General Advice
Understanding your position in the
course
After the return of each of the midterm exams, it is a great idea to
take stock of your position and direction in the course. To do
this, you will need to make sure that you have a strong
understanding of how course grades are computed in this
course; this information is found on the Course Grades page of this
website.
For example, knowing your existing midterm exam scores and the
resulting grades on the 4-point scale by way of each exam's curve
(see the Exam Curves page on this website), and knowing the weights
of those exams (see the Course Grades page on this website), you can
compute the current state of your weighted average; then the grade
bins (see the Course Grades page on this website) will give you some
idea of your current position in the course. Of course this is
not a guarantee, and your actual course grade will depend on your
grades on the other graded items that are not yet determined.
To get a sense of your direction in the course, you can simply look
at the trend of your exam grades on the 4-point
scale. For example, if your curved grades on the first two
exams are 3.71 and 2.14, this is a worrisome down trend, suggesting
that something has gone wrong with your ability to keep up with the
material; in such a case, the student should take steps to identify
with confidence what the problem is, in the hopes of turning things
around while there is still time.
You can also compute what you have "in the bank"
(this is a great feature of this grading system that averages on the
4-point scale). To do this, compute a hypothetical weighted
average for the whole course, using the grades you have, and
assuming 0 for those that you don't. The result of this
hypothetical computation will tell you the lowest possible grade you
can receive in the class. Similarly, you can speculate
on possible future exam scores to see how they would
affect your course grade. Examples:
- Suppose the weight of the first exam is 25%. A student
whose score curves to 3.64 on the 4-point scale is already
guaranteed a course weighted average of at least 0.91, which is
in the bin for a D. Of course this student will almost
certainly get a higher course grade than this, but it still
might give this student some peace of mind to know that they
have already passed the class.
- Suppose the weight of each of the first two exams is
20%. A student whose scores on these exams curved to 2.07
and 1.63 might reasonably be worried about the downward trend,
and as such concerned about the possibility of failing the
course. But this student is already guaranteed a course
weighted average of at least 0.74, which is in the bin for a
D-. Of course this student should still work hard to do as
well as possible on the remaining exam(s) to maximize their
grade, but knowing that passing is guaranteed will take away a
lot of the possible worry.
- Suppose a student has 3.35 on midterms with a total weight of
60%, and has perfect (4.0) homework and attendance grades (with
a total weight of 10%) they are confident they can
maintain. The student is hoping to do well enough on the
final exam (weighted 30%) to get an A- (whose bin starts at 3.5)
in the course. A bit of simple arithmetic shows that this
student simply needs to score at least 3.64 on the final exam.
Working together on the homework
Working together in groups on
homeworks is strongly encouraged! You
will find that the people you are working with either (1) understand
something you don't, in which case they can explain it to you; (2)
understand something that you do understand, but from a different
point of view -- these additional perspectives can prove to be very
useful; or (3), don't understand something that you do understand --
in which case you have the opportunity to explain it to
them... I think you will find that in the process of
explaining something, very often you will achieve a better
understanding yourself.
Of course, it goes without saying that even though you may work in
groups, the homeworks you turn in
must be your own work. You may share ideas,
perspectives, approaches to problems, but copying is not
allowed. Furthermore, keep in mind that the homeworks are
primarily a learning tool, and count for a fairly low percentage of
your grade. Do not deprive yourself of this invaluable
learning opportunity!
Note, because of the sheer numbers involved, usually the grader will
be asked to grade only a specific subset of the homework
assignment. The homework score will be entirely based on the
evaluation of your answers of just those select problems. The
grader will not look at the other problems -- so, you cannot assume
from the lack of any marks that the solution you submitted is
correct. Also, note that students are required to turn in a
complete homework assignment. If you turn in an incomplete
assignment, then you subject yourself to the risk that some of the
problems you did not submit might have been the ones that were
graded, resulting in a score that might be disproportionately low.
Exam Studying
This course covers a huge amount of material, and so each exam
requires an enormous amount of preparation. In fact, it is not
reasonable to do all of that preparation in the few days before
an exam, in the way that students usually think of
"studying for an exam". Students who procrastinate their
studying for the exam until the few days before will find themselves
completely overwhelmed, and are far less likely to do well on the
exam.
Rather, a much better way to prepare for the exams in this class is
to prepare for the exam continually throughout the semester.
That is, after each lecture, the student should study that material
sufficiently thoroughly that he or she feels prepared to take an
exam on the topic. Note, this requires substantially more work
than merely working the homework problems. (See the discussion
of the expectations in this course on the Learning Expectations page
of this website.)
