Answer: This is the MFAQ, the most frequently asked question.
As you know, Harvard, as many other colleges, experiences grade inflation.
This is unfortunate, because it allows students less and less to gauge their
progress and for students near the AB treshold, it is essentially a matter of
luck on which side they fall. The quantisation by letter grades is too
rough and with time, most of the grades had been pushed into the A-B range.
This discourages top students to work harder (I already have an A) and
students with a B grade (I can not get into the A grade).
A simple solution to the problem would be to get rid of the quantisation and to
submit a number between 0-100 to the registrar, forcing the distribution to have
a fixed mean and standard deviation. But the world is not perfect and so,
every teacher at the end of the semester ponders over the problem to figure out
a "curve". I keep three constraints fixed: there should be maximally 40 percent
A and A- and the average should be near the threshold of B to B+. Furthermore, the
standard deviation window should include only A and B grades. Then,
according to the distribution, the cutoffs are set. This is usually not
fixed from year to year. It depends on how difficult the exams are percieved,
where natural flat parts of the curve are and depends at the analysis of grades of a few
"gauge" students, I look at closely and where I can see what was the reason
for success or for difficulties.
Question:
Q: Why not fix and publish the cutoffs of the curve beforehand?
Answer: In other words, why not skip the "curve" at all?
It would be possible but a bit risky, especially with a resurrection
policy. It is very reasonable in courses, where the outcome of exams can
be precicely predicted (courses with a lot of memorizing material or
routine problems like in early calculus courses), but not in courses, where
the material is more complex and the exams are more conceptual. It can be even
less predicted when projects are included. In multivariable calculus, it is more difficult
to write exams with a predicted outcome than in a course like 1a. I aim for a 75 average
score. For an over 80 percent average, the exam was too easy or too close to practice exams
or exams from the previous years (which are assumed to be known) for an
average below 70 percent, the exam was too hard, confusing or there were time constraints
during the exam. Since our HW scores usually high, we end up with an over 80-85% average
final score. In this course, we had quite a typical distribution, but due to the TF problems,
it had been very hard to get close to 100%.
The multple choice parts make it also difficult to have a really low score.
Question:
Q: Why is this course so competitive?
Answer:
Some degree of competition is a good motivator.
Most Harvard students are competitive. Otherwise, they would not
be here. But too much competition can be tough too.
If you feel stressed, use your teacher more as a coach during the semester.
He or she can tell you how to learn faster or how to take short cuts. As in
sports, a good coach can help.
Question:
Q: Do you have statistics about resurrection?
Answer: Sure, we have statistics about everything.
Resurrection happened rarely. This is clear because the HW score is
in general high so that beating the average grade with the final grade
was not easy. But rusurrection happend. It could happen when
hard work with the tougher material of the third part could make
a difference. The graphics to the left shows the correlation between
average score and final score. The graphics to the left shows the
difference of the final score minus the average score.
Question:
Q: How much line integrals do the Biochem sections have to know?
Answer:
Everything inlcluding the fundamental theorem of line integrals
that is until and including section 13.2.
Greens theorem (section 13.4) is no more part of the
final exam for Biochem sections. The fundamental theorem of line integrals
13.3 might not have been mentioned also in some sections. So, 13.3 is also
not part of the exam.
Question:
Q: Will the review sessions be taped?
Answer:
We try. It is never clear however whether the videos can
be put online on time. Meddia services are running on a
smaller staff than usual during reading period because
many students working there have also exams.
So, please try to come or assign somebody to take notes
for you.
Question:
Q: Do TTh sections also have three jokers?
Answer:
Yes, this needs to be clarified: to be fair, both MWF and TTh sections
have 1 week worth of their lowest HW scores as "jokers". So, three hours
for MWF correponds most weeks to 2 HW for TuThu. We take the same percentage
off from the bottom scores from everybody and this percentage is
measured by 3 hours for MWF.
Question:
Q: Are the midterms curved?
