Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: (271) Next: Homework Assignment 25
Previous: Class Notes for the Week of March 24 (5 of 5)

What Went Wrong with Term Exam 4?

The responses I got for the "What Went Wrong with Term Exam 4" question of HW24 are below (reformatted and with all identifying details removed). I've added my my own comments in green; many of my comments appear just once though they pertain to many of your messages, so many messages appear without comment. (This especially applies to the last few).

Subject: Term Test 4

To Dror, 

To start off, I thought the test was fair. There was nothing too tricky
about it. I lost most of my marks for Problem 1. This was mainly
because the "pi proof" took so little class time, class notes, and
text book pages in relation to the other topics of Taylor polynomials,
sequences and series, so that I spent much more time studying those
topics rather than the irrationality of pi.  Therefore, I realized I
didn't know the proof as well as I thought when I was at the exam.
Well, that's what went wrong for me personally.

Many, many said the π problem (Problem 1) was hard. It wasn't, though my moral is that I should have re-iterated the fact that theoretical material, i.e. e.g. proofs and parts of proofs and variants of proofs done in class, is absolutely fair game. If I teach a proof it is not because I fear that the gods of mathematics will strike me if I don't. It's because I think it has a value. So such proofs are an absolutely essential part of the material you ought to know coming out of this class.

In general, I believe it would benefit the class if a sample test was
given prior to the test, similar to what you did for Term Exam 1. I
know that means extra work for you, but it does give everybody a better
idea of what is expected of us for the test.

Sample tests are indeed a good idea, but as you wrote, it's extra work for me. I could not do it for exams 2-4. The final is more important and there will be a sample exam in advance of it.

Finally, I have a question about the solutions to Term Exam 3. For
problem 5-4, the summability of (log n)/n^2, I did the integral test
and showed it was divergent. I can't figure out what was wrong with my
reasoning. Perhaps you can enlighten me.

Thanks, and I hope my comments were helpful,


Subject: Feedback for Term Exam #4

With the exception for the first question, I felt that the term exam
was reasonable, compared to term exams we have written in the past.

My low mark was mainly due (like all the other past exams) to my
carelessness.  Once I go through the solutions to the term exam, i
realize how not difficult it really was, and that i didnt see the
relationship  in the questions all together right away.

Regarding Question 1, (which I talked to many students about it) it was
quite difficult, I talked to a few fellow students in the class last
night over MSN, and they felt that Question 1 was very difficult, even
for the students who got outstanding results in the past term exams.

Other than that, I have no further comments.


Subject: exam 4

Hi professor, i'm NAME. About exam 4, i did very bad in question 1, and
i lost a few marks on calculation errors. so my biggest fall was
question 1...

i was thinking only stuff from chap 20, 22, 23 and possibly the
volume/surface integrals would be on the exam and so i just studied
those and that's how i ended up screwing Q1. i did understand the
proof when we did the irrationality of pi in class, but it was quite
long ago and i forgot the crucial move of proving such p(x) does not

(by the way, on the solution sheet u said 0 < int p(x)sinx < int sinx = 1
it should be 0 < int p(x)sinx < 1/2 int sinx = 1    u forgot the 1/2 )

Thanks! You are of course right, and my solution set is now corrected. I guess I loose a point.

for the calculation errors.. i had a personal problem and so i wasn't
feeling well to do the exam...didn't really think clearly

um, about the way u teach, i think it's fine~ but about "WHY" we r
doing it, i didn't get the msg clear for chap 20. i know Taylor is good
becoz we can use it to approximate or even find the "closed form",
and i have a feeling that taylor is not just for approximation.. and
now i know it's for power series..  (but actually it's not related to
why exam 4 is so bad..) i guess maybe u can tell us more as to
what/which theorem is important in later chapters, so we know what to
expect...like i still don't know why Lagrange form for the remainder
term is needed when i can find a bound from the other 2 forms...(at
least up til now)

I do my best to separate the important from the less important in class; the Lagrange form for the remainder is an excellent example - I didn't even cover it, so I agree with you it is not terribly important.

also, in our class, i don't see many people who like math, i only know
Gary likes it very much, and possibly a few others, but not the
majority..  i guess they don't see the differences between analysis and
standard University math...maybe they didn't do a lot of
exercises...i'm really not sure, just my opinion

It is the highest calculus course at UofT and my assumption will continue to be that people are in it because they want to be in it.

