years ago when I was a graduate student teaching a night
course in third-semester calculus, I got the twenty-four hour flu about
an hour before class started. My office mate, being a very kind person,
offered to substitute for me. "Just tell me which section you were
supposed to do," he said.
The next day, after I had sufficiently recovered, I asked my
officemate how things had gone. "Fine," he replied, and went on to tell
me how far he had gotten in the material. "But how did you prepare?" I
asked. "Easy. I went in to class, announced that I was substituting for
you, asked to borrow a copy of the text, and gave the students a five
minute break while I looked over the author's approach to the material.
Then I made up three examples of varying difficulty, and I went with
it. By the way, how are you feeling?"
So there it is; a basic plan for lesson plans ("a plan for plans"),
courtesy of my officemate.
Start by finding out what today's topic is supposed to be.
Peruse the text to see how the author approaches the topic; this
helps you preserve the same notation as the text, among other things.
Prepare an intuitive explanation (a "heuristic argument") as to why
the topic is important, useful, and relevant.
Next, prepare a few homework-style problems of increasing difficulty
to illustrate to the students the main concepts of the section of the
text.
Allot remaining class time to answering questions or doing old
homework problems.
Of course, this methodology doesn't solve all problems. If it did,
teaching would be trivial. So, let's discuss some of the issues raised
in the above outline more fully.
One complaint often voiced is "But I don't like the way the author
does this section. Why should I encourage bad mathematics?"
Fair enough. Even though we may not have had a choice in the
textbook, the students will still be using it for explanations,
exercises and homework. We can offer alternative proofs or better
methods, but if the students are getting their homework from the text,
they would rather not have to keep "translating" from our language and
symbolism to the author's. Thus, we owe it to the students to at least
say, "Here's how the author approaches... An easier [more common,
better, more useful, more sophisticated] way is as follows. On the
homework and tests, use whichever method you like best. I don't care as
long as you get the right answer and can explain your method.
Another common objection is that we should not use "heuristic
argument" rather than an "honest, direct, complete proof."
This suggestion may simply be a function of audience level
(discussed more fully in the section Student
Types). Clearly, if you are teaching the intermediate value theorem
in real analysis or topology, you will want to consider the roles
compactness ands connectedness play in the discussion. But, for a
freshman English or biology major, some pictures of continuous and
discontinuous functions that have positive y-values at x = 1 and
negative y-values when x = 3 will be much more convincing than an
unintelligible, unmotivated "formal proof."
This last is also not to say that you can't be lucky and draw a
class of eager students in an enriched calculus program for potential
mathematics majors but now we are back to the Student Types question.
A third objection is, "Why do examples? They're right there in the
book."
You're right; there are worked out examples in the textbook. But,
first of all, many, if not most, students donÕt read the book.
Second, not every detail of the examples is spelled out in the author's
exposition. Further, it isn't always necessary to choose the examples
in the text; many instructors I know don't choose the author's
exercises. Instead, they opt for a few problems "near" the assigned
homework problems, telling the students, "If you understand how to do
these examples I'm showing you, you'll have a great start on tonight's
assignment." The underlying message is the "great motivator": "It's
worth watching me do these problems, because they're like the ones
you'll be trying soon."
One more objection to the proposed lesson plan is often brought up:
Is the suggested allotment of time for a lecture correct? That is, how
can you leave so much time for questions and homework? Don't you need
all that class time to explain the details of the current topic?
This, too, is a reasonable objection. Different instructors find
that they take different amounts of time to explain details of a
lesson. Still, I try to find ways to leave time for student questions;
otherwise, how do I know whether they are absorbing the material I
claim to be teaching them? The best way to find out if my lecture is
being received is to give the students a chance to tell me what is
still bothering them. I will return to this topic again in the Active Classroomsection.
Return to Top
It goes without saying that grading can bring
on problems. Many students seem to feel as if they "start out with
100%, and we [faculty] must justify the removal of each individual
point." At the same time, faculty sometimes take the exact opposite
approach.
Grading is best treated as a learning
situation for all concerned. The instructor learns how well he or she
has taught the material and designed the exam, while the student learns
how well he or she has absorbed the course information and studied for
the exam.
In mathematics, you will usually be grading
homework, quizzes or examinations. Other possibilities are that you may
grade writing assignments or class projects and, of course, you will be
involved in assigning final grades.
Homework is generally the easiest to grade;
the assigned problems are usually well written out in the text, and the
solution method is fairly clear.
A common, but not universal, technique for
grading homework is to assign each problem a fixed number of points.
Some graders use a two-point system, "0" for a wrong answer, "1" for OK
but not complete, "2" for fully correct. After using this methodology
once or twice, most graders find that it doesn't have enough points to
properly distinguish among the variety of possible errors that a group
of students can make. Students also tend to sense the same problem.
Their complaint about the grading is usually to say something like, "I
only got one number wrong, and all I got was a 1".
A zero to five scale is probably better:
"0"-- didn't even try the problem,
"1"-- tried, but not even close,
"2" and "3"-- various levels of somewhat valid but mistaken attempts,
"4"-- correct answer but with some minor errors,
"5"-- the correct answer with details spelled out.
Note the last comment: Only the correct
answer with details merits full credit. There will be points early in
the semester when students will ask you to reconsider grades because
they "got the right answer" without showing any supporting evidence as
to how they did so. You can use this as an opportunity to instill good
habits into the students. Explain to the questioner that he has lost
one point on this particular assignment for not clearly describing the
way he went about solving the problem. This is much easier than trying
to convince the same student that he should have lost twelve points out
of twenty for the same approach to a problem on the second exam.
This last paragraph points out a good
general principle, both for students and assistants. Homework time
should be used to instill good habits. For the student, this means
writing correct, clear, complete solutions. For the instructor, look to
make uniform, defensible grading with useful comments.
New TAs often ask how long comments on
papers should be. My response is usually "Not long at all." I say this
because it is fairly common for newer TAs to continue the solution to
the problem in the margin of each student's paper exactly from the
point at which the first error occurred. Students often don't read
these comments; sometimes they do read them, but still don't understand
what they did wrong.
There are a least two ways to reduce the
amount of commenting you need to do on homework. One way is to simply
put an "X" mark at the place where the first error occurs, and then
after all the papers have been graded, write up solution sets of the
most commonly misunderstood problems for all the students. A second way
is to start or finish the next class with a "couple of homework
problems lots of people seemed to have difficulty with."
Quiz grading is not dissimilar to that of
homework. You can use a similar point scheme, and again you can save
some grading time by putting the answers on the board when you hand
back the quiz. One difference that sometimes occurs, however, is that
if you are the one writing the quiz, you may occasionally find that
your question is inappropriate. Even if you are not writing the quiz,
but simply (remember, nothing is ever simple) choosing a problem from
the text, you may choose one that requires a piece of information that
you actually didn't lecture on. In that case, common sense should take
precedence over pure grading issues. Did you ask a question students
couldn't answer with current methodology, say? Then maybe you should
give everyone full credit for their valiant efforts, with extra credit
for the one or two who may have actually known how to solve the
problem.
Exam grading is also in many ways like
homework, although in this case careful preparation before grading can
save much time. There seem to be two models of mathematics exams: Those
that are given to classes of up to thirty students, and those for five
thousand (Well, maybe five hundred). In the first case, you end up
grading all problems on all the student papers. In the second, you tend
to grade only one problem but you must grade until you drop, and then
get up and grade some more.
Uniformity with fairness and speed are keys
to grading exams. Nothing is more disconcerting than finding at 3 a.m.
that you have graded 347 papers, an undetermined number of which were
done incorrectly. If you are grading 500 papers, carefully doing the
problem yourself before grading any papers is central to uniformity.
When you have a complete solution, make up a credit scheme before you
grade any papers. (A sample problem solution with grading scheme is
shown at the end of this section.) While you are proctoring the exam,
you can show your answer and grading scheme to other TAs for comparison
although it doesn't hurt to remember that this will be your problem, so
the final approach is your call, as well as being your responsibility.
My point here is that, within reason, you need to find a grading scheme
you are comfortable with, one you can defend.
As you grade the first few papers,
occasionally review your scheme to see if it still seems to fit what
the students actually knew and did. This review will also help avoid
grade inflation or deflation that seems so inevitable over ten hours of
work ("This is the same mistake that I've seen a hundred times now
well, this time you get a zero!")
Uniformity and fairness are related to one
another. You may be a harder grader than your officemate, but if you
can defend your methodology to other TAs and students, they will
"generally" accept it. (Note that last generally. Some may not; see the
section on being a good
colleague.)
Most TAs see the "speed" part of "grading with speed" as only being
of benefit to themselves--"I want to get this pile of papers done and
out of here!" But speed with accuracy also benefits students, because
they get to have their problems back while they still remember what the
questions were.
To aid in speediness, try some of the following:
- Grade problems "backwards" if the answer is correct, you can scan
the earlier parts to see if the details are there.
- If a few students have a unique, strange method of solving a
problem (this happens maybe five percent of the time), put these papers
aside for a while until you can let your subconscious work on where the
isolation may have come from.
- Do not write long comments on the examination papers; use the
advice given earlier in this section.
- Make an answer sheet to hand out to all the students. Go over the
answer sheet on the day on which you hand back the examination copies.
If students feel that they did not get graded uniformly, you can make
adjustments right after class.
- Don't fight with students over problems that were obviously
graded incorrectly; at the same time, don't capitulate over every
request for a regrade just because it was asked for.
If you are grading a full class of exams--thirty students, the
entire set of exam questions, say--grade problem one for each student,
then go on to problem two, etc. In this way, you will ensure more
uniformity. Also, try to grade each individual problem in one sitting;
take a break only after you have seen all the unique, exotic
methodologies the students can come up with. (By the way, I find that I
can sometimes bribe myself into grading by promising that I'll take a
break as soon as I finish these last eight copies of problem 3. Not
being very bright, I'm usually able to use that argument to convince
myself to work ten more minutes.)
Uniformity has other benefits. Among them: It leads to fewer
re-grades, which take a lot of time. It also makes for more defensible
scores, so that students consider the grading (and the grader) fairer.
After you have graded as many homework, quizzes and exams as you can
stand, you will have to assign final grades. Each department of each
university and college seems to have allowed its own system to evolve
and each of these systems is like each other, but not quite. For the
bare-bones description of one such system, check out the grading
section of What
Should be on a Syllabus. Note, however, that this section is not
completely forthcoming as to how allocation of final grades is done in
an standard class. Well, let us lift the veil.
I am occasionally in charge of a large number of calculus sections,
for instance 26 sections averaging 20 students each. Thus, by the end
of the semester, we calculus instructors have in the range of 500
grades to assign. Assume that we give three exams during the term
(these are called prelims where I come from), each worth 100 points. We
also administer a 150 point final exam (yes, it is called a final); and
additional materials, such as homework and quizzes, add up to 50 more
points. Thus students can earn a total of 500 points. If the exam is
scheduled for a Tuesday morning, we will spend that afternoon,
Wednesday and Thursday grading. By late Thursday afternoon or Friday
morning at the latest, each instructor will have collected finals,
recorded grades, and totaled raw scores. (Of course, there are always
one or two instructors who have failed to do the above; they should
read the section Get Along
with Colleagues.) We then have a meeting at which instructors put
up the raw scores of their students. This we do in ten point intervals,
from 500-491 to 210-201. Numbers below that fit into the 200-0
category.
We find the median grade (not the mean), and assign to its ten-point
interval the set of last B- grades. Working up and down the intervals,
we then assign an A range, a C level, and a D range.
| Interval |
Total |
Sum |
| 491-500 |
3 |
3 |
| 481-490 |
8 |
11 |
| 471-480 |
14 |
25 |
| 461-470 |
20 |
45 |
| 451-460 |
16 |
61 |
| 441-450 |
20 |
81 |
| 431-440 |
16 |
97 |
| 421-430 |
22 |
119 |
| 411-420 |
28 |
147 |
| 401-410 etc. |
Having put together a curve based on class scores, we now assign
letter grades to each student in each section. We are not done,
however. In each section there are grades that are anomalous. Some
students have one grade that is much lower than the rest, say. Others
have a rising set of scores, e.g., 49, 62, 87, and 130/150 on the
final, showing that they maybe have caught on later than others.
Occasionally, students will have personal problem. I discuss each such
anomaly about fifteen- percent of the total -- with the individual
instructor, and we come to some sort of consensus. We seem to end up
raising about half the grades, but no single grade ever goes up more
than one level, e.g., from C+ to B-.
One of my general feelings about grading is that students always do
less than or equal to their best on individual exams; but still, there
must be two or three exams where they perform to expectations theirs or
mine. Further, good homework and classroom questions may show interest,
but they are a precursor of good exam performance, not a substitute for
that performance. And finally, I have a thing about the grade of A+; I
will never raise a total below 490 points out of 500 to an A+. To my
mind, an A-natural is a perfectly wonderful grade, and I won't
apologize for giving it.
Return to Top
There are various ways to approach the
methodology called cooperative learning. You can suggest that students
do their homework together. You can offer them weekly study sessions
where they can sit together and work problems while you circulate
through the room offering hints and suggestions as to how to solve
problems. (For more details of how such a program worked for Myrtle
Lewin and me a few years ago, see [17].)
After a short introduction of a
particular topic, you can stop lecturing so as to let students try two
or three of the exercises together, after which you can have them
present their solutions at the board. You can make up worksheets for
students to use to discover mathematical concepts for themselves some
sample worksheets are provided at the end of this section.
You can teach a project-oriented type
of class, and then make up some really hard worksheets that the
students will need a few days and some help from you to construct
solutions for. Then the students can work in small groups writing up
their solutions. This method of cooperative learning is very
labor-intensive for all concerned, and is not one you should simply
blunder into. There are books available, however, to help you with the
details see, for instance, [8] and [15]. At the end of this section, I
have offered some of Matt Horak's calculus projects as samples.
You can assign major projects (or
final projects, or individual projects) in place of some exam or final.
Students can then report to each other on what they have learned and
they can evaluate each other's projects.
I have no doubt that you can think of
two or three other ways to encourage cooperative learning in your
classroom.
In addition to the question of how
cooperative sessions can be done, there is the more interesting
question of why you might want to do them. Faculty often express a
desire to have students actively engaged in the learning process; what
better way to do this than to get them to work the problems and
construct the examples?
An objection that is sometimes
raised to this last is, "But when six people work together, I can't
usually tell which of them are really working, and which are copying."
OK. "And, can you tell when they hand in their homework whether they
did it themselves or had someone else do it for them?" When you get
them working in class, you can walk around observing the dynamics of
groups; you can ask questions like "Where are you stuck?" and make such
comments as when your group has a solution, I'll ask one of you [not
saying which one] to present it on the board.
Return to Top
Technology is being used more often in the
mathematics classroom, from low-tech aids like overhead projectors and
microphones through mid-tech calculators to high-tech computers. The
more time you spend in teaching, the more you will be called on to use
some of these materials.
Perhaps it is just my own bias, but I find it
particularly annoying when an instructor comes to class unprepared to
use the necessary equipment. Ho ho, well, I brought these
transparencies, but I see they don't fit the projector. [You didn't
bother to check beforehand?] "And, I can't figure how to turn it on...
can anyone help me? Oh, and I see that my data disk isn't compatible
with the software and no, I didn't prepare any backups, and..."--well,
you were at that talk too, so we both remember it well, right?