Answer: No. The final grade will be curved according to the class
distribution. You find the formulas on the website and you can do the math
yourself. Most of the time in the past, a final grade of 90 or more guarantees
a A grade. This could theoretically change if the final exam would have a 95
average but you can bet that the final exam will have a similar difficulty than
the midterms and accordingly a similar mean.
Question:
Q: I'm trying to graph an implicit surface in Mathematica,
but although I do everything as it is in the notebook, it does
not work.
Answer: This is a common problem.
The implicit plot requires you to load a library first.
If you try first to run the command without the library, it
will no more load the library. Save your notebook and start it
again but make sure that you run the command which imports
the library before. Here is a simple example: start a new
notebook and enter the following two commands:
Question:
Q: Which sections in the book are skipped in the book for this course?
Answer:
You can correlate the syllabus with the book index. But it is easier to tell,
what we do not cover:
Keplers laws in 10.4.
General change of coordinates (section 12.9)
In section 13.2, we do not look at line integrals of the form
int f ds, where f is a scalar function. These are generalized
length integrals and a million times less relevant than the
actual line integrals. Line integrals for us always involve
a vector field F and are of the form
int F(r(t) r'(t) dt. You can safely ignore
any line integral of the form int f ds. Believe us, the subject
is difficult enough without them.
In section 13.6, we do not look at surface integrals of the form
int f |r_u x r_v| du dv These are generalized surface area integrals
and a million times less relevant than the actual surface integrals
int F(r(u,v)) (r_u x r_v) du dv. you can safely ignore
any surface integral which involves scalar functions f. Believe us,
the subject is difficult enough without them.
Question:
Q: Is it true that the material in the biochem section is much
easier than the material in the regular and physics sections during
the last two weeks?
Answer:
We try to have all sections to do the same amount of work and keep
things on a similar difficulty level. Because the subjects are very
different, this is not easy. In the past, both cases occured. There
were years, where the biochem section material (a basic introduction
to probability theory) was easier. In other years it had been harder.
Because part of the final exam will be split. That is: 2-3 questions will be
different for the biochem sections than for the other sections, the
fairness of the exam is an important issue. In order that nobody has
to worry about "being in the wrong section" or having had an unfair
final exam", the final exam is split into two parts, one common part
and one which is flavour dependent.
Subtotals are computed for the common exam as well as subtotals for
the part which is split. The second part will be curved seperately
in such a way that the average of the biochem class and the regular
class are the same.
Question:
Q: What does this Stallone movie have to do with Math?
Answer:
First of all, the movie made it to the
Golden Raspberry Awards.
THen, climbing has a lot to do with gradients. The fastest way to the top is
going in the direction of the gradient, if z=f(x,y) is the height function.
If you fall, that happens along the "steepest decent".
Note that the mountain in the movie is a surface which is not a graph of a
function. In order to describe it, you would have to write it as a level surface g(x,y,z)=0, or
parametrize it.
Question:
Q: How much should I read about the chain rule?
Answer:
As little as possible! Just keep the one dimensional
chain rule d/dt f(g(t)) = f'(g(t)) g'(t) in mind. The
only new chain rule, we ever need in this course
is d/dt f(r(t)) = nabla f(r(t)) . r'(t)
It is the usual chain rule, where f' is replaced by
the gradient of f and r'(t) is the velocity of the curve
r(t).
It is an interesting phenonmenon, that if students
start learning about the multivariable chain rule,
they lose even the ability to do the one dimensional
chain rule. Chaos takes over in the brains, the end
of the world is near etc. etc. And all because of
that wicked chapter 11.5. Dont read it!
Question:
Q: Why is the second midterm also in the same time as the life science
exam?
Answer:
Cruelty of course! No, seriously, when we start organizing a large course
like 21a, the first thing we do is to organize the rooms, at least three months
before the actual exam. We did that early August for this course. All the rest of the organization
of 21a, the syllabus etc depend crucially on these dates. There are not so many days eligible for
midterm exam dates due to a few religious holidays and due to syllabus constraints.