(in MY opinion) the exam is appropriate, Q1 is the hardest, then others
r really quite routine...i wouldn't say it's harder than the homework
assignments i think Q1 is hardest because i don't remember its proof
for pi...i forgot the proof, and it became very hard to proceed...i
didn't come up with the move "consider int p(x)sinx ..."

hope this helps. 


Subject: Test

Hi Professor Bar-Natan,

For what its worth, my personal result is lower because I got a little
lazy and didn't study as much as I should...

In terms of difficulty I thought the test was comparable to the others.
Some of the questions like Q4 and Q5 everybody should have gotten since
they were almost identical to questions in the book. Q1 was difficult
becuase I didn't realize we were being tested on material from that
chapter (perhaps others thought this as well?). Q2 was nice becuase it
required a little bit of thought and then became a straightforward
calculation. Q3 parts 1 and 2 were easy and again part 3 required some

I think you presented the material concisely and cleary with


I cannot really say conclusively what went wrong as a group; personally
i did ok. I was troubled with the first question for quite a while but
i think this is more a result of my own mistake in reasoning, rather
than the exam being inappropriate. I believe the exam was quite fair.

From other people's responses i can speculate the following:

It is possible that the new material was more foreign than anything we did
previously, so people didn't really see the major points.

It is possible that people were stuck on previous material and so couldn't
understand new material.

Sorry, this is not so helpful. I did not expect the results to be quite
so low; I am likewise surprised.

Good day.

Subject: Term Exam 4

Personally, the biggest reason why I didn't do well in this Exam is
because I didn't do the homework questions unlike the other times,
when I was up-to-date with the homework. As a result my marks have
dropped significantly in this test.
Also, I haven't been attending the lectures regularly recently.
Earlier, I went to every lecture. It is harder to catch up on my own.

Class attendance is ESSENTIAL and even more essential when the material is less fun. What you like you have a chance of learning all on your own. What you don't like is completely hopeless without an external push...

But I guess, the underlying reason I identified, is the subject matter
itself.  Otherwise, if I found the topic interesting, I would have
definitely done all the things above the right way. The entire gamut of
topics on Sequences and Series seemed very bland. It wasn't as
interesting as Integration or Differentiation. But I guess I should
have been motivated enough to study it nevertheless.

I agree. Sequences and series are a little less exciting in my mind too. But there's nothing I can do about it - not every day is party day.

Hey Professor B,

I think there are a couple reasons why this term's average might've been far
lower than the previous ones.  First, i think this is about the time everyone
goes berzerk trying to juggle 10 different assignments, essays, and tests
around their schedule.  Personally, this has meant a lot less studying time for
this Calculus test, which i regret but saw no way around it.  I started
studying a week in advance for the other midterm tests, whereas this time
around, i could only start the saturday afternoon before the midterm.  

I hereby offer my sincere empathy. Exhaustion is an old friend of mine and I know her well. I wish I didn't and I wish you didn't either, and I hope things will be better near the final.

Secondly, I think the topics themselves are a little confusing for one to wrap
his/her head around.  Personally, sequences and sums are so intimate that it
becomes a challenge to differentiate (no pun intended) the two sometimes.  Mix
this in with some limits, delta's and epsilons, as well as derivatives and
integrals, and we're dealing with practically everything we've learned since
September.  It's not as bad once one understands what it is we're trying to
achieve, but i think it's this big picture that we lose track of sometimes,
especially now when we're nearing the end of the year.
As for the test itself, questions 1 and 3 were, i thought, hardest to answer. 
1 because perhaps most of the class didn't think to study for the
irrationality of pi proof (myself included), and 3 because one wasn't sure how
rigorous a proof the marker was looking for (for which i lost all marks for).
All and all, i believe the test and the way you've taught the past several
chapters have been more than fair in all respects.  I think it's just the
sheer exhaustion of the students and the increasing complexity of the topics
that is influencing the overall marks.  The end is near though, and now that
things are starting to die off, we can start paying more attention to the class
and less to the work.  Thank you for your concern.

Subject: What went wrong.

I don't feel that the test was unfair, however the first question really
caught me off guard because I thought that the irrationality of pi wasn't on
this test. I also thought that in general the test was difficult (but not too
difficult). By this time of the year everyone is really stressed and busy. I
know I didn't study as much as I had planned. The content also was different.
Everything up to now has been about continuous functions (for the most part).
The test was fair though, I'm just glad it's over.