Rule One: If you are using technology in
your class, test it beforehand. Have a backup in case the worst
happens.
Practice with the equipment before your talk or
class; turn it on, check the displays for visibility, set audio
equipment for sound clarity without feedback, make sure computer and
calculator displays are visible from the back of the room, check to see
that you have the proper cables and plugs for your laptop.
Rule Two: Make sure your examples
justify the technology you are using.
Why do you need a calculator with an LCD
display projector just to draw the graph of a parabola? Do you have to
load up Minitab or Datadesk to compute means or deviations for a sample
of size six? In other words, choose your examples to fit the equipment.
Let the calculator graph the function y = sin(x)/(1 cos(x)) so that you
can find all max and min for that function. Have the calculator graph y
= ax2 + 1 for various choices of a; then ask students to
describe how changing a affects the graph. Do the same for y = x2 +
k, for various k.
Use Minitab to find means and deviations of
census data, so as to eventually construct and test hypotheses for
yourself and the students to defend or disprove.
Rule Three: Make sure that your
overheads and displays fit.
A transparency with writing too small to read
may not (quite) be completely useless, but it certainly is frustrating
to the audience. It is not always possible to know how a particular
room or hall is going to be configured for a talk or class, but there
are many good reasons for putting less information on each individual
transparency and making the size of the type, font or print larger than
you think you will need. The next rule will discuss more details of
speaking from overheads; for now, let me continue with another
nuts-n-bolts comment.
Another frustration for the audience that is
simply solved is that of the shrinking violet who refuses to pick up
the microphone. He thinks his mumbling is sufficient, while the
audience knows that the solution is available right at the lectern, if
only he would use it.
Rule Four: Realize that teaching with
technology is not the same as lecturing.
The audience does not have time to take notes
or absorb the ideas being shown on transparencies. They are barely able
to listen to your presentation. Such problems especially occur in good
talks; it can be very frustrating for a listener who thinks that he or
she has just seen a remarkably good lecture but can't really reproduce
any but the most minimal parts of it. Such an audience has been more
entertained than taught. To alleviate this problem, you could consider
bringing individual copies of transparencies and displays for handout.
Speak slowly and allow ample time for
questions while overheads are still on display; recall that all but the
experts in the room need time to absorb what you are telling them.
Also, do not play "peek-a-boo" by covering over parts of
transparencies. If you don't want the audience to get too far ahead of
you, put less material on an individual transparency and write larger.
Another method of keeping interest is to offer people a related
exercise at the start of your presentation that (you can claim) that
they will be able to solve by the end of the talk.
Rule Five: Be prepared for total system
meltdown.
I once gave a lecture in Japan on the day a
typhoon hit. Five minutes after I began, all the electrical systems
failed. When I asked what to do, the audience said, "Just go ahead with
your talk." So I used chalk, wrote large on the board and spoke loudly.
Some of what I said must have gotten through, because afterward a
listener came up to me and very courteously pointed out an error I had
made.
Return to Top
A variety of small writing assignments
are usable in the mathematics classroom.
Have a supply of 3x5 index cards in
the back of the lecture hall for students to use to write questions
about the lecture. Answer the best or most frequently asked questions
at the start of the next class.
Ask an occasional quiz question in
class: "What's so fundamental about the fundamental theorem of
calculus?" "Describe one application of today's topic."
At the end of a solution of a
mathematical exercise, ask the students to describe the real-world
implications of the answer they just got.
Have the students write out a
description of the topics covered since the last exam, as well as why
those topics might be important or useful.
None of the above assignments takes a
long time to construct, nor is it difficult to grade. Yet each enhances
the students awareness of the usability of the classroom material.
Further, each asks the students to think a bit more holistically and
carefully about the somewhat deeper meanings of the materials they are
studying.
Of course, the above are only a small
sampling of the possibilities of writing assignments in mathematics.
For lots more, check Countryman [9] or Meier-Rishel [19].
Return to Top
Making up exams is both an art
and a science. If you do it properly, you get an honest appraisal of
your students understanding of the course and the material and approach
you have taken. At the same time, by constructing good exams, you can
avoid the pitfalls that make examinations time consuming to grade and
difficult because of post-exam complaints. If you don't do it well, you
can probably guess what I'm about to say.
So what are some of the steps
you can use to avoid pitfalls?
First, make a list, for
yourself and for the students, of the topics you have covered since the
last exam. If some of these topics are too time-consuming or not
interesting enough to test, say so. If you feel you need to test one of
the time-consuming topics, e.g., Newton's method, or Riemann integrals
evaluated by summing and using induction, you can consider testing them
by assigning a special overnight take home project.
Now that you and the students
know which general topics are to be tested, it is time for you to
decide, without the students help of course, how many and what kinds of
problems to assign in an average examination time period.
Let's say you have ninety
minutes. I sometimes tell students that I try to design a one-hour exam
and then give them ninety minutes to do it. This rather meaningless bit
of information seems to relax them. Generally speaking, stress
reduction before the exam is not a bad idea; there is a difference
between making an honest and fair, yet difficult, exam, and making one
which is simply filled with tension.
Suppose you have decided on a
five question exam, based on the fact that there were seven major
topics since the last test, and one of those is easy enough to skip,
while a second can be embedded in a later, more important topic. Make
about forty percent do-able by anyone who stayed awake long enough to
watch you show some examples on the board. (This is one of the reasons
I don't want to pass students who cannot get a 40% average on my exams.
See the section on Grading for
details.) The forty percent do not have to be just like trivial
homework, by the way; you might split some of your five problems into
easy, moderate and difficult sections, thereby spreading the easy stuff
around the exam.
Now you have sixty percent of the test left for more challenging
material. Half or a bit more of that can be similar to some of the more
interesting examples and homework problems the types of problems that
make students think, but this group had a chance to do that thinking
last week while they were doing their homework exercises. I basically
never give the students assigned homework problems on the exams, by the
way, although I do know some people who do. I just feel that using old
exam questions as homework problems often makes the students feel that
the instructor didn't really put an effort into the preparation.
Assuming that about twenty percent of the exam is still to be
constructed, it's now time for you to think of a more challenging
question -- or parts of questions. Now is the time to think, "What is
the essence of the material I have been teaching for the last four
weeks, and how can I ask the students to show whether they have
absorbed that essence?" This does not necessarily mean asking them to
formulate a proof; rather it should indicate that you could quiz them
about some fundamental points that you have been making repeatedly
during your excellent lectures. One effect of putting such questions on
the exam is to increase attendance at the rest of your excellent
lectures--"Wow! If I go to class, it might help my grade on the next
prelim!"
For many more details on how to make examination questions that hit
the mark, try the section Using
Cognitive Levels to Make Appropriate Problems.
Let me say a bit more about finding challenging problems. Early in
my career, I used to expend real energy trying to fashion a problem
that would force students to use current knowledge to discover
something new. For instance, I might be inclined to write, "You've seen
exponential growth. Well then, now I'll ask you to find out about
logistic growth all on your own." These well-meaning attempts almost
always turned out very badly. The "numbers" would turn out to be too
messy, and the concepts were too far from the students current
awareness. Further, thirty minutes or so was simply too little time for
serious thought. So, eventually, I came to the realization that at best
I could formulate a couple of problems that proceed from easy to
difficult, with the difficult part counting maybe only five points.
("You couldn't get that part? Well, good thing it was only worth five
points. OK, let me show you how...")
How do I do this? Glad you asked. Let's go back to the exponential
growth into logistic growth problem. We split it into four parts, each
worth five points:
In problem four, you found the rate of growth of a strain of
bacteria. Now let's suppose that the bacteria are growing in a lab on a
circular Petri dish whose area is 5 cm2. Thus it is fair to
assume that the area, A(t), covered by bacteria in the tube at any time
t is governed by the equation
a) dA/dt = k(5 A(t)).
If you know that A(0) = 1 and dA/dt = 0.2 at
t = 0, what value do you get for k? Is this k value positive or
negative, and what does it tell you about dA/dt?
Now writing b) dA/dt = k dt
5 A(t)
for your value of k,
solve this equation for A(t).
Your solution in part b) will have an arbitrary constant in it. Calling
that constant D, find its exact value.
Using your final solution to part c), make a reasonable argument that
A(t) is never larger than 5. What is your reasoning for this?
Notice that the above is still not an easy problem it wasn't
supposed to be. However, the first part should be manageable for any
student who understands what you have taught about exponential growth
and decay. Part b) is harder, of course, except that you have already
separated the variables in the equation the students have to solve.
Even if students found a wrong answer to b), you can still grade part
c) as if b) was correct. Thus they can still receive credit for part c)
without getting very many points for b) at all although they do have to
get some kind of reasonable answer for b). That leaves part d). It's
not easy, "but hey, at least it's only a five-pointer, right?" If you
now design one more problem with a hard five-pointer as part d) and you
are done. On this exam, it'll be easy to get at least 40%, the average
should be around 70 or 75%, and more than 90% shows that the students
have worked.
Before moving on, let me make a comment about quizzes: I tend to
make them relatively easy. For instance, if I lecture on the chain rule
on Friday, and maybe I've shown the students how to find the derivative
of sin2(3x), I might then ask them to use the last five
minutes of class to find the derivative of cos3(2x). Once
the students see that the quizzes are reasonably easy, they have
incentive to come to class and listen carefully to what I am teaching.
Further, the quiz is then easy for me; I can sometimes finish grading
in the fifteen minutes between classes, if no one stops me to ask
questions.
Return to Top
What makes examination questions easy
or difficult? To some extent, it is the students level of preparation
and their attitude toward taking tests. These are internally applied
forces coming from the test takers themselves. But, there are also
external stimuli at work here, such as the difficulty of the
examinations we instructors construct.
Some years ago, a variety of
individuals began to study what are now called cognitive patterns--the
ways by which individuals learn information. By now, a number of books
and articles have been written offering models of what are termed
levels of cognition; i.e., the levels of difficulty of concepts, ideas,
questions. Most of these texts are more applicable to the arts than to
the sciences or mathematics (c.f. [1], [2], [3]), but one which has
been successfully used in mathematics is due to Benjamin Bloom [3].
The Bloom model splits cognition
into six levels, from lowest to highest. These levels are: Knowledge,
Comprehension, Application, Analysis, Synthesis and Evaluation.
Knowledge, Bloom's lowest
category, pertains to whether the student has absorbed and can
successfully reiterate the concept being taught. For example, in a
first-semester calculus class we instructors might apply this principle
to teaching the definition of the word limit. To see whether the
students have absorbed that definition, we can ask the logical but not
particularly inspired question, "Can you define the word 'limit'?" This
is an acceptable question; for the students to master the concept, they
must be able to articulate it.
Let us now aim for higher levels of
questions by going to the next level in Bloom's model.
A note here before we go on: Bloom's
use of the word "knowledge" in reference to the lowest level of
cognition has sometimes been called into question. Many people have
suggested that they believe the word knowledge indicates more cognitive
awareness than Bloom seems to ascribe to it. Alternatives such as memorization
and recitation have been suggested. We believe that the word recall
would adequately express the concept that the student is trying to
reproduce what the faculty member has taught.
A second level of cognition
described by Bloom is comprehension. Do the students have some
understanding higher than mere memorization of the concept under
consideration? Here we mathematics instructors may ask, "When you think
of the word limit, what do you see and how does this relate to the
standard definition?"
Once the students have shown ability
at the level of comprehension, they can be asked an application
question: "If you were trying to extend the definition of limit to
three dimensions, how might you try?" With an assignment of this sort,
we are attempting to get the first-semester calculus student to extend
their intuition to third-semester concepts. Notice, of course, that
since the topic of functions of two or more variables has not generally
been introduced in the first-semester course, such a question is
probably not a good one for an examination; it is, however, a
reasonable discussion topic for a classroom situation.
An even higher level of cognitive
thought that we might ask of a student is that of analysis.
Here the student must take apart the concept in question and put it
back into context while considering the implications. For example:
1. Use words and pictures to
investigate what happens
to the function
z = f(x,y)=xy/(x 2 + y 2)
as x and y head toward zero along the two distinct
paths x = y and x = 0.
This will not be an easy question
for first-semester students to answer; they will need help in finding
the way because the thinking requires an inductive leap. At this level
of complexity, we might ask the student to write out the thinking in
natural language as a way of articulating the mathematical concepts.
This could be done at the board, as homework, as a class project or in
an individual paper, depending on time constraints. Note, however, that
for a third semester or higher calculus student, the above question
might simply be one at Bloom's comprehension level.
Synthesis refers to the
coalescing of analysis into an argumentative claim, a difficult and
complex cognitive process for most students, but one that can guide
them to a more sophisticated level of mathematics. Topics of this type
can include hypothetical definitions, various historical treatments, or
differing interpretative views. A typical synthesis topic: Discuss
historical factors that led from CauchyÕs definition of limit to
Weierstrass's definition. Such topics best lend themselves to writing
assignments.
Of course, a synthesis question can
also be quite sophisticated, and we would probably not want to suggest
it to students in a first-semester calculus course. However, we might
be interested in spending a few minutes discussing such a topic in
class so as to give the students an understanding of how mathematics is
not written in stone how it changes over time, how intuition solidifies
into definition and theorem, and related ideas.
In a synthesis assignment, we can
modify the expectations we hold for students. Claims can be low-level
or high, depending on whether the thesis and subsequent proof reach
back to the earlier levels of critical thought those of application,
comprehension and knowledge or up to a high level like evaluation and
argument. Generally, a synthesis paper includes most, if not all,
cognitive categories; since the response will be complex, the student
will address at least one high-level idea.
The highest level of comprehension
in Bloom's model is called evaluation, where the student shows
that she can embody the concept itself as well as all the cognitive
levels described previously. At this cognitive level, we might ask for
a major paper, since obviously these questions require a substantial
effort from the student. For instance, we might ask: "Why is it so hard
for people to absorb the epsilon-delta definition of limit?" An
evaluation discussion is sometimes called an argument. Argument
papers make for quality major projects, as in "Discuss how Newton's and
Leibnitz views of calculus differed, and the implications of each
person's approach for the history of thought and for the teaching of
mathematics."
2. An Example from Infinite Series.
To find out if students have been
listening in lecture, I sometimes ask a quick quiz question at the
recall level; for instance:
Find the limit of the series:
3/2+3/8+3/32+3/32+
This simply asks the students to
substitute the appropriate numbers into the formula for convergence of
geometric series.
To gauge the students at the level
of comprehension, I propose that they:
Find lim[1 - 4 - 6 + Σ(3/2)(1/4)n]
Here I am testing whether students
know the above formula. At the same time, I can find out whether they
have absorbed the fact that, as I have said a number of times in class,
adding or subtracting a finite number of terms to a series does not
affect its convergence or divergence.
The following qualifies as an
analysis question in a freshman class:
Find all a, r such that Σarn converges.
This last is a possibility for
either a longer quiz question or an examination problem.
The next question, a "synthesis"
problem, was used as a major part of a final exam:
Discuss the infinite series you have
seen in this course. Include convergence tests, and provide examples of
series for which each test does and does not work.
Although students did not exactly
appreciate the above question, I found out a great deal about what they
had learned about infinite series.
