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What Went Wrong with Term Exam 2?

The responses I got for the "What Went Wrong with Term Exam 2?" question of the Solution of Term Exam 2 are below (reformatted and with all identifying or personal details removed). I've added my own comments in indented green font; many of my comments appear just once though they pertain to several of your messages, so several messages appear without comment (this especially applies to the last few). The responses appear in the order that I received them.

I thought the test was fair. I know at least 50% of the test was
questions which required a methodolgy very similar to that of either a
test from the past 2 years, or a homework problem, and in some cases,
the question was exactly the same. Please continue to do this, as I
tend to study best from old tests and homework. I thought everyone did,
but evidently not.  I think maybe some people were expecting a set of
functions to differentiate as a question, I know I was, as well as a
few others who told me they were.

   Indeed several people complained about the lask of a straight
   differentiation problem. This did make the exam a little harder than
   in previous years. I have two comments:

   1. I doubt this is a sufficient explanation for the overall low
   2. This in itself won't necessarily change the final class average;
      the final grades are almost always rescaled.

That would have likely improved the average, as I assume anyone who is
taking this course is likely very comfortable with straight
differentiation.  Other then that I can't really think of anything that
would have raised the average across the board, but honestly I have
more fun trying to figure out these proofs then just differentiating. I
mean, if I wanted to do that, I would have gone to 137 or whatever.
Even though I did bad on the first exam, it was still fun and
interesting. So was this one.


Hey there Dror, this is NAME, one of your students for MAT 157,
responding to your request for thoughts concerning the midterm.  (This
is fairly wordy, so I hope you don't mind0

[omitted part]

As for why the mark for everyone was so low (as opposed to my specific
case), I have a few theories.

Firstly, I had looked at the previous tests, and this tests did seem
notably harder then the ones from previous years - however, not by so
much that the average mark should be a fail.

Secondly, I know some students who took 157 a couple years ago and who
had taken their OAC years, and from what I can tell, the course still
seems roughly the same (though maybe you have changed it in subtle
clever ways in which I am not aware). While ideally students are
supposed to get all the math they need in 4 years, I doubt these two
sets are equivalent. At the same time however, I don't think there's
any reason why you should have to make the course easier, so I don't
think there's really anything you can do about this.

   Indeed the course is almost identical to what was given in previous
   years.  The class is not a direct continuation of anything in high
   school; in a sense, it starts from scratch (for example, we haven't
   yet even defined logarithms). Though of course, some "mathematical
   maturity" is a prerequisite for this class. It is very hard to gauge
   how much of this extra mathematical competence is lost by not having
   grade 13, so it is very hard to re-design this class to fit with the
   new high school curriculum.  It now seems to me that some such
   re-design is necessary, but I'm still far from clear if it should be
   major or very minor.

Thirdly, I think the most challenging thing has been simply
understanding what exactly a proof is. Personally, I never did any
proofs whatsoever in high school, and the notion was entirely new when
I first started 157. While every student I'm sure is aware on some
intuitive level what they're supposed to be doing, I think that the
formal, logical, rigour that we're supposed to be working with isn't at
all obvious. The main reason I say this is because I've found this
course to be substantially easier since I picked up a crash course
booklet on proofs (How to Prove It, by Daniel Velleman), which deals
with the formal logic behind the proofs, and some ways to try and make
the process more intuitive. Of course, I'm new to to this whole formal
proof business, so maybe I have it all wrong, but I think maybe setting
aside an hour or two to talk about what a proof actually is, and what a
mathematician looks for (and how, as well) when analysing one may be
extremely helpful.

   Yes, this is a comment I've gotten a number of times - proofs are
   hard.  I'll take a look at this book by Velleman (will you lend me
   your copy?) and I have to think hard if it will be appropriate for
   future years to spend some formal study time on proofs per se. Most
   mathematicians (the younger Dror included) never formally learn
   about proofs - they just see enough examples of proofs until it
   becomes evident what a proof is. It may be that the current high
   school curriculum doesn't make it as evident as it used to.

And lastly, I'm sure some part of it still lies with the students, as
the distribution of marks was so extreme - obviously, some students are
still doing just fine, the problem is merely that it's the minority.
Anyway, I did a fair amount of homework before the test - but I am, and
was even before the test, keenly aware that I could have studied a lot
harder. I think (just like that pamphlet at the beginning of the year
said) a lot of it is probably simply the fact that most, if not all of
the students in your class are used to coasting without doing any
work.  And while they may have even been able to pull it off in mat137,
I think 157 is just something entirely different. Hopefully for those
students who did quite badly, this test will simply be a wake up call
to what they're actually dealing with.