Subject: MAT157Y

Hi, This is NAME, and I personally got not excellent, by all means, but pretty
respectable (especially with surprisingly low mean average) grade around 80 in
the exam.
I didn't think the exam was exceptionally difficult compared to the previous
ones, although I found it more challenging than the previous ones (hence the
lowest mark among the 4 exams) in a way that I didn't prepare for some of the
questions in the exam, for example the first one. I did, however, remember some
of the techniques of proving that pi is irrational and I got marks for it, but
I think that question is where most people lost the marks. 
But again, I am pretty happy with what I got and hopefully, I can continue to
be so in the final exams and the courses to follow in future.
Thank you

Subject: Mat 157

I think the time of the test was bad, right now I have essays to write, term
projects to do, and other tests to study for; and every one I know is in the
same situation. I didn't have tthe same amount of time to study for this test
as I did the last few.  I think it would have hellped if the test was a week
earlier or one or two weeks later.

Hi Dror,

I have a few ideas about what may have been different for this test as compared
with the previous 2.  For me, I thought that the infitite series chapter was
one of the most difficult chapters in terms of exercises since least upper
bounds or limits.  

I did well, so nothing really "got" me this time, but other students said
they weren't prepared to reproduce part of the pi is irrational proof.  I
guess that if one didn't remember to consider p(x)sin(x), then it's difficult
to figure it out on the spot.

Not just difficult, practically impossible.

In other classes, too, averages have been going down.  Many people in MAT 157
are also in Physics, in which the averages went from ~70 before Christmas to
~50 this term.  That's not a reason for the drop in marks, of course, but it
may be that students are just feeling the pressure to do well more this term
and it's hurting all their classes.

That's all that I can think of right now.  I think that you've been doing an
excellent job- consistently throughout the whole year.


Subject: Feedbck on Test # 3


  After speaking to several of my classmates, it became clear that the
unsatisfactory results of the test are not due to:
1. Possible unfair complexity of the test. [test covered exactly the material
we studied]
2. Larger amount of theorems or concepts to learn. [actually there were less
than usually]
3. Lack of examples given in class. [there was no deficiency of examples, and
all of them were really helpful]
  However there seems to be a consensus that this test:
1. Covered material never previously learned by most of the students. [example:
series and sequences applied specifically in analysis]
2. Was considered hard by some students due to their absence from lectures and
tutorials. [it is almost end of the semester and attendance had sadly dropped]
3. General exhaustion of majority of the students. [End of the semester is
filled with tests, assignments, essays, and exam preparation]
  I sincerely hope that at least some of these ideas may help determine the
real cause of the test results.

Best Regards,


My name is NAME and I am a student in your mat157 class. I have 
noticed that with the weekly assignments some students copy each other or copy 
straight out of the solution manual associated with our textbook. And thus 
many students were not practicing the new material and once the test came 
around they found they had a lot of work to catch up on. This may be a reason 
for the low average. 

The reason I do nothing about students copying from the manual or from each other is that there is nothing I can do about it. And you are of course right that it hurts the people who do it.

I am sorry I do not have any suggestions on how to improve the average.

Ps I personally liked the weekly assignments because it forced me to review 
the material that was taught in class.

Subject: term test 4 feedback

a) the proof for pi irrational was quite daunting, i don't think i understood
what function you were looking for and how you picked the needed criteria for
that function

b) Taylor series were great, but i started getting confused when the remainder
and error stuff was added

c) the following was confusing and definitely not intuitive:
lim x-->a of [ ( f(x) - Pna(x) ) / (x-a)^n ] = 0

d) on the exam, problem 4 part 2: i don't know where to begin! perhaps more
examples concerning sequences in class?

I agree a class example would have been helpful, though notice that a similar problem did occur in the HW: Chapter 22 Problem 5.

e) determining the limit of a sequence or series, or determining whether or not
a series converges -- i didn't practice enough on those

Commentary on Mid-term #4:

Reasons why I faired badly on the exam.

  1) I was pretty sure that the chapter on pi would only be on the 
     final exam, (not on the midterm/I believe you said this in class).

I don't recall saying so. I did say it wasn't on Term Exam 3; for Term Exam 4 I wrote which chapters where in and I didn't exclude this one.

  2) I did not study well/long enough, (very busy writing essays, others

  3) With regard to comments about yourself, I sincerely doubt it had
     anything to do with your teaching.