An "evaluation" project, much too
involved to be considered a mere examination question, is one given in
a mathematical exposition course: Trace the historical development of
the concept of convergence. Include some analysis of Euler's work on
defining the exponential, along with a discussion of the Riemann
integral, Cauchy's tests versus Cauchy's "proofs," Weierstrass' work,
and G.H. Hardy's contributions to series tests.
3. Levels of Cognition and
Tutorials.
Once we understand levels of
cognition, we see that we have been using them in everyday situations:
A student walks into our office and
says, "I can't factor 10x2 x 3." In Bloom's model, this is a knowledge
question. We show her how to factor the polynomial, then suggest, "Now
you try the next one, 15x2 x 1." We are testing comprehension.
Next, we wonder if she can find a
way to factor a cubic. Here, we are asking for application, and also an
analysis. The student now wants to know, "Is there a formula for
finding the roots of all possible polynomials?" She's having us analyze
and synthesize our knowledge of algebra.
Finally, when she blurts out, "Why
do we need to know this? Can you give me a real world application?
What's algebra good for, anyway?" she's asking for evaluation; she
wants a reasonable argument as to why she should learn the subject.
Of course, we have been dealing with
such students all our academic lives. The point here is that when we
understand the cognitive levels of the questions and answers we are
dealing with, we will better understand the cognitive levels of the
students. Then we can design our curriculum, questions and exams to fit
those levels.
4.Course Design and Cognition
Although the content and
methodologies of calculus and algebra courses are usually (thought not
always) quite well defined, the same is not necessarily true in other
areas of mathematics, such as history of mathematics or geometry. Here,
too, knowledge of cognitive levels can aid in course construction and
application.
Tom has previously, but less
completely, described a geometry course for students who "know no
geometry" [23], which he has constructed around the Bloom model:
Start by asking the students what
geometric words they know or remember. At this point, they are working
on the level of knowledge, offering words like point, line, plane;
names like Euclid and Pythagoras; such concepts as theorem or axiom;
vague terms like shape and solid. Spend some time asking the students
to catalogue their randomly chosen terms into categories; what makes
words like point or Pythagoras different from surface or theorem? This
allows students to show how much they comprehend the terms of geometry,
and to analyze differences and similarities in the concepts.
Now ask: "What is geometry?" This is
a question that requires a good deal of synthesis and evaluation. Allow
students to find a definition that appeals to them and suggest that as
a "working definition" of the concept. Then give them an assignment
that challenges their definition.
For instance, students almost always
describe geometry as earth measurement. A good assignment which
confronts this definition is to have them measure the height of a large
building--an application of their knowledge of geometry--and then find
out by subsequent discussion (analysis of their methodologies) that the
technique they have used does not work on the surface of the earth.
Then go on to the move sophisticated problems that occur because of the
counterexamples that can be constructed, questions like: "What do
triangles look like on the surface of the earth?" Such questions are
very analytic and give the students a feel for how the rules of
mathematics are constructed.
The course is now opened out upon a
plethora of analytic questions that lend themselves to writing and
discussion:
"What is an analogue to a straight
line on the surface of the earth?
"How would you measure angles on the earth?"
"What is the difference between the surface of the earth and the earth
itself?"
Further, you can address synthetic
and evaluative questions:
"How many surfaces are there? And
what does the word surface mean?" "If Euclid knew the earth was round,
why did he say he was studying geometry? What was his definition of
geometry, and how might it differ from ours?" "What are 'straight
lines' in space? Can we even talk about them when there is no 'grid' to
compare them with? And if we cannot talk about straightness in space,
how do we know how to get to the moon?"
None of these questions is trivial,
yet all come out of only a few hours of simple discussion and seemingly
trivial writing assignments given to moderately skeptical students who
are supposedly not at all mathematically inclined.
Return to Top 5. Some Final
Considerations
Although we have chosen Bloom's vertical model of cognition, there
are others that can be useful in the processes described in this paper.
For instance, see Chaffee [7], who offers a horizontal model based more
on student writing strategies. Another model, more linguistic in
format, is offered by Vygotsky [26]. See also Piaget [21] for a
discussion of cognitive strategies in young learners.
Another simple cognitive model from composition courses, somewhat
overlapping with Bloom's, consists of just three classifications of
writing: personal, informational and argumentative. One personal paper
might be "I chose to do mathematics because I found it as creative as
art"; another is "My mind is more inclined to algebra than geometry."
Most mathematical papers fall into the informational category: the
"large-scale geometry of the universe can be partially explained by
curvature in two and three dimensions"; or "Messages can be sent with
(almost) complete security." Thesis statements formed around business
applications of linear programming and fractals and fractals dimensions
can also fall into this category. Argument papers, as much rhetorical
as mathematical, have either soft or hard theses. A soft thesis
requires only information to prove its claim through a lower level of
cognition; a hard thesis, unless it employs sophisticated statistics,
must be supported by more analysis and interpretation. Examples of soft
theses might be the following: "Calculus students are ill-prepared by
their high school experience" or "Writing assignments provide an
effective form of evaluation in the mathematics classroom." Some
examples of hard theses, depending on how they are approached, are:
"Statistical analysis shows that cancer rates highly correlate with
cigarette smoking" and "Better socialization in middle school leads to
higher retention rates of women in mathematics and science programs."
By using cognitive models as guides in our mathematical teaching in
ways that our writing colleagues have long done in composition, we can
move students to deeper levels of understanding of mathematics. In the
classroom, where students often ask, "What's it good for," the use of
cognitive techniques can help them, and us, find answers to this kind
of question.
References.
Bloom, Benjamin, ed. Taxonomy of Educational Objectives. New
York: Longmans, Green, 1952.
Chaffee, John. Thinking Critically. Boston: Houghton Mifflin,
1997.
Piaget, J. The Child's Conception of Number. London: Routledge
and Paul, 1952.
Rishel, T. Writing in the Math Classroom, Math in the Writing Class;
or How I Spent My Summer Vacation Using Writing to Teach Mathematics,
ed. A. Sterrett. MAA Math Notes 16. Washington: The Mathematical
Association of America, 1992.
Vygotsky, L. Thought and Language. Cambridge: MIT Press, 1986.
Return to Top
Show up early, maybe by five minutes.
Say hello. Cheerfully. Start handing out homework or new handouts. Ask
a general question, like "How's it going?" Or, "Was the homework too
hard this time?" If the response is "I couldn't do number seventeen,"
say, "OK, I'll do that one on the board when class starts." If it's
"Yes, the homework was too long," then tell them, "OK, I'll do some at
the beginning." If you're doing a recitation and not lecturing, you can
ask if the lecturer is up to date on the syllabus--you may well know
the answer to this question, but at least you will elicit a response.
This is preferable to telling the students where they're supposed to be
in the text and which homework you'll be discussing today. By this
time, more students are filtering into the classroom, and you can bring
them into the same preclass conversation.
If the students want to talk about
their flu, the last class, or last night's basketball game, that's
fine--until the prescribed time for your class to start. Then say,
"Well, it's about time to start. Did anyone have a homework question,
or something from the last lecture that you didn't understand?" Or tell
them, "Jenny said she couldn't do number seventeen, so I promised I'd
do that one. Did anyone have a question before that?" Suppose no one
does. Ask Jenny to remind everyone what question seventeen says--that
way she can talk--then ask all the students present if "anyone has an
idea how to start the problem." If no one says anything, don't just
start solving it, offer a hint, like: "...this is a section on
parametrically defined functions; what might that have to do with the
problem?" At that point, someone usually suggests, "Well, I started
with the formula for 'parametric derivatives'." Respond, "Right. That's
the one that goes dy/dx = (dy/dt)(dt/dx), just like the chain rule,
right? So how does that fit the problem? " In this way, you are getting
the students to tell you what they know, not just about this problem,
but also about their comprehension of the recent material from the
course.
There are a number of general ways to
keep the classroom active: Ask leading questions--ones whose answers
are not simply yes, no, or "square root of two." When a topic depends
on some earlier concept, ask the students to provide the earlier
information and formulas so as to show you what they remember. Once you
show how to do one problem, choose a similar one and ask students to
work on it communally. Then, when they have all had a chance to begin
solving it, go to the board and write down what they tell you is the
method of solution.
You can also tell the students a bit
about the history of the topic or one application of it, and then ask
them to bring in more of the history or another application for the
next class. Then you can start off the next time with what you have
found out. Don't rush through your own answers as if time were the
enemy. Give everyone time to think of whether they understand your
calculations and whether they need to ask about the seemingly trivial
steps you thought too easy to write down there you wrote 2+1, then over
there you wrote 3, why?
Every topic was new to each of us at
some point; we had to think hard about what made it work. Then when we
found out, we began to think it was trivial. But it isn't--not for the
new student, not for the person who didnÕt see it well the first
time out--so we should give everyone a chance to ask all the questions
the smart ones, the wrongheaded ones, the ill-thought out ones, the
ones we should have asked the first time we saw the material. If we can
set up a classroom atmosphere where our students can ask all these
questions, then we will be a long way toward being a "good teacher"
whatever that concept may ultimately mean.
This has only been a minimal
presentation a first--case-scenario--of methods for making a classroom
more responsive. More interesting and advanced techniques include the
use of worksheets and cooperative strategies like having students work
together. Students can also make presentations, both small, like
individual problems, and large, like final projects, of work they have
done.
As you get further into this topic,
you will want to consult such references as Bonwell [4], Cohen [8] and
McKeachie [18] for more advanced suggestions.
Return to Top
When you lead recitations, you will find that you
open yourself to all sorts of questions. This situation is one of the
most anxiety-producing ones in teaching; "I have absolutely no idea
what they will ask. How can I handle that?" Let's examine some of the
types of questions you will be called upon to answer:
The standard question.
An example: "Can you do number twelve?" If you
prepared to before going to class, you can even answer, "Sure." Just
make sure that you then do it.
The question that makes no sense.
Everyone asks such questions on occasion; resist
the impulse to put the questioner down. Instead, think about how to
turn the question into a good one, maybe by responding with, "Maybe
what you are asking is...?"
Of course, it is also possible that the listener
asked a meaningless question because what you thought was a perfectly
clear explanation was opaque to him or her. Or else he or she was
daydreaming though part of your previous discussion. In any case, you
now have been given an opportunity to reinforce points you (thought)
you made earlier.
The silly question.
Don't make a big deal of it. Act as if it's an
honest question. Answer it quickly, then move on.
You will occasionally have a student who seems to
specialize in asking silly questions. Other students will roll their
eyes as soon as they see his or her hand go up; resist the impulse to
"side with" the others by smiling, joking or answering with a smirk on
your face. Such behavior on your part is simply unprofessional, even if
you know that some of the students are going to downgrade you on
evaluations for your "allowing too many stupid questions."
The unintelligible question.
You might simply say, "I don't quite understand.
Could you rephrase that?" Or, "...Are you asking about...?" Then try to
rephrase the question into something sensible.
Alternatively, you can ask someone else to try to
rephrase the question.
The "challenge to your authority" question.
I often get these in first semester calculus.
"Let's see if he knows what they taught me in my last week of intensive
calculus at my high school." Or, "Let me ask him how to do the hardest
problem in this section, even though it wasn't assigned for homework."
I answer these questions slowly and carefully, if
I can. If I don't remember the answer, I will respond, "That wasn't
part of the assignment, but I'll be glad to show you in the next
class." Then I make sure to do so. I resist the impulse to turn the
question back on the student by asking, "Did you try it? Then
what's the answer?" This last might set up an adversarial situation,
one where you are either perceived as knowing the solution but
unwilling to show it to the students, or as being someone who isn't
really open to answering student questions.
The "good question."
Hooray! A good question. Say, softly, honestly,
"That's a good question." Then answer it. By publicly recognizing good
questions, you encourage more of them.
The question you don't have any answer for.
This is everyone's nightmare, and this nightmare will sometimes come
true. So what? Just respond, "I don't know." Then ask the audience what
they know about the topic. You might just learn something new.
Remember, anyone can ask anyone else a question they can't answer.
On the topic of questions and answers, I am
reminded that for some years I have had a large cartoon poster on the
back of my office door. A large beaked avian in a dress--"Ms. Bird,"
perhaps--is standing amid a circle of cute, fuzzy, small animals, who
are looking wide-eyed at Ms. Bird as she intones, "There is no wrong
answer, Malcolm, but if there were, that would be it."
Questions and answers are an integral part of
learning. Our method of handling them is important to our effectiveness
in our teaching, and ultimately in our careers. It behooves us to get
used to them, to think about them, to encourage them, and to enjoy what
they can teach us about ourselves. Sometimes we will even be surprised
at how much we actually know!
Return to Top
Too many faculty interpret the word
"motivating" as "pandering"; dressing up as Isaac Newton, say, telling
silly jokes that are out of character, or giving out A's as if they
were jellybeans. Now, far be it from me to claim that I've never told
silly jokes, and I don't give out A's like jellybeans, but I have given
out Pringles potato chips to illustrate hyperbolic manifolds, and...but
I digress.
To me, motivating means addressing the
history, culture, and usefulness of mathematics.
You don't have to get a second
bachelor's degree in history to insert a bit of information into your
calculus class about Newton and Leibnitz, or about Bishop Berkeley and
his feud with Newton over infinitesimals. You can also spend a few
minutes reading up about Euler's treatment of exponentials, Cauchy and
Weierstrass on limits, and Bolzano about continuity. There are a number
of references you can use for this material, for instance, Boyer [5]
and Klein [16]. For a more advanced approach, try Edwards [13].
Cultural aspects of mathematics are
also related to the history. Students enjoy hearing about the ancient
Greek approach to infinity, and how it would have affected their
willingness to accept the eighteenth and nineteenth century approaches
to calculus, infinity and the infinitesimal. Further, the fact that
such ideas continue to be questioned and refined into the twentieth and
twenty-first centuries makes the students feel that their own
skepticism about these concepts is relevant and valuable.
Students are also highly interested in
how mathematics applies to their own fields of interest. Here, the
principal of "Show, Don't Tell" takes over; start a section on second
order linear ordinary differential equations with a model of a shock
absorber from a car. Discuss the principle of damped oscillation. Then
derive the differential equation for the model, discussing possible
benefits and shortcomings of the assumed linearity of the system of
equations. Once you have solved the system, don't stop there discuss
the meaning of the various constants as it applies to the comfort level
of the ride of individual automobiles.
The same principle applies to using
Fourier series in solving the heat and wave equations, to using linear
algebra in describing inventory control, to exponential functions in
drug prescription, And but you get the idea. If you can't think of
examples, just look at some of the harder problems in your current
textbook; chances are that there are some good applications there.
In closing, recall that students are always asking for motivation; you
are likely doing the same with respect to your first-year analysis
course. If you are skeptical of what I just said, simply take note of
the number of times you hear -- or, in your analysis class, you think
-- that famous question, "What's this good for?"
Return to Top
One of the ways in which you can be helpful
to your students is to offer general suggestions as to how to solve the
types of mathematical problems they will be seeing throughout the
semester.
Many years ago, George Pólya wrote
a book called How to Solve It [22], in which he addressed the
same question. Many people have used Pólya's model since then
(only a few with attribution). Pólya suggests the following. To
try to solve a mathematical problem:
- Read the problem.
- Read the problem again.
- Draw a picture or diagram.
- Find and label the unknowns what are
you looking for.
- Find and label the known quantities.
- Write down all the formulas and
relations between the known and unknown.
- Solve the problem.
- Check the answer.