In addition to spreading out the effort, there are more advantages
to this strategy. First, the concepts in question will have enough
time to "sink in" -- this is a phenomenon of learning,
that it just takes time for a student to become comfortable and
fluid with an idea. If you wait until the day before the exam,
that "sink in" time just will not be there.
Second, by being thoroughly comfortable with the content as soon as
possible, the next lectures will make more sense to
the student because the foundations have already been
understood. Remember, this is a largely "vertical" course in
that most of the ideas covered in this course depend on an
understanding of those presented previously.
If a student does this consistently throughout the semester, then in
the few days before each exam the student can concentrate on
memorizing needed formulas, refreshing ideas that have already been
thoroughly learned, and the total effort is something that is
reasonable to do in those few days.
Improving Your
Performance
Sometimes, students receiving low scores on exams feel that they
understand the material better than the score would suggest.
There are several possibilities that might explain this.
The most common explanation is that the student simply does not
understand the material deeply enough -- and very likely, not as
well as he or she might think. These students need to find
ways to "raise the bar" on their comprehension. There are
several ways that you might do this.
- Most obviously, some students need to increase the
amount of time that they spend working on this
class. Sometimes, this can be achieved with better time
management techniques. Sometimes this time must come at
the expense of other classes. And of course sometimes the
time can be found by reducing time spent on social and/or
extracurricular activities.
- Students should also make sure to spend study time
efficiently. Make sure that you are not just
memorizing algorithms, but that you are also understanding the
underlying ideas, developing comfortable familiarity with their
use and with their connection to related ideas. Remember,
this is a course about ideas and reasoning, not
memorization.
- Some students might simply not have a good idea of how to
gauge whether they understand the material well enough or
not. One good way to gauge your understanding of some
given topic is to contemplate the prospect of giving an
oral presentation to a group of people on that
topic. If you feel that you can give a comfortable and
thorough presentation of the topic, including the background,
relevant derivations/proofs, connections to other ideas, and
example applications, and that you would be comfortable
answering potential questions from the audience, then that is a
good sign that you do have a good understanding. If the
prospect of such a presentation makes you nervous, this might be
a sign that you do not have the necessary level of understanding
yet. In fact, the part of the presentation that you would
be the most nervous about might give you specific clues to
suggest what you need to study more thoroughly.
In fact, as discussed previously, you might consider actually
giving such presentations. If you work in a group, the
rest of your group might be willing to listen to your
presentation, ask questions, and give constructive criticism
afterwards. This could be a good learning experience for
everyone involved.
- Some students might not have absorbed the material thoroughly
enough to allow them to solve problems quickly and
efficiently. Note, exams in this class will
(unavoidably) involve a time constraint. This time
constraint also serves a purpose, which is to create an
incentive for students to learn the material to a sufficiently
deep and thorough level that they can solve the problems
quickly, which requires greater proficiency and comprehension
than merely being able to solve the problems in a comfortable
amount of time. That level of proficiency and
comprehension is something that we want to achieve by creating
exams that reward it; and it is a higher level than that which
is required in lower level courses (and of course high school
courses).
If you feel that you could have solved all of the problems on
the exam but you just didn't have enough time, the above point
might be relevant to you. Be sure to learn the material to
the point where you can solve problems more quickly and
efficiently.
- Make sure to use all of the resources for this course.
If recordings of the lectures have been made, you should
re-watch some of those lectures, particularly those on material
that you do not feel as comfortable with. While watching
those recordings, you can of course make extensive use of the
rewind button and the pause button to give yourself the exposure
and time to be able to absorb the ideas completely. And
note, if you are stuck on one topic, it is very likely that this
is a consequence of not having a sufficient understanding of
some earlier material; before continuing then, you should make
sure to go back and solidify your understanding of that earlier
material.
- If you feel that you need more practice materials than are
presented in the book, remember that in many of my classes there
are some old exams from previous semesters available for your
use. Most of those old exams also have solutions available
-- you might want to try working the problems first on your own,
and then comparing your solutions with the posted
solutions. Make sure to compare both the final answer, and
the method, and the clarity of presentation.
You can also generate your own practice materials,
especially if you study sometimes in a group. Each member
of your group could generate some hypothetical exam
problems. You can then exchange the problems, work on them
separately, and then come together as a group to discuss
them. Good questions to discuss would be:
- Was this problem too hard or too easy?
- Does the problem focus on important ideas from the course,
or is it tangential and/or irrelevant?
- What important idea does this problem allow the tester to
gauge in the student?