  I believe it had mostly to do with lack of time.
  (hope this was constructive.)


Subject: Term exam 4

In my case, I would have to say that I didn't study appropriately for
this exam, ie. I didn't do enough questions to be comfortable with the
material and I guess it's obvious how to fix that problem.

Dear Sir,

I do not think the last exam was inappropriate.  The questions on
Taylor polynomials, cosmopolitan integrals, and convergence of series
were definitely at the right level.  I can assure you that you
presented the material very well, so teaching was not the problem.  I
think one of the main reasons for the low class average was the first
question.  I know some people that did not study the proof for
irrationality of pi because they thought it was "just for fun".
Another reason could be that some people stretched themselves too thin
while trying to study for the MAT240(algebra) test along with this

Besides the more general reasons I gave above I can also give some
reasons for my less than stellar peformance.  I misinterpreted part 3
of question 3 and showed what was required for "a small value of x"
instead of for "small values of x".  It would have been clearer if the
question stated what it meant by "small values of x", but at the same
time that could have given away too much.  Another reason has to do
with the proof for irrationality of pi.  The "back-of-the-envelope"
proof that was provided was little to go by.  For example, it would
have been helpful if it stated that we're only interested in what p(x)
is on [0,pi].  It took me a while to figure out why p(x) was "clearly
positive yet small", because if my understanding is correct, without
the restriction on the domain of p(x) it would be easy to have a value
of x for which p(x) would not be positive all the time.  I'm sure all
of this was explained in class, but I just assumed that I could learn
the proof from the textbook which usually explains material quite

I guess the moral ought to be that both class attendance AND reading the book are ESSENTIAL.

I hope this is more or less what you wanted as feedback.  Thank you for
taking the time to listen to what we thought about the test.  It's an
opportunity that we don't usually get, unfortunately the algebra
professor doesn't seem to be concerned with low class averages.

He/she must be more senior than me... :-)


    With respect to the results of the term 4 exam, some of the low
results can be attributed to the fact that it's the end of the year and
there are a lot of assignments, essays and term exams going on.  I
personally feel that first two questions on the exam were a little
confusing and hard.  The first problem was confusing because you wanted
to know whether or not you could find such a polynomial and the way in
which you proved that pi was irrational was that you could find such a
polynomial. So I thought there was one because I guess during class it
wasn't made clear that the polynomial itself doesn't exist.  For the
second one, it was confusing because the previous questions that you had
assigned for the cosmopolitan integral i believe didn't use inequalities
but equations so one could just use the equations for volume.  So since
it hadn't been seen before, it was difficult and personally, it seemed a
little inappropriate for an exam for that reason.  Just in general
though, the last little while I've been a little confused with series and
sequences and the convergence of a series versus a sequence  converging. 
There wasn't really a problem with the way you taught the material, it
was just difficult distinguishing between the two.  So, in sum, I feel
the first two questions were difficult and there's a little confusion
about the difference between series and sequences.  If when pi was proved
irrational it was also made clear that the polynomial thus, overall
doesn't exist and if inequalities were dealt with beforehand in the
assignments and more explanations and examples of the difference between
sequences and series (ex. the difference between having a cauchy sequence
and the cauchy criterion for a series), the results would have been

Generally speaking, it is quite fair to have an exam question which isn't quite like class or quite like HW - we are trying to acquire a certain body of knowledge; not merely a set of routines for solving repetitive problems. So in studying, beyond solving HW problems and memorizing proofs there has to be a stage where you stare at your notes and try to really really deeply deeply understand what is going on. See for example what I wrote About the Second Term Exam.

Subject: Exam feedback

I'm sorry for sending this late. On analysis of Term Exam 4, I have found that
many of the mistakes which caused me to lose marks were computational in
nature. For instance, I computed the wrong Taylor Polynomial in Question 3 and
I forgot to change the bounds of integration when I made a substitution in
calculating the volume for Problem 2. Problem 5 was more time consuming than it
should have been, because I had to think quite a bit about which convergence
tests to use. The last one on that problem was particularly frustrating.

In terms of conceptual difficulties, Problem 1 and Problem 4 were the most
problematic. I felt the most clueless about Problem 4. I remembered that the
first part of Problem 4 corresponded to a theorem in the textbook, but I could
not remember the proof of it. I do not know if I would have been able to figure
out the second part, because it did not look familiar to me, and I ran out of
time since it was one of the last questions I tackled.