And here I might add a suggestion:
Think about how you might generalize the
problem.
In his book, Pólya offers a number
of individual problems -- geometry problems, word problems and related
rates problems from calculus, and others that he solves by using his
methodology. Many calculus texts, for instance, Stewart [24] and
Thomas-Finney [25], do the same in their discussion of word problems.
Instead of my taking up space showing how they do this, I will just
recommend that you take a look at these books for details.
Let me make a few suggestions about how
you might approach the teaching of such difficult topics as word
problems and related rates problems for the students:
Exercise:
Choose two problems from the textbook you
are using next semester. Solve these problems slowly and deliberately
using Pólya's method--no shortcuts allowed! Does his method fit
these problems? Is the fit perfect, just adequate, or not at all? (For
instance, people often complain that "a picture just isn't needed for
this problem.") How would you modify Pólya's approach to fit
your problems?
Return to Top
No doubt you have been on the student
side--the "giving side"--of course evaluations. Now you will see the
"teacher side."
Course evaluations can be extremely
useful in telling you how your individual class of students has
perceived your teaching during a specific semester. Did the students
see you as "organized"? Even if neither they nor anyone else can tell
you exactly what "organized" means, the students have an opportunity to
offer an opinion.
Were you "helpful"? Does that mean,
"did you answer questions without insulting the students'
intelligence?" Or that you had lots of office hours, even though most
students never came? Or that, heaven forbid, one afternoon you showed
them a "preview copy" of the exact same exam as the one exam
administered that evening? Now that last would be "helpful"--but not in
the sense that a good faculty member would like to see.
Were you "knowledgeable" about the
material? Of course, you could be successfully completing a course in
complex analysis with a grade of "A+", and still have the students in
your calculus section saying that you weren't.
In view of all the above skepticism,
what is the function of the student evaluation process?
Well, first of all, the process does
have uses other than the ones just discussed. Take, for instance, the
question of "knowledge." If you know lots about functions of several
complex variables yet can't give an intuitive response to the question,
"Why does the ratio test work?" then your calculus students are of
course going to see you as someone who doesn't know much about
calculus. ("He's really a nice guy, but...") Alternatively, if your
response to the question about the ratio test is to give a rigorous
proof of the test, then the students are likely to give you high marks
for knowledgeability--and low ones for helpfulness. ("She knows a lot
of math, but she can't bring it down to our level.")
As to the question of "organization,"
this one is tricky. I have personally thought a lot about what it
means, because I am consistently rated lower in this category than any
other in my over thirty years of college teaching. I believe that when
students talk about organization they seem to mean, "He has a plan in
his head as to where we'll be at the end of each class, he tells us
what that plan is, and he gets there almost every time." I must admit
that, if this last is the students' "definition," then I don't conform
to their ideal. Instead, I am willing to take questions at (most) every
opportunity; I am happy to revisit earlier concepts if students show
that they don't know them; I sometimes make up more than one lesson
plan in my more "nonstandard" courses, and then let the students
questions and interest dictate which one I use on a particular day. In
return for this last, I occasionally begin class by outlining where we
have been, and I do periodic reviews to show the students where we have
come to.
I also take some comfort from the fact
that one of our college's previous teaching award winners once told me,
having seen my evaluations, "Don't worry, Tom, I always get 'low marks'
for knowledge, too."
Student course evaluations are thus
useful, without being a complete determiner of teaching ability. They
tell us what the students expect of us. They teach us something about
the expectations of our audience. Maybe most importantly, they allow us
to find out our own classroom goals, and how these goals conform to or
conflict with the ones enumerated on our evaluation forms.
In short, we need to know how we are
being evaluated, where those evaluations go, and what each of the
qualities listed on the evaluation form has to do with our approach to
teaching.
At Cornell, we give a nuts and bolts
evaluation form to all TAs and faculty. They are asked to hand out and
collect this form very early in the term, usually around week three,
but they are specifically encouraged not to turn it in to any official
or semiofficial entity in the department. The function of this
evaluation is to let the instructors find out how I'm doing in various
categories, from "Do I speak loudly enough?" through "Do I give enough
time for questions?" to "Are my answers intelligible?" The full form is
given at the end of this section, along with a few other typical
evaluations.
Return to Top
If there is only one piece of
advice you should take to heart from this entire volume, it is that
embodied in the title of this section: Get along with your colleagues.
When you are told to come to the
"calculus one" meeting, do so. Grade in a timely fashion, and do it in
such a way that others will not have to field two-hundred student
complaints.
Do your share of the busywork
involved in administering the course; offer to give occasional makeups;
to run off five hundred copies of exams; to teach once or twice for a
sick colleague. Show up for office hours; don't leave it to the other
TAs to offer excuses for you and then have to help your students in
your place.
Be nice to people, even if they
aren't nice to you who knows who they are, or what kinds of problems
they may be having at the time. And, if they really aren't
nice, then at least be formal and professional. You have every right to
choose your friends, but colleagues are more like family you have
little or no choice in the matter. If you really can't stand someone,
ask yourself why. If you think you have a good reason, fine; chalk it
up to experience, and move on to get the job done as quickly as
possible so as to be out of the way.
Don't speak ill of fellow TAs,
about faculty or about administrators to your students. As a friend of
mine said the other day, "Oh my goodness. I didn't realize that that
person I was complaining to about the dean was his spouse!" If you have
a professional problem with a colleague or co-worker, take the
problem to a supervisor if that person is trustworthy. If not, try your
graduate student director or the chair. Say something like, "I don't
want to cause a problem or get anyone into the middle of an argument
with another TA, but something has come up about (say) an ignoring of
some possible cheating [say] on the last exam, and I need to talk it
out."
By the way: Yes, I have seen a
(very) few TAs who were successful in the classroom, but who were so
difficult to deal with as colleagues--always arriving late, leaving all
their TA duties to others, skipping classes, etc.--who were told to
leave. So "brilliance" is no guarantee of support for a teaching
assistant. And, believe me, it usually gets even more difficult for an
uncooperative faculty member. So learn to be cooperative now,
or expect to pay a lot more later.
I have also seen a number of TAs
and faculty who are "selectively nice." If a tenured faculty member is
asking them for a favor, that's fine, but not a secretary. (You have
seen this too, where a faculty member treats you badly because you're
"just a TA.") Please don't be like this. In many ways, the staff runs
the university. They were at their desks before the chair became chair,
and they will be there when the chair has gone back to being a lowly
tenured professor. They know how to cut corners, and they can be
helpful to people who are courteous--and coldly uncooperative to those
who are not. More to the point, they too just want to be treated like
human beings, just as TAs do. So treat them that way.
'Nuff said.
Return to Top
The first time you step in front of a
class, you cross an invisible line. You don't see it, but the students
do; you are no longer one of them. That's why they look at you
quizzically when you ask, What should we do today? They also don't
appreciate your little jokes about exam grades. And, when you write a
cute comment on their homework about how this work is more like high
school stuff they see the comment as acerbic, and they let you know.
A professional is one who speaks
for and has responsibilities to the discipline he or she is teaching
and to the other practitioners of that discipline. Some of these
responsibilities have been described in the section called Get Along with Colleagues,
but not all. There is more to being a professional than speaking
courteously to an officemate, as important as that is.
You have responsibilities to students:
- Don't discuss their individual grades in public, and don't
compare the students to each other. It is one thing to say, "You're a
very strong student"; quite another to comment, "I thought that Joe
would be better than you [or vice versa], but..."
- We have all met people who are very likable, but favoring them
with "hints" or "extra help" that others don't get is not fair.
- Socializing can lead to difficulties, even in the most benign
situation--See Case Study V for an
example.) So, if you know deep down that you are not going into a
"benign" situation, do not participate. A night of binge drinking with
your undergraduate class is "definitely contraindicated," as a friend
once said to me.
- If you are not sure how much fraternization to have with students
(after all, we don't want to be totally standoffish), ask trustworthy
colleagues and faculty for their advice.
- Be careful the kinds of jokes and comments you make in front of
students, who can be sensitive in very unusual ways. For instance, I
once teased a student who knew an arcane fact about Galois theory that
he "must be reading the same kind of weird stuff that I am." When he
obviously bristled, I had to apologize to him for my comment.
You will also have responsibilities to the faculty and other TAs:
- Do not insult or belittle others' teaching styles, or their
approach to research. For instance, in a discussion of methodology,
"Here's how I teach word problems" is clearly more tactful and better
received than "Students tell me they don't like the way you do that
topic. " And, you don't need to tell your officemate that Professor
Jones "can't be very competent, since he's still writing papers on..."
- If you have an honest disagreement with a colleague, keep it on a
professional level: "I really think that problem might be too hard for
these students," said directly to Professor Jones, is a professional
comment. You may be right, you may be wrong, but at least you have had
your say. The alternative of going to your office mate to gripe that
"Ole Jonesy's just trying to nail as many freshmen as possible," maybe
true, but not a professional alternative.
- Pitting your class against every other instructor's is not
professional. The fact that "My class had a 73% average, but Joe's was
67%" does not make you a better instructor. You may have overlooked the
information that your class met at 11 a.m., while Joe's was right after
lunch. Then there was also the fact that you asked two students with
low averages to switch out of your section after the first exam.
- Similarly, you don't sit in the department lounge bragging about
how much better your course evaluations are than others--or how yours,
bad as they were, at least beat out Joe's. If someone wants to make an
honest comparison of his or her evaluations with yours, you certainly
don't need to lie, but you, as a professional, know that there are many
factors involved in various ratings of classes, students and even TAs.
For instance, is it really true that you passed out donuts on
evaluation day and then told the class that your job was on the line?
Well, that method seems to have worked!
Most importantly, you have responsibilities to mathematics itself:
- Prepare the material. Read up on it (Yes, even "precalculus" has
a history). You needn't be a cheerleader, but you should be ready to
make an honest reply to "Why do we need to know this?"
- Show some interest in your teaching assignment, and in
mathematics in general. If you can't find any reason for teaching that
is more compelling than drawing a paycheck, is this really the way you
want to spend the next forty years of your life?
VJust as no one can ever know all of mathematics, no one ever
knows all it takes to be a professional. But, through a combination of
talking to trusted colleagues, thinking before acting, and using common
sense, we can avoid most pitfalls. If it feels wrong and sounds wrong,
act carefully, because it likely is wrong.
Return to Top
Without trying to be exhaustive in my list,
let me say that there is a variety of teaching methods, among them:
Each of the above has positive and
negative aspects. For instance, the lecture method imparts lots of
information in a short amount of time; in the proper hands it is
usually organized well; and it makes use of the expertise of the
lecturer. On the other hand, students can easily "nod off" in a
lecture, and information imparted is not necessarily information
received.
Socratic dialogue is often touted as an
active, open form of learning that gets students involved in the
educational process. On the other hand, such discussion can lead
nowhere. It is also often "falsely democratic" in that a few speakers
can dominate discussion, either crowding out other viewpoints or
allowing for participation by only a few.
I claim that a guided discussion is one
that has more structure than the Socratic model, perhaps by using
information sheets or reading lists. Handled properly, such guided
discussion can overcome many of the objections to Socratic dialogue
listed above, and can achieve the objective of finding out what the
student knows. Yet, students and faculty sometimes complain that such a
technique isn't fast enough in imparting knowledge. Better prepared
students, especially, often object to having to wait for students who
"just don't get it."
Then there is "recitation," or
student-guided learning. This methodology makes maximum usage of the
student and of the text, putting the "burden of proof" (not to mention
"burden of effort") on the students. This method doesn't necessarily
make effective use of the expert, and if guided discussion operates
slowly, this method can really operate slowly. On the other
hand, say those who advocate this technique, once students do "get it,"
there is no doubt they have it--whatever "it" is. A final argument
usually offered against student-guided learning is that it can miss the
most salient items involved in a course.
Thus, what we need to ask ourselves is:
What is the goal we want the students to achieve, and what is the best
methodology for achieving that goal?
Return to Top
As a TA,
While doing homework before class, think
about:
Also as a TA, you may also be called on to
discuss:
Some pitfalls TA's can experience:
Generally speaking, students just want to
get through the course with a good grade and go on to what's important
-- to them. They will accept, for the most part, work that is relevant
to their goals, and their questions will reflect those goals: "Why is
this important?" "Will this be on the exam?" You, for your part, should
see both these questions as fair at any time in the course, and should
be prepared, within reason, to answer them. Possible answers to the
first question: "It's used later, in..." "There are engineering
(business, economics, architectural) applications, and if you learn
about this technique, it will make you better in that field." "The
author needs it later in the chapter (or in the text), when we
solve...." And the last "fair" answer: "I'm really not sure. Let's see
if we can find out why the author and instructor want us to do this."
If you think about irrelevance while you
are preparing your recitation at home, you will be ready to answer most
of the questions that come up in class. Plus, you will be active in
your preparation, which will make you more interesting to the students,
and your material more interesting to all of you. You'll eventually
become the kind of instructor the students "inadvertently praise": "I
didn't think I'd like the material or the course, but he/she showed me
why it was useful. I still don't like it that much, but now I see why
it's important."
Return to Top
Chances are, if you are a sensitive, caring
instructor, students will begin to see you as a "lifestyle advisor." A
typical first reaction is to feel honored--but then an almost immediate
response sets in; "How can I give advice to this person--or anyone, for
that matter?"
Good. You should feel skeptical. Of course, you
are an intelligent person and a trained mathematician. And that's the
point; you are trained as an intelligent mathematician, not as a
counselor or therapist. When someone comes to you to say, "I'm afraid
I'm suicidal," how should you respond? How about as the intelligent
person you are: "Thank you for coming to talk to me about it. It's
pretty obvious that this causes you some pain, and I'm glad
youÕre willing to speak to someone about it. I'd like to help
you--would you mind if I called the university counseling service for
you? I think they can provide some real support."
The vast majority of student problems are not so
serious, and you can deal with them fairly easily:
In the first situation above, I am making a
strong suggestion, and any normally intelligent student will see the
wisdom of taking it. Of course, he or she might really be broken up
about the death of a goldfish ("But I had Sleepy since I was twelve!").
But if the student must go ahead with his or her decision rather than
taking my advice, then it does become something he or she must accept
the consequences of. I, as instructor, am absolved from the
responsibility of having forced my will on this poor student. "Well,
I'm sorry you failed the makeup, but I did warn you about what might
happen if you took it." Outcome: The student was offered a chance to
think like an adult, and rejected it.
In the "death of the grandparent" case, I am
again leaving the choice up to the student, but this time I am not
implying any penalties. I emphatically do not get into any
discussion of whether the grandparent (parent, uncle, second cousin,
etc.) has died--I don't ask for "proof." To me, there are two
possibilities. In one case, the student is upset and doesn't need to be
challenged. On the other hand, in the event that this is the sixth
grandparent of this student who has died in the three years he has been
at the school, then the student still has a problem, only this problem
is of a different sort.
If you do find out later that the second student
has been using "death in the family" excuses a lot, you might want to
suggest that he see a counselor or examination specialist to learn how
to cope with examination stress.