Anyway, that's what I think about all of this. Personally, I still find
the course material fascinating (if infuriating sometimes), your
lecture style to be excellent, my TA (Shay Fuchs) to be helpful, and
overall, the course to be highly rewarding. I don't think there's
anything in particular that anyone is doing wrong, I think it's
probably just a lot of small things which have coincidentally lined up
to produce an unfortunate test result.


Hi, after reviewing previous tests such as, 0203,
I would conclude that this years test was unfair. As an example I would
have to draw from the fact that the tests from the links I just
provided, ask to evaluate limits and calculate dy/dx, along with a
graphing question which was present on our test. This I think, is
totally unfair as the difficulty of those tests was much lower than
that of our own, given those questions. Those questions would have
allowed a great majority of the class to pass this test, and this
situation would thus be avoided.

   I may agree that this year's exam was somewhat harder than last
   year's, I don't think it was unfair and definitely not "totally
   unfair". There's always some variation; in itself, there's no
   unfairness in that. And if you compare the solution sets you'll find
   that they aren't so different.

                                 Also, I think applying the method of
solving this problems, or their analysis was not concentrated on in
lecture, as proofs were just presented in a routine fashion. I found it
difficult to visualize solutions for certain problems.

   I really am doing my best to present everything in the most visual
   way possible.

However, as the fault of the student in poor studying methods cannot be
discounted this may have affected our ability to analyse those
questions. Yet as I stated before, I found this test too difficult, and
considering the class average, I would like to know if a bell curve
would be considered?

   Grades are always rescaled to some extent.

Thank you for the opportunity to respond,

Hello sir

As you've requested, I will provide constructive criticism for what
happened on the exam. I wouldn't say you made mistakes in the studying
material. In fact, I also don't think there has been a problem with the
way I studied. I mean almost all my assignment marks are pretty high.
What I would add though, is the way you teach. I'm not saying it's bad
or anything, but I don't quite understand things. It goes SO fast. And
I wouldn't mention this here, with two exceptions.  #1. I have liked
math since I was a young kid. (I did math in another country, much more
advanced than the high-school system here). It's always been my
passion, so I'm here because I like it, I don't just want to "barely"
pass. And #2.  - all the people from my group of friends have failed it
(about 4 people), and they had pretty good marks before. So I'm just
going to be honest with you - I don't understand what you teach.
Anything we do in class, I go home and try to understand by myself.
That being aside from my actual studying time of the subject.

   This is particularly painful to hear. I love math and the last thing
   I want to do is to push people away from it.

But I will stop wasting your time right now, I hope this helped.  One
more thing - if you can, please do something about this, it basically
killed our marks. A bell curve, a make-up something, making this test
worth less, anything.

Thank you for your time, and once again sorry for the length of this


Epsilon delta proofs are a horror to me and most of the exam was on it
even though chapterwise 9-11 dont have them. Looking at the previous
term tests I was hoping for the majority of the test on those chapters.
But i still think its more of my fault than anything else.


Dear Dror,

My greatest problem in this calculus class was that I felt I was
lacking the knowledge base needed to succeed.  Between my high school
algebra and calculus classes, I could not have done more than 10 proofs
in total.  In the beginning, I found this class overwhelming.
After a less than stellar mark on my first exam (13% to be exact), I
went back and taught myself the background knowledge I was lacking by
studying basic calculus and proofs rigorously.  I found it pretty hard
to catch up, but it helped me to achieve a 61% on this exam.

   That's great!

So for me, what went wrong the first time can be summed up by lack of
preparation.  Lack of preparation in high school, and my inability to
prepare for the exam because of an inadequate knowledge base.

   To add to what I already wrote - it may be that highschools have
   changed and that in universities we've under estimated how much.



I have seen nothing significantly wrong with the way you are teaching
now, anyway, this is the way it is in university, what else more can
you do?  Like what the Chinese proverb says, we as students have to
work harder and harder, knowledge won't be absorbed until we have spent
sufficient time in studying them. [sentence removed]

[paragraph removed]

Well, excuse is no use. The only way to go is to spend much much more
time on it. So firstly, my question is: for this particular course,
basically at least how much time do you think should be spent after
class? Before I was told that in university, you have to spend 3 times
of in-class time after class, but I am not sure if this will work for

   There are no rules; different people need different amounts of times
   to learn the same things.  "3 times of in-class time after class"
   doesn't sound unusual, but again, some may need more and some will
   need less.