But the main issue is finding an exam room for 250-300 students. The life science exam happens
during a regular class and it would room-wise have been possible to move those dates because
the rooms are anyway booked for classes. But changing an announced exam date is always
problematic, after the semester has started, hundreds of people have organized their
calendars already. We would not have changed the exam dates even in September, since
sport coaches or art directors might already have had a look at the exam schedules.
To prepair for a day with two exams, we advise start early with both exam preparations.
By the way, during final exams, you will most likely have two exams at the same day.
Take the opportunity to fine tune your time management skills now.
Question:
Q: What do we do if we have to compute the directional derivative
D_u f when u is not a unit vector?
Answer:
The book defines the directional derivative only for unit vectors. We extend
this definition for all vectors u, so that the formula D_u f = nabla f . u
holds for all vectors u. The reason why calculus books define directional
derivatives for unit vectors only is that unit vectors provide a "direction".
The reason for extending the definition is not only mathematical elegance but
also that with a narrow definition, students always want to divide by the
length of the vector.
Question:
Q: How much do I have to know about PDE's (Partial differential equations)
Answer:
The website has two one page handouts on PDE's. Read them. You should know,
what a PDE is, especially be able to distinguish a PDE with an ODE (ordinary
differential equations). You should also know a few PDE's like the
heat equation u_t = u_xx , wave equation u_tt = u_xx, advection equation u_t = u_x,
Burber equation u_t + u u_x = u_xx, laplace equation u_xx + u_yy=0,
Schroedinger equation u_t = i h u_xx + V(x). You should be able
to verify that a specific function is a solution to a given PDE. Finally, you should
also have an idea what these PDEs describe.
You have not to know in this course, how we find solutions to PDE's. The
next math course Math21b will teach you that for the above PDE's.
Question:
Q: I have trouble graphing surfaces, especially parametrized surfaces.
Answer:
Graphing surfaces by hand is difficult. But can be fun and challenging to draw
a good picture of a surface. Traces can help you to visualize the surface.
Use Mathematica for more complicated surfaces. Here is a list of surfaces, you
should be able to draw by hand: plane, all quadrics, surfaces of revolution,
the torus (doughnut), graphs of simple functions f(x,y).
Question:
Q: I did not get the book yet. What should I do?
Answer:
The Coop had temporarily not enough books on hold. Contact
math21a@fas.harvard.edu, so that we can get you the first weeks
homework set in a different form.
Question:
Q: Can I take the course Pass/Fail?
Answer:
While this was ruled out at the organisatorial meeting,
it can be an option in very exceptional cases. Please contact
the course head for advise in this matter.
Question:
Q: Can I use the old edition of the book?
Answer:
We need the third edition. The problems are different from
edition to edition.
Question:
Q: Do biochem sections need a second book?
Answer:
Technically, it is optional. But since it costs 9+ dollars
only, why not get it?
Answer: This is the MFAQ, the most frequently asked question.
As you know, Harvard, as many other colleges, experiences grade inflation.
This is unfortunate, because it allows students less and less to gauge their
progress and for students near the AB treshold, it is essentially a matter of
luck on which side they fall. The quantisation by letter grades is too
rough and with time, most of the grades had been pushed into the A-B range.
This discourages top students to work harder (I already have an A) and
students with a B grade (I can not get into the A grade).
A simple solution to the problem would be to get rid of the quantisation and to
submit a number between 0-100 to the registrar, forcing the distribution to have
a fixed mean and standard deviation. But the world is not perfect and so,
every teacher at the end of the semester ponders over the problem to figure out
a "curve". I keep three constraints fixed: there should be maximally 40 percent
A and A- and the average should be near the threshold of B to B+. Furthermore, the
standard deviation window should include only A and B grades. Then,
according to the distribution, the cutoffs are set. This is usually not
fixed from year to year. It depends on how difficult the exams are percieved,
where natural flat parts of the curve are and depends at the analysis of grades of a few
"gauge" students, I look at closely and where I can see what was the reason
for success or for difficulties.