The above analyses also apply to such common
situations as breakups with significant others, the "three exams in two
days" phenomenon, and the "field hockey road trip" model. I consider
the "significant others" problem reasonably serious, and usually adopt
the "grandmother" approach to this one. As to the "three exams"
problem, I'm much less sympathetic with that. "Sorry, but it must be
the same for lots of the students in the class--and you do have a week
to study for the exams." Then I continue, "The exam will be harder, and
I couldn't give you more than one extra day." One of the main reasons
that I am harsher in this situation is that, as soon as I give this one
student more time to study, thirty-four others will be along with the
same excuse. And, if I try to accommodate all of them, then most of the
thirty-four will also have problems with the makeup date, etc. Then the
grades will not correlate well between the original exam and the
makeup, and that will lead to even greater inequities.
On a lighter note, students are apt to ask some
really rather bizarre questions:
I have been asked all the above, although it has
been a while since someone told me I was too young to be the professor.
In the case of your being surprised by a
particular question that you haven't thought of, rather than give an
immediate reply, you may want to think about it for a day or two. Tell
the student you will respond by e-mail as soon as you can, and then go
ask someone you trust how he or she would handle the situation. This
avoids the problem of making a bad decision on Tuesday and then having
to either live with it or rescind your ruling on Wednesday. (Still,
rescinding a bad ruling is better than causing real inequities over the
long run.)
One more general statement about making
decisions: Over the course of your teaching career, you will be called
on many times to decide in a fair, quick and accurate manner on a
matter of some importance to a student or group of students. At first,
this process can seem daunting, but after a while, you will have seen
almost all the types of questions you will be asked. While this
experience can make most of the decisions easy, some will never come
smoothly, and just when you think you know how all this is done, you
will make a blunder that will make you question your own
intelligence--if not your sanity. The only thing you can do in such a
situation is admit your mistake, and go on to try to learn from it. In
the main, all you can do successfully is try to be fair and honest to
all the students, to your colleagues, to the department, to
mathematics, and especially to yourself.
Don't worry too much; you'll do fine.
Return to Top
The audience is important. Know, as far as you
can, the motivation and interests of the group you are teaching.
Are they future engineers? Then pure theory
is not always appreciated, or necessary. For this group, it helps to
have a few mechanical or electrical applications for them to chew on.
Are you talking to pre-meds? Then they are also likely to be taking
lots of biology and chemistry courses, and would appreciate hearing how
mathematics is applicable to these courses. You could do problems
related to bacterial growth, for instance, but you could also model
drug dosage problems, spread of disease questions, and applications to
heart pumping problems.
Precalculus students are usually encouraged
to hear that "this algebra and trig you are learning will show up a lot
in calculus," but they are more convinced if you tell them about sine
waves in music, heat exchange or building earthquake proof buildings.
You shouldn't necessarily try to show the details; you might not know
them all yourself. Your job is to teach the precalculus. But at least
you will be giving the students a small glimpse of a possible future.
Sometimes instructors interpret the
statement "consider the audience" as meaning, "pander to the audience"
or "only teach the fun stuff," whatever that may be. I am definitely
not suggesting this course of action. Your goal should be to teach the
course material that you have been assigned--but there is nothing wrong
with including some direct, well-chosen applications that apply to the
material and appeal to you and the students. It is not "pandering" to
the students to use the course materials and content in a meaningful
way.
Return to Top
- Act as if you don't have a "real job."
It's just a TA.
- Don't show up for class. Several
times. Without an excuse.
- Be insolent to faculty (especially
your advisor), the students and the staff members.
- Call in sick every Monday. Leave for
the city on Thursday afternoon.
- Never plan what you are going to do in
class, this stuff is too easy, anyway.
- Make clear that research is
everything; you are going to solve a great problem and join one of the
top departments where you will only teach one graduate course a term.
- Skip office hours. Your officemates
can take care of any of the students who'll come by.
- Make it clear from the start that you
don't intend to do anything extra--in fact, you won't do anything that
you don't get a salary for. And, you only do that under duress.
- End all your classes early. Can't do
this, you say? Too many questions? Simple to solve; just belittle the
students who ask them--that'll ease up on class time.
- Leave for vacations, breaks, and end
of term early. Aruba awaits!
- Don't hand in grades on time. Got some
graduating seniors? So what! They didn't like you anyway.
- Offer "grades for favors"--only
"jokingly," of course.
- Show up at undergrad parties. They're
so much more fun, anyway. Drink a lot. Leave at 2 a.m. with one of your
students.
- Tick off the TA supervisor. He isn't a
real mathematician, anyway. He stopped doing research about the time
you were born.
Return to Top
It can be very difficult to leave one's own
culture and come to another, only to be asked to teach to "foreign"
students, maybe in a "foreign" language.
Earlier, I mentioned briefly how a new
graduate student has a wealth of first semester problems: setting up an
apartment, finding a roommate, opening a checking account, finding a
grocery store--and, all the while, trying to learn how to be a graduate
student and a teacher. And, when you have just come from another
country, these problems are compounded exponentially: how do you get a
social security number so that you can receive a paycheck, what is an
"I-9" anyway, do you have to carry a passport all the time, where do
you find ramen noodles, how much money do you really need to
live on in the States, etc.
I am very sympathetic with your
problems--I lived through this same experience, not once, but
twice--but here I will concentrate on the teaching aspects of your
situation. First, I will address the question of how to get through the
first few days in the classroom in the United States, and then I will
talk briefly about the cultural aspects of teaching in a foreign
country.
A First TA Assignment--What to Do,
How to Cope
First, it is all right to be nervous; I
would be surprised if you weren't. All new TAs--at least, all the ones
who care about the job and want to do well--are nervous. Just remember
that, although you may not have taught before, at least in the United
States, you have been recognized as an intelligent human being
by those who chose you to be a graduate student. They did not select
you for failure; instead, you have been recognized for your potential
for success. One of the early measures of this potential is the number
and quality of the questions you ask.
Some questions to consider about your
TA assignment:
Can you spend the first semester or two
grading in a more advanced class rather than having to stand in front
of a group of freshman calculus students? If so, you will have an
opportunity to take some English language classes and watch some
(terrible) television to get the language down better. You can also
have a couple of office hours to do some one-to-one tutoring to
increase your language abilities.
Is it possible to get a class that is an
"easy job," not necessarily in the sense of material simpler to
understand, but in the sense of material "easier to explain to
students?" By this I mean students in Fourier series, for instance,
have fewer misunderstandings than those in a supposedly trivial
freshman algebra class; thus Fourier series, although harder to
understand, can be easier to teach.
If a grading assignment or an easy job
is not possible, there are still ways in which you can help yourself
get through the semester with your sanity, and your dignity, intact.
Find yourself two "mentors" you think
you can trust; one from your home country (if possible), and another
from the United States. Ask them to advise you, in a semiformal way.
Then, before your first class, after you have done the first homework
assignment, have a mock class with (at least) the U.S. mentor. In this
mock class, go to a board and do some problems as you would in the
first day's class. Concentrate on pronunciation--especially of
difficult words like "continuity" and "theorem" that will appear in a
mathematical context -- write a lot on the board, and ask for advice.
See how well you understand the comments, questions and advice that the
mentor is giving you.
Next, go to your countryman and ask some
questions about what you should expect the students to be like. This is
information that your countryman will understand, but the U.S. TA,
having lived and taught only in the States, will not. (I will have more
to say about cultural questions later in this section.)
Then go to your first class. As part of
your introduction, explain to the students that you are from another
country and new to teaching. Tell them that, to help them and yourself,
you will be trying to speak slowly, that you intend to write a lot on
the board, and that you would appreciate it if the students would help
correct your pronunciation. When someone asks you a question, if you
simply do not understand it, ask the other students for help in
rephrasing it. If someone says that a problem can be done another way,
offer him or her the chalk and ask if he or she will show you “at the
board.” In this way, you can see the solution in writing, rather than
trying to understand your student's "Noo Yawk" (i.e., New York) accent.
If a student says, "I don't understand," ask where on the board is the
step that isn't clear, and fill in missing details at that specific
point. If the student's response is that "None of it is clear," try to
do the problem a different way, or start it again, but this time add in
all possible details. If nothing is making sense, try to get the mentor
to observe one of your classes to see where the problem lies. And, if
all else fails, start writing up solution sets for the students--but,
don't do this from day one, because they will become a crutch for you,
and the students will expect them all the time.
After the first or second week, you
should not need to use your mentor to practice every class. However, he
or she can still be useful for occasional discussions of how to
approach individual topics, how to pronounce new words that come up,
and how to deal with situations you have never experienced in your home
country.
Some other suggestions as to how to
treat the first few days of class can be found in the other sections,
for instance, in Day One.
Cultural Aspects of Teaching
Learning about the culture in which you
are teaching is at least as important as knowing how to speak the
language, yet this aspect of teaching is often overlooked.
When I moved to Canada, the first
lecture I gave was on my research. Since I spoke English and they spoke
(an only slightly better) English, I didn't feel that there would be
any particular problem. At one point in the lecture, however, after
some people asked me about details, I asked, brightly, "Any more
objections?" The response, by a number of people, was "Oh! We weren't
objecting!" Of course, I also hadn't thought that they were really
"objecting"; I was just using a word I had used many times before to
indicate the fact that I would be willing to answer more questions.
Then, after a few more incidents of this sort in restaurants, banks and
the like, I suddenly realized that, although I may be a reasonably
polite "American," I'll never be a polite Canadian-American.
When you walk into a classroom, you
carry with you many assumptions; for instance, about what students are
like, how they should address you, whether and how they should ask
questions, and how they should dress and act, just to name a few. Many
of these assumptions come from the way you were taught in your home
country; you will be expecting similar treatment from your students
here.
Sorry, but let me tell one more story
from my past:
When I was in Japan, I had a chance to
observe some college mathematics teaching. In each case, when the
instructor walked into class, the students stood. After they sat back
down, the instructor then delivered a lecture at the board from notes
without any questions from the students, who spent their time quickly
taking as many notes as they could. Then, when the instructor had
finished his fifty-minute lecture, the students again rose as he
quickly left.
My point in telling this story is not to
claim that a particular method of instruction is better or worse than
another (although I do express some opinions as to the efficacy of
interactive methods elsewhere in these notes), but rather to show how
different such methods can be.
In the States, most students will have
come from high schools where they were encouraged to ask questions in
class, sometimes merely by interrupting the teacher; and where they
were often required to work in small groups in rather noisy classroom
settings. Thus, if you have come from an atmosphere like the one I
described seeing in Japan, you may be shocked by what you think is
insolent behavior on the part of your American students. Occasionally I
feel as if I should apologize for some of what I see in the U.S.
classroom; at the same time, I can understand how our students can
profit from being able to ask a question when they want to--within
reason.
As usual, I have some suggestions. If
you find, as I do, that students should be able to ask questions, but
in a more mannerly way than by just yelling out, "That's not right! The
answer should be five," then I propose that you tell them on day one
that "I am open to questions. But, please raise your hand, so that I
know who is asking the question, and so that I can finish my thought
before answering you." (For more suggestions as to how to make your
classroom more active, look at such sections as The Active Classroom and Motivating Students.) If, on the
other hand, you believe that students should hear what you have to say
before they start asking questions, ask them to hold their questions
until you have fully explained the topic, the example or the exercise
you are working on. In short, remember that the culture may not be
yours, but the classroom is yours, and it is your right to decide, within
reason, what is the best way for you to get the material across.
A short list of thoughts about teaching and living in an alien
culture:
- Although it is usually easier to live with, room with, and deal
with people from your own country and culture, you will find yourself
learning so much more if you make the effort to spend a fair amount of
time as you can in the "native culture."
- There are a number of advantages to TAing in the U.S. You will
learn the language, which can help you in your studies as well as in
you teaching. Further, if you decide to stay in the States after
graduation, you will need all the language skills you can get for your
job.
- If you go back to your home country, your knowledge of English
will also be helpful there in your job and your research.
- If you find that some classroom situations are bothering you,
talk them over with your mentor from your own country. The problem may
turn out to be easily solvable, or it may really reflect some cultural
differences. In either case, you can get advice on dealing with the
situation.
- You do not have to accept what you perceive to be improper
behavior on the part of your class. If you find the students acting in
a way you think is unacceptable, talk it over with your U.S. mentor. In
this way, you can gain some valuable insight into how standards vary
between your own culture and the one in the United States.
- And finally, enjoy your teaching experience. View it as an
opportunity to learn about another culture. Try to develop a classroom
atmosphere that is in keeping with your personality, while at the same
time is relaxed. Why not have some fun while you are getting a degree?
Return to Top
[This section recapitulates some of the material
from the earlier Checklist section, but in a bit more detail.]
Everyone who starts a new job walks into a
place that has a life of its own; there is all sorts of "silly stuff"
too small to consider, and at the same time too important not to
consider. And, if you ignore all the little details below, a lot of
people will get annoyed with you.
For instance, where are the bathrooms located
in your new building? If you cut your finger, can you get a bandage?
Where do you get keys to the buildings and
offices? Do you have to pay a deposit for the keys?
Where do you get chalk and erasers? Carry an
extra piece of chalk to the classroom. (I know an instructor who once
used his shirt to erase the board; his students never forgot.)
Find out the methods and "rules" of using the
copy machines. How many copies may be made for your class? Can you make
personal copies for a fee? Wouldn't it be nice if you didn't have to
bother the staff each time the machine runs out of paper?
Ask where and when you get your paycheck. When
is the first payday? The last?
Learn how to search for and check out books at
the library. (I'll bet the librarians know, and they might even help
you.)
Find out whether your keys to the building
work on evenings and weekends.
Who decides where your desk is located?
What are the rules about textbooks? Do they
have to be returned at the end of the semester? Can you go ahead and
write all your notes in the margins?
Where do you turn in grades at the end of the
semester?
If you are giving evening or Saturday review
sessions or exams, are the rooms generally open? Do you have to reserve
the room? From whom?
Are there syllabus files? Or files of old
exams? Who can use them?
Do you have to pay for your own pencils and
paper? How about transparency sheets if you use an overhead?
What do all the staff members do? How much of
it "concerns you?"
What are the general rules about how much time
you should be spending on your teaching? How many office hours are
considered appropriate?
When students ask you advising questions you
don't know the answer to, to whom should you send them?
Who is your immediate supervisor, and how
often should you report to him or her, if at all?
Do you have to go to the calculus lecture
you've been assigned to TA for?
What is expected of you at "vacation time?"
Can you go home during study week, say, or during that part of exam
week before your class takes its final? Or will the course overseer get
angry because you aren't available for office hours and to help with
the makeup of the exam?
Hey, there's lots of "silly stuff" you have to
ask about! I'll bet you can think of ten questions I missed! And every
school is different; donÕt assume that what you did when you
TAed as an undergraduate or when you were working on that master's
degree will apply now.
Return to Top
There is bound to come a time when you
have a disagreement with your course leader. I can think of many ways
in which this can come about; for now, I will discuss two hypothetical
situations.
In the first case, you find that
Professor Dimble, the lecturer in calculus this semester, seems to be
having an inordinately difficult time explaining the text to the
students. He is a former teaching award winner at your institution,
Grand U., but his eyesight, hearing and most importantly his memory
seem to be failing him. All of this is causing the students a great
deal of consternation, and costing you and the other TAs a great deal
of time. Students are coming to you for additional lectures and they
are bombarding you and the other TAs with lots of questions outside of
class. When you demur over extra evening sessions, they respond, "But
we need you--Professor Dimble just doesn't make sense!"
Meanwhile, you are in the last full year of your schooling; you're
writing your thesis, trying to finish up a paper for publication, and
looking for a job. You don't need the added stress of basically
teaching calculus to freshmen for a graduate student's salary.