Subjectively, I felt a lot more pressure on this exam than I usually do. I am
not sure why. I stayed up late studying the night before and so I was very
tired while taking the exam, which probably explains the computational mistakes
to some extent. Many of the things I studied in depth were not on the exam - I
spent a lot of time studying the proofs of theorems in Chapter 20, since I seem
to recall a good deal of emphasis on those in the lectures. Ideally, I would be
able to study everything with equal depth and have everything internalized, but
given a finite amount of time, this is very difficult to manage. Internalizing
a proof seems to consist of one part understanding and one part memorization.
The latter gives me some trouble, as I often have trouble recalling things that
I have even proven by myself. But had I spent the same amount of time studying
Chapter 22 as Chapter 20, I probably would have done much better on Problem 4.

Another stress factor was that I was worrying about exams in other classes,
such as Algebra.

Besides figuring out a better study technique for myself, one thing that I
think might have been helpful would have been if I had a practice exam to study
before the taking the actual exam. I do look at the previous year's exams, but
more and more, I find the emphasis of these exams to be different than both the
material taught in class and the material on our exams.

I guess that's why HW23 says "for what it's worth" before "review last year's fourth term exam...".


Subject: test problems

Sorry this is getting to you a few hours after the deadline but this is the
first chance I have had to get to a computer today.  I have spent some time
thinking about what went wrong with the test.  I knew when I had finished
writing it that I did not do very well, but it surprised me to see that the
class as a whole felt the same way.    

I found that in previous tests, more of the questions seemed to be very similar
to questions I had come across in reviewing the homework and tutorial problems. 
Solving problems similar to ones I have already thoroughly understood, although
perhaps not as useful, takes far less time on a test.  I don't think the
questions were unreasonable, but just that they might not have been what the
class was expecting to see or had time to work out.  I might have a little more
success if we had more time or less questions.  Maybe others might have found
the same.

Studying might have also been a problem.  Although I studied class notes and
the chapters in Spivak, I spent a lot of time practicing homework and tutorial
problems.  I found most of the homework consisted of questions on how to use
the theorems rather than understanding them (questions like #5 from the test). 
I found myself practicing a lot of those types of computations and remembering
theorems, so perhaps I (and others) were sinking into a routine and forgetting
to see the bigger picture as you said.  Maybe suggesting a few extra problems
similar in structure to ones we could expect to see on an exam might be useful. 

I would be very interested to see what other people thought went wrong as well.

Dear Prof. Bar-Natan,

As you requested, here are my comments about Term Exam 4(I hope the deadline
wasn't strict).   I guess it's worth mentioning that I got around 60 this
time and 80-100 on exams 1,2, and 3 respectively

The level and adequacy of the exam: Besides perhaps sub-questions 4 and 5 of
problem 5(which tested problem-solving abilities rather than the knowledge
of the material) the exam was about the same level as the previous ones, and
these, in my opinion, adequately tested the knowledge of the material.

The lectures: I think the lectures are comprehensive and adequate in terms
of teaching the material.  I missed a lot of lectures in the Winter term,
though, the main reasons being the fact that they are held in the morning,
and an excess of homework that made me stay up late at night.

My excuse: I had two consecutive tests on Monday, so I had to study for two
tests at once. The results of these unfortunate cirumstances are apparent.
Particularly, I didn't have time to review the proof of the irrationality of
PI, and couldn't even approach problem 1. Perhaps a mandatory homework
assignment(I think there wasn't one) on this topic would have helped or,
rather, forced me to understand the proof.

General excuse: It appears that, as a rule, full-year courses become more
demanding towards the end of the year. Consequently, the results are lower
towards the end of the year. This, of course, would infleunce mat157 results
even if the level of mat157 were about the same throughout the year(which, I
think, is the case).


the reason I didn't do so well was because of 2 reasons. Firstly, the 1st 
question which was based on the proof that pi is irrational was unexpected 
since in class you had said that you MIGHT ask for a part of it on the FINAL 
exam. This lead me to believe that you would not ask anything about it for this 
test, hence I hadn't looked at it and had no chance to get any of the marks for 
question 1. Secondly, the question with the recursive sequence was too much out 
of the blue for me to figure out. I guess since I hadn't seen one before I just 
got confused.  I think that in the future you should tell us exactly what parts 
from the text will be tested on, and try to give an example in class of for 
example a recursive sequence just so that it isn't too confusing.
Stefan Banjevic
ps, I'm sorry for the lateness of this response, I couldn't get to a computer 


Sorry this is late, but I naturally thought the feedback was required
at the same time as the rest of assignment 24 and I only just noticed
it was dated for the 28th.