         And secondly, I have a suggestion:  We do have notes published
online for every lecture, why don't we also publish the solution for
every homework assignment, TA's grading doesn't seem to be enough for
understanding what a correct solution should be, because very often
they just comment with "why is it true?" or "explain".

   There are many solution sets on the web page of 2 years ago.

although I know MAT157 is a more rigorous version of calculus course,
which means there must be lot of proofs in this course and hence in the
exams, lack of some common calculation problems still doesn't seem to
be reasonable, for example, there was no direct problems about
calculation of limits in TE1, and no calculation of derivatives in TE2.
I wouldn't say it's unfair though, I would like to know your idea
behind this choice.

   To some extent it's a reflection of the fact that I dislike
   calculations. Proofs convey ideas. Calculations are better left to

Thank you.

Best regards,

Hi Prof [Bar-]Natan:

I think my problem is that i didn't prepare well enough. I didn't
finish all the readings and practices. For example, problem 4's proof
is in the book, but I only got 5 for that one. I should have worked


I've been in class since the beginning but my free time is extremely
limitted and i was too slack to actually do the studing properly.

Yesterday AFAIK is the 3d time (since the beginning) that I skipped
class.  I never missed a single tutorial either. The moral of the story
is that I shouldn't have being so slack in the beginning - and
apparently now I have to pay, the hard way. I guess there is a
deviation in the above statistical assumption.

I feel ashamed of my score - but it will get better, it a lesson i
learned university teaches you one thing best - discipline. Rest can be
found in the library.

   I wish "shame" hadn't been a part of that - I doubt it is a
   constructive emotion to have.

just my comment.



    Test # 2 was a huge surprise to everyone and I peronally felt that
there were many reasons that contributated to the low avg of the class
as well as my own preformance. All of the comments are meant to give
suggestions not critisizm, no offence intented.

1) I believe the test was harder then last two years.
2) The TAs seem to be marking pretty hard (cant compare to last
3) The fact that most of the students in the class this year has not
   taken any OAC classes unlike the previous years.
    I also think that it would be very usefull if the tutorial before
each test is spend taking up last years test or some other test from
previous years.

   Will be done.


Hello Dror,

In the event that it might be pertinent i will first say that i scored
a XX% on term test 1, and actually improved my mark to a YY%. I should
also say that i have attended every class and every tutorial with the
exception of one this week.

Here are some things that you may want to consider:

1) was there a considerable difference in the marks of students who
attended different tutorials? in other words, did those who were
assigned to Mr.  Krepski score lower than those assigned to Mr Pigott?
I would wager that this difference is statistically meaningful as

   The average for Shay's group was 45.66, for Derek's 50.31 and for
   Brian's 41.34. Though this data may not be reliable - I only know
   where the students ought to go, not where they actually go.

2)Well, there are some questions here with some context first. 
   As i consider what went wrong in my studying, i have been thinking
about the extent to which i have benefitted from my attendence of the
lectures. I was rather surprised to come to the conclusion that i have
not really benefitted much at all.  I have never had such an
interesting and warm and thoughtful and intelligent
teacher/professor/instructor in my life so how could this be?  Much of
this is my fault.  I must admit that the notes that i take in class
often sit for weeks without my looking at them - i have been too
focused on the Spivak text and homework assignments. Furthermore, i do
not prepare for the classes.  Again, i am so overwhelmed with the
homework assignments that i do not have time to read, the night before,
what you will be covering in class the next day.

  Not looking at your notes for a long time is of course not an
  excellent idea, but I believe that no matter what, sitting in the
  lectures will help you. It is true that almost everything I say is in
  the book (and if not in this book, then in many others), but people
  are not robots; you can't keep up reading without it having some human
  component. This is a significant part of what you get in class.

Q: Why not ask questions in class if i don't understand?
A1: It is like someone handing you a chinese newspaper and asking if
you have any questions: you might, if only you understood what it

  Perhaps my moral is that I should push students harder to re-read in
  between lectures. This way the material will look less like a Hebrew

A2: You are very approachable; however, we all sense (rightly so) that
you have a program set and that as much as you ask if there are
"questions, comments?" we know there really isn't the time for you to
answer our questions.

  Yes, I have a program. But it is certainly interuptable.

The lectures sometimes feel like the theorems from the book are just
being copied down on the blackboard giving the bigshots at the back of
the class who had time to read the text the opportunity to "guess" what
you are doing next.  The corollaries of the 3 fun-theorems certainly
felt like this for me.  What could i learn from watching them being
written on the board?