So what do you do? Well, one
solution would be to make your feelings known to all. Complain to the
students, to your thesis advisor, to other TAs, to the chair, to the
dean! You could also refuse to give the students extra help; "After
all, I didn't create this problem." You could even call and tell your
family. This would certainly get your position across; even Professor
Dimble would probably find out, his hearing problems notwithstanding.
What will be the effect of your
complaints? Well, the students will know how you feel. But, if you
don't offer them some help, they won't see you as part of the solution,
just another obstacle in the way to their learning calculus.
Your thesis advisor will see you as
a "young researcher" who doesn't understand the professional culture,
who will mature eventually, maybe after you get to your first real job.
He or she won't say this to you directly, of course, but you'll hear
about it later through the grapevine.
The other TAs will think that
you're a prima donna, that you're more worried about your thesis and
your career than about your fellow TAs or your students.
The chair will try to explain to
you about Professor Dimble. "You have to understand, he's getting old.
We know about the problem, and maybe by next year we can do something,
but right now, weÕre in the middle of the term, and right now it
would be really unfortunate to pull him out of class. But thanks for
the information." [Yeah, sure!]
Meanwhile, the Dean's position will
be "Why doesn't he keep this at the department level? That's what a
chair is for."
And poor Professor Dimble; he'll
feel saddened, surprised, shy, confused, improperly maligned.
Personally insulted.
Well, "solution one" didn't work
too well. Is there another?
Let's back up. What's wrong with
giving the students a once a week "review session?" You could trade off
with two or three of the other TAs, so that you only have to conduct
one of these sessions once a week. Then, if you quietly go to the chair
(not the dean; keep it at the department level) to explain the
situation, he or she will be grateful for your extra work, and may even
be able to "compensate you and your fellow TAs," if only with a reduced
load in the next semester.
If your advisor asks why you aren't
getting out that twelfth draft of your joint paper, you can explain the
details. He will then tell other members of the department, and more
importantly, will write in his recommendation letter to other schools
that you are "a mature young mathematician who willingly took on extra
duties to solve a small teaching problem we had here at Grand U." This
will make you sound like someone another school would be happy to have
in the tenure-stream position it is advertising.
You will have preserved your good
relations with your fellow TAs except for that one person who refused
to help you and Professor Dimble will finish up his illustrious career
with his honor intact.
Now to the second sticky situation:
Having survived Professor Dimble, you are given what the chair
considers an easy assignment. Again, you will have a section of
calculus, but this time it's with Professor Aggress. From the "git-go,"
the good professor is after students. "Are they all as dumb as they
seem?" is the first question he asks you. He calls your homework
grading into question; "You're too easy on them, and you didn't grade
number fifteen properly" he tells you in front of a group of TAs and
faculty in the department lounge. Then he insults other faculty in
front of you, and you hear from your officemate that he is saying some
unkind things about you to them. Further, Professor Aggress gives
really difficult exams (the students say that they're "impossible"),
and wants to have no curve. "As far as I'm concerned, if they can't get
a '60' on this exam, they just shouldn't be taking mathematics."
Ok, you survived 'Ole Dimble, so
you should be professional enough by now to know that it will do no
good to simply scream at Professor Aggress. After all, shouting and
being insulting is his stock in trade; he's been at it for the last
thirty-six years. How about trying to think like a professional,
instead?
First of all, ask yourself if you
really did grade number fifteen incorrectly. If so, apologize and offer
to do it over. Then, when the next exam comes around, when you make up
your grading key, ask Professor Aggress if you can show it to him to
see if it "looks like what he wants."
When the students come to complain
to you about Professor Aggress' attitude, do not talk badly about him.
You and the entire department may completely agree with their
complaints, but whatever you say will not change the situation one
iota. Further, if it gets back to the professor, it will only end up
causing you to be castigated by the chair or someone else in authority.
Instead, concentrate on making an
effort to keep the exams and grading scheme as reasonable as possible.
To help the students, look back at the first exam to see whether the
problems were simply chosen from among the really hard ones, or whether
they were totally out of line. If the former is the case, occasionally
use class time to discuss with the students some of the later problems
in each section. If it's the latter, maybe, after you've done
your best to get into Professor Aggress' good graces, you might be able
to suggest some possible problems for the later exams ones that the
students have some chance of working.
Remember, there's a huge difference
between liking a person and working with that person.
You do not have to sit in the coffee room and defend Professor Aggress'
attitudes. In point of fact, the professor's insolence seems to be
coming from some deep seated problem that no one in the department is
likely to be able to solve. But, that isn't your job your goal is
simply to be an honest, straightforward, professional person.
Of course, one of the ironies of
this situation is that, if you really are professional, courteous,
helpful to the students, and sensitive to Professor Aggress' attitudes,
he may just ask to have you as a TA again the next semester. This is
one of the difficulties you will have to guard against; on the other
hand, you have done such a great job, that I certainly don't need to
tell you how to handle this problem.
The Perry Model
William Perry [20] published a
theory of student development that has been very influential in higher
education. The basic model has been modified in various ways. Belenky,
et al [2] have revised them to fit gender differences, and Culver and
Hackos [10] have done the same for the engineering disciplines.
Perry’s original model consisted of
nine stages of development. I will describe a simplified (some might
say oversimplified) four-stage model and briefly discuss possible
implications for mathematics
The four stages of student
development are:
Basic duality,
Multiplicity,
Relativism, and
Commitment.
In basic duality, the world is
split in two; the only possibilities are right or wrong. In this mode,
students see their instructors as authority figures who have all the
answers. Instructors who do not instill "the correct answers," who are
open to discussion, are merely bad teachers who are playing tricks on
the students.
The next stage of development,
multiplicity, is, in Perry's model, actually three stages. In the main,
"multiplicity" means here that the student now believes that "it's all
a game." At first, the students are sure that there is still a right
answer and that the instructor is just making them play along until the
correct response is found. Later in the multiplicity phase, the
students may decide that the instructor doesn’t know the answer either;
"He can't be any good as a teacher because he doesn't even know what we
know." Later yet, the students may decide to believe that "anyone may
have their own opinion."
From this stage, students then jump
to the position of relativism, where "anything goes--all positions are
equally valid. No one can argue against my outlandish position (however
badly reasoned), because there is no absolute right or wrong."
In the final stage, commitment,
students take into account positive and negative aspects of decisions
they are considering, and then make balanced judgements as to how to
follow. In Perry's nine-stage model, he splits this process into parts
that depend on the depth of the commitment.
The implications of the Perry model
for mathematics and its teaching are important. I will discuss just a
one:
Students who are in the basic
duality and multiplicity phases of growth sometimes say that they like
mathematics "because in math all the answers are known." They often use
mathematics and science as models of "correct" worldviews when they are
challenged on some point in a sociology or English class, say. When
these students find out that "ambiguous assumptions" are also made in
theorems, they can become disillusioned with mathematics itself or the
way it's being taught. Their subsequent problems with ability to prove
theorems can thus possibly be embedded in their view of education
itself, and not simply in their refusal to get the "theorem-proof"
concept. One possible implication of this last, if true, may be that
courses in proof theory for sophomores must be constructed to take into
account the question of students" belief structures.
But I am delving too far into a
theory of learning for a text at this level. The implications of the
Perry model are intriguing, however, and you may wish to pursue them
further.
Return to Top
August 27-28
You spend three hours looking for your lecturer. You meet with him or
her for an hour, then you realize that you forgot to ask a really
important question. When you go back to ask, he or she is meeting with
undergraduate advisees. You have to wait forty-five minutes for the
answer. You go home exhausted.
August 29th or 30th
Meet your classes for the first time. Take roll. Half the class is not
on the roll sheet, the other half isn't in the classroom.
Students ask lots of questions only answerable if you've been at your
institution for fourteen years:
"Is this the right class for
me?"
"Are you gonna do transcendental functions?"
"How do I transfer out of here?"
"Will you sign my schedule?"
"Did I buy the right book for this course?"
"Where do I go to change my registration?"
"I can't make the final exam at the scheduled time. Can I have a
makeup?"
[Possible answers: "Maybe.
Let's talk about that after class." "I dunno; ask me again next time."
"Go to (the appropriate) office." "Not at this school." "Maybe." "Our
book is..." "Go to (the appropriate) office." "We have some time to
figure that one out, don't we?"]
Week Two
Two new students arrive. They want you to tutor them on what they
missed.
A third student keeps coming
to your office for at least an hour every day for help with his
homework. Meanwhile, your own first big homework set in analysis is due
tomorrow.
Week Five: Exam One
Thursday night is the first exam in calculus. You have to do an extra
help session on Tuesday night, 7 until 9 p.m., for your and two other
TA's sections.
Fifteen minutes into the exam
itself, you realize that proctoring is one of the two or three most
boring things you have ever done in your life.
You spend the next two
minutes calculating in your head how many times the lecturer must have
done this in his or her lifetime.
After the exam is over, you
spend the next two days grading over five hundred copies of one
problem.
Week Six
You try to catch up on the class work for your own courses, but your
students keep coming to you outside office hours for regrades on
various problems from the first exam.
Week Seven
You get behind in your own work, so you decide to try a new teaching
strategy called showing up unprepared. That goes so badly that you
decide never to try that again.
Week Eight
One of your students asks when you are teaching next semester, because
he wants to get you again. You can't say why, but this makes you feel
really good for the rest of the day.
Week Nine: Exam Two
This week is like week five, the "exam one" week. It would have gone
better, but you have an analysis midterm of your own the day before
your students' test, and a huge algebra problem set due two days after
the calculus exam. Instead of grading for twelve hours, it only takes
you ten. And, the numbers of regrades goes way down. But it's still way
too much work.
Thanksgiving
Attendance goes down drastically; students head home even though
there's an exam the week they return. Later, they will say on the
course evaluations that you and the lecturer didn't motivate them in
the course.
Week Thirteen: Exam Three
Students are anxious and surly because they didn't get enough time to
study. The fact that this week's exam was announced in August
doesnÕt seem to matter, nor does the fact that they did skip six
classes in the last two weeks. But it's your fault that they didn't
know enough about triple integrals.
Week Fourteen
Break before finals. Students you haven't seen for seven weeks show up
for review sessions. Afterwards, two come to ask you, "What can I do to
pass this course?"
You read your course evaluations. You thought you knocked yourself out
helping students; they say you're "average," except for the category of
"organization," where you are below average.
December 15th
Like week five, but much, much worse. You have two take home exams of
your own to hand in, plus one in-class final on the day after your
students' final--and you have to grade over five hundred papers again.
December 20th
You have a grading meeting with all the TAs and the lecturer. One TA
forgets his grade book; you have to wait forty-five minutes to
calculate the class median. Then you discover that five of your
students are from one to four points below the cutoff for an A.
Further, two of your class attendees are not on the roll sheets, while
two other students you have never heard of are.
December 21st
As you turn in your last take home in algebra, your most annoying
student walks up to you and demands to meet with you today about the
fact that he got a B+, when "I obviously deserved an A-, at least!"
January 2nd
While watching a football game, you suddenly realize, "I did it! And it
wasn't that bad. And, I could do it again. But not just yet."
Return to Top
Maybe you haven't heard, but there aren't any tenured
TAs. Thus, today is a good day to start looking for a job.
Go to a word processor. Put together a resume. Of
course, your name, address and all the standard categories are on it.
If you are a new TA, you will probably have lots of white space where
your teaching, research and professional activities
should be. This white space is what you need to fill up over the next
few years; and, if you are an older TA, you should have done so
already.
So, under teaching, you should try to get as
varied a background as possible. Don't always TA in the engineering
calculus section, even if you like that best. How is it you know you
won’t like the business students instead?
Ask to teach your own section of some course, then
put together a small portfolio with a couple of your lesson plans and
all the quizzes and exams. Put copies of your good (they are good,
right?) course evaluations in that portfolio, along with any nice
e-mails or letters from students.
Grade an advanced course in your area rather than
teaching all the time. Ask to give a lecture or two in the course, then
put that on your resume.
Write a teaching statement to put into your
portfolio. You will need such a statement later on, when you go up for
tenure, so now is a good time to start to hone one.
In the research category, once you have
reached the third year, you will probably have picked out a specialty.
You can then write a short description of your topic for the resume. At
the same time, you should write a couple of pages for the portfolio on
the recent history of your topic, what the important theorems are, who
is working in the area, and where your thesis topic sits relative to
the work of others in the field.
If you are in your first or second year and haven't
done any research yet, maybe you can describe an REU paper you wrote or
your undergraduate thesis at your previous school.
In terms of professional activities, you can
list any talks you have given. If you have never given talks, now is
the time to start planning to do so. You have the possibility, from
your first year in graduate school, to speak to the Math Club or the
graduate student seminar. You can also call up you old professors at
your undergraduate college and ask them if they'd like you to speak at
their seminar the next time youÕre in town.
You should also join the AMS and the MAA. Doing so
shows that you have a real, professional interest in mathematics, both
as a research area and in its teaching aspects.
Another professional activity that is more important
than you may think at first is to go to MAA and AMS sectional and
regional meetings. These are usually within a couple hundred miles of
your home institution, and they give you a decent idea of what the
current teaching issues and hot research topics are.
Return to Top
The first time you are asked for a letter of
recommendation, you will probably be surprised. As the surprise wears
off, you will experience a sinking realization that you have no idea
what to write.
If you really don't know the student, of if you
cannot write a generally positive letter, you may suggest that he or
she "might want to find someone who knows them better and can write a
more specific letter."
On the other hand, if you do remember the student
and can write a positive letter about his or her potential, by all
means do so. Here’s how:
First, note that you are being asked for a
professional assessment, so write as careful a letter as you can. To
help yourself and the candidate, find out what the position is he or
she is applying for, and what the reason is for his or her needing the
letter. Is it for a summer job, maybe? Or is the student transferring
to another institution, or maybe asking for a letter to a graduate or
professional school, or for a scholarship? Each position suggests the
qualities the applicant will be asked to show, and will thus affect the
letter you write. Ask for a resume, and ask the student his or her
reason for interest in the position. A description of the job can also
be helpful; get one from the student if you can.
Start your letter with: "[Student] has asked me
for a letter of recommendation for [your job]."
Then: "I have known [student] since [date] in
[capacity]."
Next: "My specific comments as to what I saw..."
Here you can address such issues as "came to class," "asked good
questions," "got a good grades," "worked well with others," and the
like.
Now address the qualities described in the job
description as being desirable: "motivated," "self-starter," "team
player," etc., fit here. If you don't know what qualities the school or
company is looking for, try to intuit what the future employer would
want.
Finish with a comment as to the student's
potential.
Keep a copy of the letter, by the way. The student
may well be back with another application in hand. And, the employer
may call you for a follow up comment. "Did you really mean what you
wrote?"
Return to Top
Generally, you will be asked for one of
two types of mathematical talks during your time as a graduate student:
the "long talk," of about an hour--and the "short one," about ten to
twenty minutes. The long talk also splits into two: research-related,
and expository.