Anyway, I don't think there is any one thing I did that went wrong with
this exam, I think it was an accumulation of things.

As an idea of the level of student (such as it is) that I represent, my
previous exam marks were approx. 60 and now around 30, so I have been
struggling all along (usually just below the average), but obviously
this last exam was much worse than the others in terms of my

For the other exams, even though I did not do particularly well, I
always felt I understood the material and also understood the places
where I went wrong, in the sense that AFTER the exam, with the leisure
of time on my side, I have always been able to come to the correct
solutions or at the very least understand the provided solution.

In this exam, however, this was not the case, which suggests that there
is something fundamental that I am not understanding.

Issue 1:

I believe the main problem I have is that the newer material does not
build on the previous material in as linear a fashion as previously, or
at least I am not able to percieve it as such, which also could be the

For instance, derivatives built directly on limits and boundedness,
etc., and integrals are very directly related to derivatives, etc.

But the cosmopolitan integral section, and the irrationality of pi
represent a break from the path we were on, and I was not able to
assimilate this material very readily.

Likewise, taylor polynomials, sequences, and series, although we'd seen
examples of each previously in the text, were in fact something of a
digression I felt.

I mean this in the sense that yes, the tools we have developed are
necessary for these chapters, in the sense that we needed to go back to
our concept of limits to prove convergence of sequences and series, and
we needed derivatives to formulate taylor polynomials; so in that sense
these chapters do build on previous material because we need those
prior tools to formulate our newer concepts.  However, the newer
chapters are actually a break from the previous material in terms of
the concepts themselves, their purpose, computation, etc., which are
all very new and alien compared to what we have seen thus far.   I
think part of the problem is that all of our previous concepts related
obviously to geometry, in the sense that a derivative was a limit of
seconds, and integrals are the area under the graph, etc., whereas in
these sections the 'picture' was not so obvious, and in some senses was
quite abstract (how does one 'picture' infinity anyway?).  Even the
cosmopolitan integral, which was quite obviously geometric, was so in
such a way that it was quite unfamiliar to anything we've done before.
I don't want to suggest this was the whole problem, but only a part.

The other half of this is that I felt I understood the new material as
we progressed, when in fact I did not.  This is always the worst
situation to be in, because if you think you understand the material
when you don't, you can't even address the problem because you don't
knot the problem exists.

As an example, the cosmopolitan integral section I thought I
understood, and I was able to do the recommended and required homework
questions for this, although in most cases only after discussing the
problems in tutorial or with other students.  However, in hindsight I
discovered that I never fully understood the section at all, because I
never realised that the formulas we developed for the disc and shell
methods and for surface area were universal; I thought we derived them
only for those particular shapes.  As to why I thought this, I can't
say for certain because it is quite obvious in hindsight.  Thus, on the
exam, I tried to develop a formula from scratch which a) I screwed up
and didn't quite get right, and b) ate up a lot of time.  So even
though my reasoning was correct and I was on the right track, by not
remembering and applying the right method I really screwed myself up;
but prior to writing the exam I don't see how I would have known this
since I didn't even know I had a problem.

Issue 2:

I believe there was a lot more memorization required for this exam.
Perhaps this is not true, and is an indicator that I didn't understand
the material.

But in previous sections, such as 'significance of the derivative',
even though there were many things to memorize, they felt very natural
and inter-related, and we had much repetitive practice with each, so
that it did not seem like we were memorizing so much.