  What they say and how they are proven, just the same as what you could
  have learned from the book. But for the long term, the book in itself won't
  be enough, no matter how good it may be. See my previous comment about

If there was less writing down on the blackboard what is already in our
texts and more "examples" done, i think that that would be beneficial.

3) All of us came into this class knowing calculus, but many of us had
never seen proofs before.  I had thought that this class was going to
be teaching, first in a general way, how to construct proofs, and then
applying that in the context of calculus (which we already knew). I
find that i am do the homework assignments still unsure when something
is actually proven.

  I'm not sure I know how to teach "proofs" in the abstract. One
  possible way of viewing this course is as the training arena on which
  we learn "proofs" on the relatively calm grounds of "calculus" before
  miving on to harder mathematics.

4) I feel like i never learned epsilon-delta proofs.  It is still not
clear for me. I read and re-read Spivak.  I attended the lectures.  I
read and re-read those notes.  I still however, do not feel comfortable
with them.  And being a reasonably intelligent person i am surprised
about this.  After surfing the net and landing on a calculus page that
showed examples of epsilon-delta proofs, i found that i learned much
much more in 2 hours than i had in class or with the textbook.  I
really think that the weeks we were doing epsilon delta were critical
and that we, as a group, just didn't grasp it.  Perhaps if a wider
variety of examples were done in tutorial it would have been

5) The nature of analysis test questions - you know this - is such that
a fool can often score the same as someone who knows there stuff.  All
it takes is a flash of inspiration for a 0/20 to become a 19 or 20/20.
I for example got 0/20 on question 2.  Does this reflect my knowledge
of functions or limits or continuity or the intermediate value theorem?
Certainly not.  Leaving the question blank deserves 0/20; however, is
there a way that the question could have been written that would have
allowed me to demonstrate that i at least had some of the required
knowledge and comprehension?  I think that there is too harsh of a
threshold here, if you know what i mean: an all or nothing result.

  You have a point there and I'll try not to forget it for the next exam.

6) was the test too hard? well, besides my last comment and question, i
will say that it might have been, but only slightly.  Besides the
graphing question, i do think that there should have been some
derivative questions which would account for 20 additional marks, and
that one of the proofs should have been deleted. I also do not think
that it is a cheap way out to have the option of choosing which 4
questions you would like marked, as was done on the first term test.
If we were doing a test full of practical mechanical application style
questions this option would be ridiculous; but the nature of analysis
questions - for the reasons i mentioned earlier - is such that this
approach may be appropriate.

i hope i didn't throw or eat too much dirt!

I'll see you on tuesday.


Dear Professor Dror Bar-Natan:
Here are my few comments on the Mid-Term Exam :
    1.) "Pay  more attention to theorems in the text book than homework
assignments." I learn this lesson from harsh punishment . It is good for
me and I appreciate it .  I wish you will take one question straight from
the assignment on the next exam as well. We worked  hard  on  assignments
and we need some encouragement,don't we?

  Yes, but it is not a coincidence that the theorems in the book are in the
  book - they really are the most important.
    2.) Problem 4 seems simple,but trouble many of us, If you go from lim
( F(a+h)-F(a) / h ) to lim F(x) = F(a). It is very difficult to
figure out the small trick (h/h) .Even with a very good concept of
derivatives you might still stuck.   I know what the beginning is and
what to do the near the end. I mixed up in between. I only got 0 out
of 20 on  this question. In another hand, problem 3 is a good example to
help us to develop the skills to break up the big question,I think.

  Problem 4 is straight from class material.
  3.)I guess that many of my classmates will agree with me :
the grading of this exam is too harsh. It is intimating. If we didn't
take math like MAT157, we surly would have never been introduced to the
true beauty of math. For once, I find math really interesting and fun. If
that is the fact  "God has not seen fit to distribute evenly the gift of
intelligence", then we all expect professor to lower the bar for some
untalented but determined and struggling first year math student.
Thank you for reading my comment
Your student                                   

I got [high] percent on my second term exam, so things didn't go that
wrong for me!  Although I thought I could chip in my ideas maybe they
are of some help to you.  I think the amount of work that we are
required to do for this course is a bit too much, not to say that it
should be less but it takes a while before high school students adapt
to university level studies. Although you talked to us about studying
strategies, which were of great use to me, I think you could again
remind us that we need to think about the intuition behind a
problem/proof when we are learning it. I hope I have been of help.