Believe it or not, of the three types,
the research-related one is the easiest to give. This is mainly
because, first of all, you wouldn't be asked to, nor would you accept,
an invitation to speak about your research unless you were actively
doing some. Further, now that you are actually doing research, you are
somewhat of an expert at the topic. Thus, describing the history,
stating the main theorems, discussing a couple of relevant examples,
stating and "heuristically proving" yours and others' results, and
listing some open problems should all be no problem for you. The
hardest part of such a talk is making sure you keep it interesting for
the listener: What makes the problem significant? How can you make the
audience care? What kinds of examples will give us a feeling that we
understand the broad outline of the topic? What other examples or
results will indicate the depth or power of the problem?
Putting such a research talk together
is useful for you because it will offer you insight into your own work.
Further, you will need to have such a talk for when you go onto the job
market, so now is the best time to start preparing it.
A good expository talk is not
really hard to put together; it just takes time. What is a good topic
for undergraduates that isn't usually covered in the traditional
curriculum? Squaring the circle, say? Nonstandard analysis?
Nonorientable surfaces? Cryptography? How about "Three new uses for
matrices"? Or, "The statistics of baseball"? All these, and more, have
been discussed in books and papers that you can refer to, and build
from. These make good talks for sectional meetings of the Mathematical
Association of America, or to graduate student seminars and colloquia,
or to students at the colleges at which you are interviewing for a job.
As with the research talk, the
expository talk should be coherent and interesting. Here, however, you
should emphasize the history, examples and open questions of the topic,
rather than precise definitions, proofs and other "dirty details"--not
that these aren't important, but the interested listener can look them
up later.
The short talk, which I will
call the "ten-minute talk," is the most difficult, even though most
neophytes think it is the easiest. The reason it is difficult is clear
when you think about it: there is very little time in which to say
anything important or useful. Many people reason, "Since they aren't
giving me any time to say anything meaningful, I'll just 'wing it'."
This is a big mistake; most young mathematicians end up having to give
short talks, and the worth of their work is gauged by the quality of
their presentation. (My first talk was on the last day of a national
meeting. I had an audience of three: the previous speaker, the next
speaker, and A.H. Stone. Stone took notes on my talk.)
So, how do you give a quality
ten-minute talk? Start by "thinking backwards": What's the most
important point you need to make? Make sure that point comes in the
talk somewhere toward the end. Often people will think, "It's that new
theorem of mine that's the really exciting thing," but are you sure of
that? So often, the audience thinks that it's corollaries or examples
that come from the theorem that are important, or maybe the questions
that get answered or don't get answered that are really the reason for
the talk.
So now you know where your talk is
heading; what do you need to get there? Is there a definition you need
for comprehension? Then put it in. Similarly, an illustrative example
will usually give the audience more sense of the topic than proofs.
Also, make sure you indicate why the topic is important; whose theorem
are you extending? What question are you answering? And what new
questions have you uncovered? What are you going to try to prove next,
and do you want to tell the audience this? (After all, you may want
more time to get your own proof.)
Of course, in ten minutes you won't
have time to prove things--at most, you'll have time for a statement or
two of intuition--and then, usually only in response to a question from
the audience.
So your talk starts: "In 1987, Jones
proved... but he left open the question of.... I have a partial result
in the hemi-demi-semi-typological case. First, I will define.... A
simple example is.... Here's the main theorem.... It is proved through
the use of Smith's theorem on... An example of a space where the
theorem worksis... and let me finish with a conjecture and some
comments."
Now you have a talk, and you think it
will be ten minutes long. Try it out on your office mate, and don't be
surprised when it comes in at eighteen minutes and you feel pressured
all through the practice session.
You can save some time by putting as
much information as possible on transparencies; this should be worth
about three to five minutes. Then there's the second example that you
thought was so good, but your officemate found redundant; drop it. The
fact that it could give the audience some new insight is a point you
can use if there's a question period, or perhaps when someone stops you
in the hallway two hours later to tell you how much they liked the
talk.
Now you're down to twelve minutes. You
can drop a definition that your officemate didn’t know but everyone in
your area of mathematics does. When you also remove two of the three
"intuitive comments" you were going to put in at the end, you think you
have nine minutes. So you try the talk on your officemate one more
time, and sure enough, it is just under the ten-minute mark, although
you still wish you could have put in one more of those last intuitive
comments. At this point, you must note that it's OK to feel that you
could give the audience more; that intuitive comment could also come up
in the question period, if people are interested. And if they aren't
interested (sometimes audiences just aren’t), well then, one more
comment will just make them that much less excited.
What else? Well, in general, put as
much as possible on transparencies. This will save time, and it'll keep
you from forgetting to say something important. Another general point:
try not to appear rushed. If you need to talk fast to "get it all in,"
well. OK, but if you look worried, people will have less confidence in
your work.
Enjoy the experience as much as you
can; after all, you know as much about the topic as anyone in the
room--except, maybe, your advisor and one other person. And, those are
the people you can learn from about where you should be going
next--that's really why you came to speak, yes?
Return to Top
For you as graduate student, the
prospect of getting a faculty position at a college or university may
seem like entry into paradise, and to an extent it is. But, before you
decide that being a faculty member is all wonderful, let me explain
some of the new challenges you will encounter.
The relief you will feel from
having found a job will be almost immediately supplanted by a sense of
aloneness, insecurity and avoidance.
When you were a graduate
student, you had a cohort of other "nth years" who were
going through roughly the same crucible you were. Now, however, the
other person who was hired at the same time as you may well see him- or
herself as vying for the lone tenure slot. Your conversations will
become guarded, if they exist at all. The older faculty, at the same
time, will seem distant. Maybe it's because they have their own
problems to contend with, maybe they really are "just quiet,"
and maybe it really is because you replaced the person they wanted to
have hired. In any event, you may be spending a lot of commuting time
in the mornings and evenings trying to "psych out" other people's
motivations.
It helps to keep telling
yourself that the institution you are going to has already made a real
investment in you--they do want you to succeed, even if it doesn't seem
so at times. Try to ignore what seem to be snubs, keep to the track of
doing what's best for your students and for mathematics, and save your
disagreements for extremely significant issues, whatever those may be.
Try to build a coterie of colleagues with whom you can work, share
ideas, talk about mathematics, and--most importantly--go to lunch or to
the gym.
Speaking of investments, you
will probably think that the fact that you are now earning about three
times what you made as a TA makes you rich. Well, not as much as you
thought, because your deductions from income for taxes and social
security, for your medical benefits, for your retirement (forty years
hence), and maybe even for your campus parking place, will probably
take about forty percent of that paycheck. That leaves you with almost
double what you had as a TA. But wait--you forgot that you used to live
in "collegetown" with four roommates, and that's going to change, too.
Not to mention the car you'll buy to fill up the parking place you're
paying for, right?
Since you are new to the
institution, you won't know (and sometimes you won't respect)
the campus culture. If you are careful and politic, you will have an
opportunity to redirect the culture--within reason. But first, you must
build the confidence of others:
You must listen first, before
offering advice,
You must make realistic suggestions that come out of the current modes
of operation, and
You must be willing to accept criticism or correction.
Remember, your new colleagues
can very quickly get tired of hearing you say, "When I was at Famous U.
we did it this way...."
At your new institution, you
will be evaluated based on the quality of your research or scholarship,
your teaching, and your service.
Once upon a time, research was
all-important. This meant publications in journals like the Transactions
or the Bulletin. Textbooks, pedagogical papers and even
articles in the MAA Monthlydidn't really count. Recently,
partly through the impetus of the Boyer report (see reference [6]), the
word scholarship has come to be more emphasized, and the more
educationally oriented types of work have come to be recognized as
activity worthy of consideration for tenure.
As tuition has increased on
college campuses, teaching has also become more important; more
students and parents ask, "What am I getting for all this money?" This
is one reason why it is important for you, as a graduate student, to
get as varied a set of TA assignments as possible, and to take teaching
issues seriously before you get to your tenure-stream position, and
tenure-issue problems.
Service has often been
misunderstood as a criterion for tenure. There are some schools that
consider the presidency of the local runners' club as "service," but
for the most part, colleges interpret the word to mean work in your
specialty. Thus, if you are a math educator who gets elected to the
school board, that's service. The word also refers to committee work in
your department or at the college level--being on the arts college
bylaws committee, for instance.
Service is important,
and others do evaluate it. If you do not take it seriously, people on
the tenure and promotion committee will hear about it.
I could say a lot more about
what life will be like after you get a PhD and a job in academia, but
this is enough for now. After all, you're still trying to get out of
graduate school, right?
Return to Top
I was talking to a full professor a few days ago
when suddenly he asked, "What does a provost do, anyway?" I hope I
don't insult this person by saying that I am surprised that people can
spend upwards of thirty years in colleges and universities and still
not know very much about how governance works.
Of course, every school has a different way of
administration. For instance, colleges, which usually concentrate on
undergraduate education, are not governed in the same way as
universities, which have graduate schools and thus graduate deans. I
will describe one model, but there are many others. You can use mine as
a template for comparison.
Many people make the mistake of thinking that
governance starts with the president of the university. It does not.
For example, read the back of the ballot for trustees of Cornell
University: "The University Bylaws vest supreme control over the
University and all its divisions to the Board of Trustees." The board
then delegates its responsibilities through a central administration
and committee structure. At the top of the central administration sits
a chief administrative officer, usually called the "president," who is
responsible for academic quality, fundraising, public relations,
financial management and institutional integrity. And, often, the
success of the athletic program. Vice-presidents, for research, for
finance, etc., are then subordinate to the president.
Most universities also have a chief academic
officer, called a provost, who often oversees the day-to-day workings
of the various colleges within the university; e.g., the arts college,
the engineering school, the business school, and the like. Some
institutions have separate provosts for medical and business schools.
Again, provosts tend to have a number of vice-provosts whose
responsibilities are spelled out in their titles.
Next come the deans of the various colleges
(arts, engineering, etc.), along with a couple of deans you may not
have thought of before: dean of the faculty and dean of the graduate
school. The dean of the faculty usually represents the interests of the
faculty. The dean of the graduate school is most important to you
because he or she has great sway over such matters as how many people
are accepted into various graduate departments, how TAs are allocated
by department, and what are the requirements for the PhD.
In small colleges, the administrative structure
can be quite compressed; I know of some schools where there is a
"president" who acts as president, provost, and vice-provost. The next
officer in the chain is then the dean, of which there is only one.
There is no need for a separate dean of faculty when a college has
(say) forty faculty members, and of course, a college without a
graduate program needs no dean of the graduate school.
After the associate and assistant deans of
admissions, advising, and "dean of students," come the department
chairs. Although elections may be held in departments, chairs serve at
the behest of the dean. Thus, although a department faculty may vote to
recommend a particular person, that vote is not binding on the dean.
Departments serve important functions. They
determine their own courses of study, subject to college approval; they
initiate the tenure process for faculty; they determine the
requirements for the major and for graduation. Most importantly for
you, departments admit graduate students and offer them financial aid.
Your department is very likely to have a
director of graduate study (DGS) who oversees your academic life from
start to finish. You will also have a thesis advisor, of course, but he
or she will be much less involved in the administrative "paper trail."
So now we can figure out how you got a paycheck:
Last year, you applied to "the graduate
school," which means that a secretary in the graduate school office
opened and collected your application form and recommendation letters.
Those materials were then sent from the graduate school office to
another secretary in mathematics. That person handed them to a
committee supervised by the DGS. After a lot of internal departmental
deliberation and consultation with the dean of the graduate school
about how much funding would be available this year, the committee
chose you as one of its new students [Congratulations!]. At some
institutions, this admission included an offer of support as a teaching
assistant, in which case your letter of acceptance said something like,
"You will receive full tuition and fees and a TA stipend worth [some
amount of] dollars." Alternatively, you may have been offered
admission, but told that TA support was pending "depending on graduate
school funding." [Now you know what that sentence meant.]
A few months later, you showed up on campus,
where you were immediately handed a flurry of forms to fill out ("I-9.
What's that? Of course I'm a U.S. citizen.") by the same secretary who
sent on your application to the DGS. (Now you know why that secretary
looked at you as if you were an old friend.) You found out what your TA
assignment was to be, and the secretary filled out more forms for the
accounting and payroll offices. Two weeks later, what do you know! You
got paid! Easy, right?
Return to Top
I have been called on a
number of times over the years to sit in on classes for the purposes of
evaluating TAs and faculty. At first, I had no real idea of how to do
this; I sat and listened, made a note or two and at the end said,
"Thanks, nice class." I felt mildly uneasy about the whole system, but
there didn't seem to be any role models for me to learn from or compare
myself to. Only after a while did I start to develop my own techniques
for dealing with the process.
Now when I am asked to
evaluate a graduate student or faculty member, I first make sure to do
so at the convenience of the person I'm going to be watching. I ask
them which day they would prefer, given the way they are teaching their
class. There is no reason for me to show up on an exam day, for
instance. On the other hand, there is every reason to attend when the
lecturer is giving a lesson he or she is particularly enthusiastic
about.
I come to class early, ask
the instructor if it is still OK if I sit in (just in case plans have
changed at the last minute), and then take a seat toward the back of
the room. Although I prefer to be unobtrusive, students usually notice
that I am in the room, and I am aware that that fact sometimes changes
the dynamic of the class.
I take notes as the class
proceeds, recording the topic, the instructor's approach to it, and
especially the dialogue that takes place: S: "I didn't get that last
step." I: "OK, I'll go back." When I see what might have been a missed
opportunity for the instructor, I put a comment in brackets in the
margin of the paper for later mention. I also record seemingly trivial
facts, such as the time the class starts and the numbers of students
the instructor has. If class starts late, that is a point to discuss
later; if it is because of a late bus, that's one thing, but if it is
part of a pattern that comes from the instructor's nervousness or
dislike of teaching, then we have a matter to talk further about. If
there are too few students in class, it may be because today is the day
before semester break, or perhaps students are generally not coming to
class because they see it as useless.
After class, I ask the
instructor if he or she would like to have some coffee and talk. I like
to chat right after class, if possible, because the material is still
fresh in our common memories. I always start with what I liked about
the class, and I begin with the trivialities--"You speak loudly enough,
and I like the way you wrote everything on the board so that students
could take notes more easily." Many of the people I observe are better
teachers than I was when I began, and I tell them so.
Then I move on to more
serious issues. If students asked lots of questions, I consider that to
be a "real plus"--it shows that they are not cowed by the instructor,
even if he or she couldn't answer all the questions. I tell the
instructor this: "It's good to see how you encourage the students to
respond to the material by letting them ask you questions. I really
liked the way that you were willing to field that question about
[something you messed up]. It showed the students how you were
thinking, even though you still need to work on the answer for next
time."
If the instructor has done
a good job with "mechanical skills," I then feel free to discuss deeper
issues of teaching. Referring to my notes, I sometimes offer
suggestions as to how to approach the classroom material from a more
sophisticated perspective.
Let me interject some
points here that I have come to over time. I classify teaching into
three levels. At the first level, the instructor has an awareness of
and an ability to handle the most basic aspects of teaching. He or she
writes clearly, doesn't stand in front of the board, speaks loudly
enough, comes to class fully prepared to discuss the assignments,
treats students in a courteous manner, and can understand and give
basic responses to questions asked in class. A new instructor should be
able to learn these skills through a decent TA training program and to
perfect them during the first semester of teaching.
A level two instructor is
able to motivate the material being taught, perhaps by bringing in some
relevant additional material, thinks on his or her feet, is able to
answer without difficulty simple questions on lecture material and
homeworks, and is willing to spend some additional time with individual
students.
