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...
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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?
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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.)
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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?
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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.
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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.
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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].
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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.
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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.
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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?
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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."
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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