    Sorry to leave this to a bit later, but I've had a lot to do these
past few days.  I believe a synopsis followed by a conclusion will most
effectively convey my thoughts on the matter.

    On Term Exam 1, I was short on study time (other exams), and as
suggested, managed to go through the text and the notes, but for
additional practice problems (and past test problems), I only mentally
"made sure" I knew how to do them... resulting in a XX%.  My foundation
with delta epsilon proofs was shaky, and my sense of mathematical logic
was less keen than it is now.

    Determined to do better this time, I made sure to budget time more
efficiently, allowing me to go through the text, notes, and do the
additional practice problems as well.  However, I only had time to
glance over the past exams.  Nonetheless, I felt very confident with
the material the night before the exam, and managed to get a good
night's sleep.

    During the exam, things began as usual, reading through the
questions, and selecting those I knew how to do right at the beginning
to do.  With what I believed 40% of the test guaranteed (which turned
out to be the case), I seemed to be slightly behind on time.  This was
made worse by the fact that I could not think where to start on the
remaining problems (for a moment I feared I would get 40% on the
exam...).  Nonetheless, I pulled it together, and in around 5 minutes
of thought I had starting points for the remaining problems, and
managed to have some answers I was satisfied with, some I knew were not
logically sound, and some left unanswered, as it turned out I simply
was not moving fast enough to go back and fix or answer them.  After
time was up, while flipping through to make sure I'd written my name on
each page, I estimated a mark of 75% on the exam... something I was not
happy with, but which was an improvement on the previous mark, and
certainly not the 40% of an hour previous.  My estimate seemed to be
quite good, as I received a YY%.

    So, after that rather detailed synopsis, I believe the test was
indeed a fair test.  A few of the problems were a bit challenging, but
that also made them more fun to solve.  I believe that the problems
were more oriented towards the past tests than the problems in the
textbook.  The fact that I had no initial idea where to go on some of
the questions on the test (whereas I had ideas and used them for
textbook problems) is where I draw this belief from, but come to think
of it, the test should not just be a simple rattling off of known
manipulations of the concepts.  Such is the great balance in tests, and
although I noticed the lack of straight derivatives, I was not
particularly surprised seeing how little emphasis they received on the
problem sets, and that the point of the course does seem to be more on
the deep mathematical understanding of the concepts than simply doing

    The fact that problems that I consider to be solvable were not
solvable quickly and with good logic indicates a lack of preparedness
on my part.  Evidently I simply need to do as many practice questions
as I can (both from the text and past exams.)

    There is no doubt that the material is difficult, but you are doing
a good job of delivering it in my mind, and combined with the tutorials
and my own independent review, I am understanding it and being able to
apply it.  I just think I need more practice, and I shall now rebudget
my time accordingly.  Hopefully in this fashion I can continue to raise
my mark up to my standards.

    Sorry this is so long, and hopefully it's been helpful,


I believe that the main reason for such a result is because of the
curriculum of Ontario's high school math. I think Ontario's current
grade 12 "U" math courses (math for students in university stream,
including advanced functions&introductory calculus, data management,
and algebra & geometry) do not prepare you well enough for courses like
MAT157.  Personally, Ontario's high school math felt more like just
memorizing stuff, rather than understanding concepts. I noticed that
MAT157's curriculum is completely different from high school's. It
requires you to understand the concept and to be able to use it as
he/she is desired, which high school doesn't prepare you well enough. I
think Ontario needs to make a math course that will prepare students
well for this kind of course.  Also, because Ontario got rid of OAC,
students only have 4 years to get ready for the university. This
automatically brought down graduating high school students' standards a
bit. However, university's standard toward students haven't changed
much. This obviously would cause test results go down.

Well... That's about it for my comment.

Happy Holidays and see you on January! :)


 Hi professor! Started reading other people's comments and decided to
compose a letter of my own.
 So what went wrong? Well personaly i didnt find the test that very
bad.  The proofs were done, limits and differentials discussed and the
general idea understood. So why is the average so low? Perhaps its just
a hard transition from any other kind of math that was done before.
 Some constructive criticism. Sometimes i feel that the way proofs are
done in class is a bit confusing. I REALY like and appreciate the idea
of trying to describe math in simplier, more understandable terms but
sometimes it get a bit confusing.
 Besides that i think that the class is realy great.

    Best Regards, 

Subject: What went wrong...