At the third level, the
instructor now knows what the student is "really asking" when he or she
asks a particular question. This instructor can also say "where the
course is going," and can give solid, coherent responses to questions
of the "what's it all good for" variety. He or she acts professionally
in all classroom situations.
Now let me return to the
topic at hand. When I observe an instructor, I try to get an idea of
which of the three levels he or she is on. My goal is then to reinforce
good habits by complimenting this person on having attained the
appropriate level (I don't say it this way, of course), and then to
suggest one or two ways in which he or she can continue to get to the
next level. I always work with positive reinforcement and suggestions
built from what I have seen in the classroom. "I really like the way
you were able to do all the problems the students asked. You were
obviously prepared. By the way, remember that question twelve, the one
about integrating the trig function? I was noticing how you had a
chance there to tell the students that they'd see those kinds of
problems again if they take the next semester course."
I'd like to point out
that some people strongly dislike being evaluated, and most people have
problems with some aspects of evaluation, yet the process can be made
to be a helpful one. Since more and more schools are requesting
accountability from their faculty, there is a reasonable possibility
that you will be asked to go through an evaluation. You can help make
this a more salutary one by realizing that it is ultimately designed to
help you. After all, your students are watching you all the time, and
their critical faculties are not turned off; why not let a colleague
watch you, too. He or she may be able to say just what you need to
improve your teaching to level three--and beyond.
Return to Top
It is commonplace to hear others
say, "You can't teach teaching. It's ingrained." While I agree, to some
extent, about teaching being ingrained--part of one's personality--I
can't agree that teaching cannot be taught.
Maybe it's because I have
been trying to teach people how to teach for over fifteen years now,
but I do believe that some aspects of instruction are definitely
teachable. For instance, if you go back to a previous section, What Does an Evaluator
Evaluate you will see that I talk there about levels one, two and
three of teaching. Certainly, such level one aspects as speaking
loudly, writing clearly, not standing in front of the material on the
board, coming to class prepared, and treating students with respect are
all teachable. In fact, many people would say that they are such common
sense that they need not be taught--but we have all seen too many
instructors who seem to have skipped that lesson. My belief is that all
the level one aspects can be taught during a one-week TA training
session, and then reinforced throughout the first semester's teaching
so that they become close to second nature.
Further, I would continue that level two aspects can also be taught.
When I look at these qualities, however, I do not see "common sense"
principles, but rather teaching traits that must be developed over a
period of time, through teaching itself, but also through mentoring,
peer suggestion, and perhaps also through taking some teaching courses.
Unfortunately, college teaching courses seem to be a rarity these
days--we must hope that they will grow in number.
So simple teaching, "good teaching," I would claim, is teachable.
Any graduate student who uses his or her time in graduate school can
become a better than adequate college-level instructor. There, I said
it, and I'll say it again: Good teaching is teachable.
"Great" teaching; now that's something else. Although to me some
aspects of great teaching are approachable by us mere mortals, there is
also a sense in which teaching is an expression of personality, and
just as some of us don't really want to be stock brokers, others simply
aren't geared up for teaching. Is this bad? I don't think so, unless we
find ourselves having to teach in order to live; in that case, I still
think we should "give it our best, and not apologize for our supposed
shortcomings."
I know that I myself was an incredibly shy child who never wanted to
be called on to recite, and at times I still have more-than-normal
problems with the concept of standing in front of an audience. Yet, I
have managed to teach classes of up to five hundred undergraduate
students and give serious mathematical lectures to working research
mathematicians (usually the latter is easier than the former). But I
think that I can do so only by wrapping myself completely in the
mathematics. Early in my career, I used to try to memorize every detail
of my lecture, hoping, I guess, to fool people into thinking that I
knew all about the material. That would work only as long as I never
had to look at my notes; once that happened, it was all over for any
"quality exposition." Later on, I realized that I could use
transparencies, notes, even full text, whatever it takes to get my
point across, and that, by trying to use my memory, I was often
depriving my audience of the gist of my talks.
Some people have told me that it is possible to find a model of good
teaching in those who have taught us well in the past. Well, maybe. I
recall some people who taught me well; while they have definitely shown
me many things about teaching, and about their fields of study, they
didn't seem to conform to any single mold.
One of the first college instructors I had, and who I thought was
"gifted," broke most of the rules I might try to enunciate. I won't
give you his name or his school, for reasons that will be clear from my
description. First, Professor E., an English professor, would drink his
lunch, as we used to say. Then he would start a seventy-five minute
class at 1:30 in the afternoon with a thirty- to forty-minute standup
comedy routine with no basis in the classroom readings or discussion.
At some point in the routine, he would stop, sigh, pull out some old,
yellowed papers from a severely beaten up briefcase, and say to the
assembled multitude, Well, I guess I have to say something about
Hawthorne. Don't feel you have to listen; you can go to sleep now, if
you wish. Then he would proceed to offer a careful, lucid analysis of The
Scarlet Letter and its implications for Hawthorne's life and the
sociology of early New England. In case you think that Professor E. was
uniquely sensitive to students or the community's concerns, I'll just
point out that he told us one day in class that he stayed in our
backwater town only because, "as I have told the faculty many times,
this is a place to which culture is coming. Although when, I don't
know."
Another professor at a different school, a mathematician, was
incredibly shy; when he wasn't teaching, he seemed incapable of
conversation. Yet, when in class, he gave the kind of mathematical talk
that made every student sure that he or she understood every
detail--until we tried to do the exercises. When we would come to ask
him about the problems we were stuck on, he would say, "Oh yes, that
one got me for a while, too. Let's see if we can figure it out again."
Professor M. was an incredible motivator who allowed us into his mind.
He took apart proofs as if they were watches and then put each piece
back together exactly where it should go.
A third instructor was a stickler for proofs in an engineering
calculus course. Somehow, he was able to convince the engineering
students that "You need proofs to understand why things work; otherwise
your bridges won't stand up!" And, he had the force of personality to
make his opinion stick. Thus, when he took the class through the
difference between a hypothesis and a conclusion, when he showed us by
examples how each hypothesis was necessary to the proof, when he
counted hypotheses in his proofs, we listened, and listened
carefully--and not because the material was going to be on the exam.
What have I learned from these instructors? Well, definitely not
that I should drink to excess before going to class. Nor do I do
standup comedy for my class--although I do sometimes exhibit a sense of
humor, I can't remember, let alone tell a canned joke in any
circumstance. While I am shy, I don't walk around hoping that people
will not talk to me, nor do I try to convince engineers that proofs are
"where it's at."
I guess what I have learned from all this is that great teaching
comes in all forms, but that mainly it comes from the delicate
interaction between two personalities: That of the instructor, who
somehow conveys a love of learning, and the student, who comes ready to
absorb and apply what the instructor has to give--no matter how
imperfect that instructor may be outside his or her domain of
expertise.
Return to Top
Case Study I
You have arranged your calculus class so that
you collect homeworks in class each week for grading. You have told
students that ten percent of their final grade will come from these
assignments. After five weeks of the semester, you find out that
complete homework sets are sold in the campus bookstore, and that they
are also available in the library. What do you do about this? Do you
change your grading policy? Do you stop collecting homework? Do you
give credit for homework that you have already collected?
How do you change your examination and
grading policy for the next semester? Should you try to change
department or university policy about what kinds of books and course
materials are sold in the campus bookstore?
Case Study II
Assume, for the sake of argument (It may even
be true!), that you are a female TA. You have been assigned an
overwhelmingly male class of calculus students who are, to say the
least, rather boisterous. One student in particular has a tendency to
mention your clothes, your hair, your personal appearance at or near
the start of class every day. How do you respond? Do you publicly
reprimand the student in class? Do you change your manner of dress? Do
you change your behavior? Do you ask the lecturer to intervene?
Exactly what is going on in class, and how do
you handle it?
Case Study III
You are proctoring an exam. You notice that
Student A is looking at someone in the row ahead of him, but that the
person he is looking at seems too far away for him to be copying
anything from. Occasionally during the exam, you go back to the general
area where the students are sitting, but you see nothing unusual from
either him or from Students B and C sitting one row ahead and about six
and eight seats away. At the end of the exam, you take Student A aside
and mention that he shouldn't spend so much time "looking around." He
responds, "Didn't you see what was going on? Those students were
cheating! I want you to take them to the judicial administrator, and I
want to testify!"
How do you respond? What do you do about the
situation? Do you take the case to the judicial administrator? Do you
have Student A testify?
Case Study IIIB
You took Case Study III to the judicial
administrator. Sure enough there was cheating going on, which you were
able to prove by using copies of the exams. You did not ask Student A
to testify. Student B admitted "looking over" the exam booklet of
Student C, but Students B and C both claimed that Student C was not a
participant in the malfeasance. Student A comes to ask about the
outcome, and when he finds out that Student C has not been convicted,
he is again upset. "But she was involved, too! She was showing him the
answers!"
Now what do you do?
Case Study IV
It is Sunday night; the second exam in your
second-semester calculus class is scheduled for Tuesday. You are in the
middle of a review session for three of the sections of the class.
After an hour's worth of student questions, you realize that your
answers are not getting through. Students do not understand power
series, a topic that will surely be on the exam. On Monday you are
supposed to go on to a new topic, so you feel some time pressure--after
all, the next exam will also be important.
What can you do to help the students prepare
for this exam? Should you simply ignore the syllabus? How could you do
this without incurring the wrath of the course leader?
Case Study V
On Tuesday you decide to go to a movie on
campus. While standing in line with your roommate, you meet one of your
students. You begin a conversation, which you continue in the theatre.
Two days later, the TA coordinator receives a dyspeptic letter from
another student in the class, saying that you were "out on a date" with
one of your students, that the student you were with is "the soon-
to-be former" boy- or girlfriend of the letter writer, that you are
trying to break up the relationship, and that the TA coordinator should
not tell you about the letter because you will just try to fail her or
him in the class, "...which you've been trying to do all term anyway."
The TA coordinator calls you in anyway to
discuss the situation. What do you do?
Case Study VI
You hear from one of your students that one
of the other TAs in your large-lecture class is "favoring his own
students." In particular, your student claims that "the other TA is
changing grades for his own students but not for others, and he
actually showed his students how to solve two of the exam problems
before they appeared on the test the other night."
How do you handle this complaint? Should you
confront the other TA? How do you verify the accusations? Can you
verify them? Should you verify them?
Case Study VII
You are doing a review session for tomorrow
night's exam. One of the students asks you to solve a problem that you
know will be on the test. What do you do?
Case Study VIII
During your discussion of final grades, the
instructor you are working with announces a new grading policy: no
student can have a final grade raised unless another student’s grade is
lowered by an equivalent amount. You immediately think about one of
your undergraduates who spent two weeks in the hospital recovering from
surgery. She then got a terrible grade on the next exam, but righted
herself enough to score 97% on the final. Should you give her the A you
think she deserves, even if this means finding another student whose
grade will have to be lowered from, say, a B to a C? Should you just
forget changing any grades? Or, should you argue with the instructor
about his grading policy, and if so, what do you say?
Case Study IX
You are TAing another large lecture course.
At the final grading discussion, the faculty member in charge announces
his "ironclad policy" that no student with less than 490 points out of
500 is to be given a grade of A+. You have a student who received 486
points, but who would have done better except that she had a death in
the family. You decide to give her an A+ anyway, which you do without
consultation with the "czar."
Two days later, another student from the
class comes to see you. "I got 488 points, higher than my roommate, to
whom you gave an A+. I demand an A+, too, and I intend to fight this
through the administration if you don’t give me one."
Now what do you do?
Case Study X
It is week three of the semester, and your
class has shrunk from twenty-five to nine. You happen to see one of
your former students outside the cafeteria; he seems not to want to
talk about it. After some prodding, he says, "It was your accent," then
he walks away.
What do you do? Do you think it was your
accent? What else could the problem be? Where can you go for advice?
Should you discuss the situation with your class? With fellow TAs? With
an instructor?
Case Study XI
You have won one of the college's TA teaching
awards. Your response is happy, but also ambivalent. It is nice that
your students like your teaching, but at the same time you feel your
teaching isn't that special. As you pass the coffee room, you hear
another TA saying loudly, "Yeah, but he's just an actor. It's all show."
What do you think? Is it all show? Are you
"just a performer?" And, how much of teaching should be performance?
Return
to Top
References
- Angelo, Thomas and K. Cross, Classroom Assessment Techniques.
San Francisco: Jossey-Bass, 1993.
- Belenky, M.F., B. Clenchy, N. Goldberger and J. Torule. Women's
Ways of Knowing: The Development of Self, Voice and Mind. New York:
Basic Books, 1986.
- Bloom, Benjamin, ed. Taxonomy of Educational Objectives.
New York: Longmans, Green, 1952.
- Bonwell, Charles and J. Eison. Active Learning: Creating
Excitement in the Classroom. Washington: George Washington
University, 1991.
- Boyer, C. and Uta Merzbach. A History of Mathematics. New
York: Wiley, 1989.
- Boyer, Ernest. Scholarship Reconsidered. San Francisco:
Jossey-Bass, 1990.
- Chaffee, John. Thinking Critically. Boston: Houghton
Mifflin, 1997.
- Cohen, Marcus et al. Student Research Projects in Calculus.
Washington: The Mathematical Association of America, 1991.
- Countryman, J. Writing to Learn Mathematics. Portsmouth,
NH: Heinemann, 1992.
- Culver, R. S. and J. Hackos. "Perry's model of intellectual
development," Engr. Educ. 73, 221.
- DeNeef, Leigh and C. Goodwin, eds. The Academic's Handbook.
Durham: Duke, 1995.
- Eble, K. The Craft of Teaching. San Francisco:
Jossey-Bass, 1988.
Edwards, C. The Historical Development of the Calculus. New
York: Springer-Verlag, 1979.
- Highet, G. The Art of Teaching. New York: Vintage, 1977.
- Hilbert, S. et al. Calculus: An Active Approach with Projects.
New York: Wiley, 1994.
- Klein, Morris. Mathematical Thought from Ancient to Modern
Times. New York: Oxford, 1972.
- Lewin, M. and Thomas Rishel. "Support Systems in Beginning
Calculus." PRIMUS. V(3). 275-86.
- McKeachie, W. Teaching Tips. Lexington, MA: D.C. Heath,
1994.
- Meier, J. and Thomas Rishel, Writing in the Teaching and
Learning of Mathematics, MAA Math Notes 48. Washington: The
Mathematical Association of America, 1998.
- Perry, William. Forms of Intellectual and Ethical Development
in the College Years: A Scheme. New York: Holt, Rinehart and
Winston, 1970.
- Piaget, J. The Child's Conception of Number. London:
Routledge and Paul, 1952.
- Pólya, G. How to Solve It. Princeton, 1945.
- Rishel, T. "Writing in the Math Classroom, Math in the Writing
Class; or How I Spent My Summer Vacation." Using Writing to Teach
Mathematics, ed. A. Sterrett. MAA Math Notes 16. Washington: The
Mathematical Association of America, 1992.
- Stewart, J. Calculus: Concepts and Contexts.Pacific
Grove, CA: ITP, 1997.
- Thomas, G. and Ross Finney. Calculus and Analytic Geometry.
Reading, MA: Addison-Wesley, 1996.
- Vygotsky, L. Thought and Language. Cambridge: MIT Press,
1986.
PAGE 96
PAGE 128