But in these chapters there is just as much material to memorize, but
it all feels unrelated, arbitrary and abstract (again, perhaps
indicating there is something I am not 'getting').  In 19-appendix
there are three integrals to remember plus all our old geometric
formulas, which may not be fresh in the minds of many (such as surface
area of sphere, etc.).  In chapter 20 we have to remember the form of a
taylor polynomial and how to find the coefficients, 3 different
remainders, etc.  In chapter 22 there are many similar things to
remember, as well as sequences.  In 20 and 22 much of this material
seemed unrelated (not just from each other, but from the rest of the
book as well), and its purpose and application were often vague or
misunderstood.  Ie., I'm still not sure I understand the purpose of the
remainder, except as a measure of error, but if that is it's sole
purpose why do we have three versions of it and how do they relate to
the rest of the material?  In chapter 23 there was again a lot of
similar issues, and although the material built quite directly on
chapter 22, I found the volume of material quite daunting: there were 9
new theorems as well as lemma's, conditions, criteria, etc.  In
addition to THAT, the convergence/divergence properties of many series
had to be memorized in order to make the comparison test a useful

Issue 3:

Here is a breakdown of why I did poorly on each exam question (you may
want to skip this I guess since it is more specific to me and not
necessarily representative of the problems the class had as a whole):

Question 1 I don't think I could have gotten even with extra time,
which probably means I didn't really understand the irrational pi proof
even when I thought I did.

Quesiton 2 I discussed above.

Question 3 part 1 I got wrong, but it was a simple arithmetical issue,
where I couldn't multiply 2 by 3, etc. and ended up getting my sign
confused (multiplying by 1 instead of -1, because I took the derivative
of x instead of -x), which are simple mistakes I made under the
pressure of exam anxiety.  3 part 2 I didn't realize was supposed to
build on part 1, and since I got part 1 wrong there was no way I could
do this.  I didn't even attempt it correcty however since I thought you
were looking for some completely general formula when you were not.  3
part 3, again I didn't realize it was building on part 1 so I could not
have gotten it right however long I took, and since part 1 was wrong,
it wouldn't have helped anyway.

Question 4 part 1 I could not have answered.  I didn't see how to
approach the question at the time and now that I do I'm not sure I
could have come up with it on my own, even though I'd looked at the
theorem and its proof several times.  4 part 2 I could not have gotten
either, since I did not think to apply part 1,  although in hindsight
the answer is quite simple and obvious, it's a case where I don't
believe I would have come up with the answer in an exam setting.

Question 5 was quite hard I thought.  This is the one section where I
felt I had a chance to make up some marks but did not.  I spent a lot
of time and effort looking at series from the book and determining if
they di/converge.  However, with a couple of exceptions, the examples
on the exam were quite challenging (even if they were from the text).
Part of my problem was simply exam anxiety and part was arithmetic


Unfortunately I do not see any simple answers to these issues and even
if I did they might only be applicable to myself.

Issue 1 might be helped by more examples, by more references to how the
'picture' looks, by more references to the relevance of the new
material and how it relates to or developed from material we already
know. (and then again, maybe it won't help)

Issue 2 might be helped by improving issue 1.  Otherwise I'm not sure
there would be any way around this.  Possibly by doing more numerous
but less challenging homework questions? (or maybe the opposite is
true, and we should do fewer more challenging ones?  I'm not sure).

Issue 3 is my own problem, but as a guideline I study by re-reading the
text, re-reading my class notes, and by re-doing the homework along
with all recommended problems.  I then repeat as necessary until I run
out of time or feel comfortable (more often the former).  But I feel
this is a reasonable study method and ought to yield good results, so
hopefully I will improve naturally if I am able to properly grasp the

As a next to final note I think in general you do an excellent job of
presenting the material and teaching to the level of the class (this
isn't sucking up but is meant to indicate that perhaps the fault is not
with the classroom presentation per se, but simply with the material or
our class as individuals, and that by changing too much it might make
things worse rather than better (ie. if it ain't broken, don't fix it,
and just because the exam mark suggests something is broken, perhaps it

As a final note, I found this exam the most challenging yet.  Possible
difficulties with the material aside, I felt the style of the exam was
quite different from the previous 3.  In the past, the format for the
tests was Q1 was a difficult proof, Q2-Q4 included some graph or
application style question, an easier proof, and a variation on a
homework problem, while Q5 was usually a computation of some kind.
Exam 4 I felt strayed quite dramatically from this format and part of
the problem may be that it took people by surprise.  Q5 was
computation, but much more difficult than usual; there were no real
homework Q's as such; there were no application Q's except perhaps Q2
which was a very difficult reworking of the type of problem we had
faced in class and text, etc.

Well, sorry for the length of this, but I hope some part of it helps!


Thanks for your very thoughtful notes! I read them carefully, but at 6PM the evening before I have to teach, I can't write an as-thoughtful response. I'd be happy to read your notes again with you at my office hour, though.