Sorry for my late feedback. I don't know how useful my thoughts are to
you now, but as I was reading some of the other comments on what went
wrong I had a couple of my own comments to throw in as well.

  Useful, and further comments by you or others will remain useful
  thoroughout the term.

I guess I can't speak personally for most of the other students as I am
[an advanced student]. But none the less I feel that I've already
learned much more at a deeper level than I did in either of those
courses. I'm not doing outstanding in the course as yet, but I enjoy it
very much and like the current level of difficulty. In fact my scores
are lower than I'm sure that I could attain if I were to focus more
(and I will). I felt both tests were not overly difficult, but I made
mistakes that I shouldn't have.

Anyways on 680 news this morning there was a report that was
criticising Ontario highschool education for cramming students with
lots of knowledge to compensate for the lack of grade 13, and at the
same time the students were lacking in deeper understanding of the
subjects. And perhaps this is reflected in the desire of many students
in this 157 class to have more 'calculation' based questions.

So it's a shame, but perhaps for the time being the loss of grade 13
will have that negative effect on students. I fast tracked from
highschool so I also did not have grade 13 and indeed found mat157
overwhelming (that was two years ago) and I lacked the mathematical
maturity for the course.

I do not think that this test was significantly more difficult than in
past years and it would be a shame to make the class 'easier' (or to
focus more on calculation type questions). I apologize for having lots
of comments and opinions and no solutions :)

Happy holidays,

Subject: Test 2

   I am one of the students who did poorly on test 2. In my opinion,
the reason for my failure and likely the poor results of others, is not
related to how the material was lectured or how the tutorials were
done. I think that it is a combination of the lack of the 13th grade
and the bad habit of being used to not having to do much work and get a
good mark in math. To prepare for this test I only spent the weekend
before it looking over the text book and some questions without
focusing on the material too much during the term.  Evidently, this was
not nearly enough effort put in so that I could comfortably understand
everything as the handout said. Although this course does not require
previous knowledge of calculus, it certainly requires a strong
mathematical history, the lack of the 13th grade put students entering
the course with less than the previous years for which it was initially
geared for. I think the best solution to this problem as suggested by
others is to make sure the problem sets are mainly finished before the
tutorials so that they do not go to waste, extra practice for the tests
at least a week and a half before their date, and consulting many extra
sources within the context of the course.  Although what I said was
mainly mentioned by others I wrote this feedback to enforce these ideas
so hopefully we can do much better in the future.


Regarding the midterm; I thought it was fair and even quite generous.
I say generous because there are many assigned problems for which I
could not reproduce a proof quickly on an exam, and you didn't ask any
of them.

I noted that the "locally bounded" question (which I feel was the most
involved proof), was taken from a previous exam.  So, even if one
didn't prepare by "thoroughly understanding the material", but instead
only studied particular questions, the hardest one could have been
reviewed ahead of time.  (Moreover, it was divided into parts on our
term test, breaking down the proof into the requisite steps, making it

I was surprised the "prove continuity given differentiability" question
was weighted so heavily.  However, the proof is straight forward and so
I assume that was your "nice" question.

I think it was just a matter of preparation, as always.

Hello, Sir.

Sorry I send this email late, but here are my thoughts regarding

First of all, I don't think there is any unfairness in this exam.
Numbers 3, 4 are questions taken out of past exams and class notes, and
number 5 is a curve sketching question, which most of us have learned
in high school.  I think many more of the class would have passed by
preparing more for the exam (reviewing class notes, textbook, and past
exams).  The fact that I lost marks on number 1, 2 showed that I still
didn't really get the idea of how to approach a proof.  And the reason
for this is the lack of practice.  Although we get recommended problems
besides questions required to be handed in, I personally never really
did those recommended problems.  So the first thing I can think of now
to improve my mark is to start trying some of these problems.

Secondly, I think your teaching style is pretty good.  You present
material carefully and with explanations.  However, I found that most
of the time in class for the term, you only taught us proofs of the
theorems but not examples of proofs that use these theorems (except
corollaries).  I understand it is not really possible for you to give a
lot of examples in class but maybe we could do that in the tutorial.

And also we all noticed that the averages of these two exams for this
year are much lower than the last two years.  As the instructor, did
you notice any difference between this year's students and the last two
years'?  (For example, how we react in class and etc?)

  Yes, I do feel there is a difference, and that's why I'm concerned.
  Other than the obvious difference in grades, it is hard to pinpoint
  the difference - it's more the "feel" that I get when I turn back
  towards the class and see too little interest than anything clear and