- What other ways might this problem be rephrased, while
still being basically the same problem?
- Are there other types of problems that might be created
that test this same underlying idea?
I think you will find that in trying to generate hypothetical
exam problems, you will be forced to think about the underlying
concepts in a way that will help you organize the ideas, and
uncover areas you do not understand thoroughly. Of course
working on your group members' problems will also give you good
practice. Finally, the discussions afterwards should help
generate good dialogue about the ideas in the
course, which should give opportunities for some students to
hear explanations of ideas they don't understand, and
opportunities for other students to practice giving explanations
of ideas they feel they do understand.
Another possible explanation is that the student might not be using
good test taking skills. These are unavoidably
important, and every student should make sure to have a reasonable
competence with them. For example, don't spend all of your
time on one question on the exam. If a problem is taking a
long time to work out or if you don't think you know how to do it,
consider that you might be able to get more points per unit time
working on other problems first. Make sure to flip through the
entire exam to have a general idea of how many points are available
for what problems, and take the best of those opportunities (the
ones with the most points, that you can solve in the least amount of
time) first.
Yet another possibility is that the student might be allowing nerves
to affect their ability to think efficiently and creatively during
the exam. If this is the case, you will need to find a
way to take control over your thoughts so that you can
concentrate and work efficiently and creatively during the exam
time. Of course there is no substitute for full and
appropriate preparation; if you are well prepared then you are less
likely to have anything to be nervous about in the first
place. Also, it is important to block out all thoughts that
are not directly relevant to the problems on the exam. Don't
allow yourself to think about what is going to happen after the exam
is over, what your grade might be, how that will affect your final
letter grade for the course, what you are going to major in, how
much you are going to need to study for some other class -- for the
duration of the exam period, all of these topics are nothing more
than distractions from what you should be doing, which is to be
thinking actively, efficiently, and creatively about the problems on
the exam.
You might also do some of your own independent research on how to
overcome nerve related issues on exams.
Whatever the explanation might happen to be, students should be sure
to understand though that letter grades in this course must be
determined strictly from the grades on graded items.
Exceptions to this rule would be unfair to the other students in the
course. So, one way or the other, if your scores are not in
the range that will lead to the letter grade you are hoping for, the
only solution is to find a way to bring up your scores.
Remember, it is your responsibility to identify and solve the
problem or problems that are preventing you from
achieving your goals.
I try to be available to
students as much as I possibly can. But students should be
aware of the fact that I tend to be extremely busy on the day of
exams, and also on the day before -- I have the exams to write (in
all of my classes, not just this one!), to be copied, checked,
solutions to write, and lots of other essential minutia that must be
dealt with.
Tragically, many students leave it until the day of the exam, or the
day before, to come to me with their questions. Again, I make
every effort to be available, but sometimes the reality is that I
just don't have the time to answer questions so close to the exam
when I too am so very busy. All too often such students find
themselves with very little time left before the exam, and
significant concepts not yet understood.
Please do not procrastinate like this and create for yourself a
no-win situation.
Of course I have already promoted the idea that students should be
preparing for the exams continually throughout the semester;
students who take my advice on this will have the further advantage
of being able to come and ask me substantial questions well before
the exam, when I am far more likely to have time to answer those
questions.
Try to make sure that you get good sleep on at least
the night before the exam. Inconvenient though it may be, the
reality is that sleep is an important factor in a student's ability
to think analytically and quickly, both of which are critical to
doing well on a math exam. Obviously study is critical too,
and students have to make their own decisions about the trade-offs;
but do not underestimate the importance of sleep. In fact it
would be best to get good sleep every night, and having a regular
and full sleep schedule makes it easier to get the sleep when you
need it.
Ideally of course students should not have to make trade-off choices
between sleep and study. With good time management,
you should be able to plan your study time well in advance, and
arrange to be thoroughly prepared still in time to get a full nights
sleep.
To a great extent, this is a writing class.
This might sounds like an outrageous statement, but the fundamental
point here is that this course, just like a history course or a
political science course, is about the comprehension and
communication of ideas. Obviously the types of ideas in
question are different, but like many or even most other courses on
a college campus, this course requires students to understand
concepts, and then to communicate that understanding through
writing. In this course the writing comes in the form of the
answers that the student gives to homework and exam problems, rather
than traditional essays.
Students should take very
seriously the idea that the solutions they write for the homework
and the exam problems should reflect this perspective.
All too often students think of the writing they do on an exam to be
simply a personal convenience for them -- that is, a tool to keep
them from having to work the problem entirely in their heads.
This is NOT the right point of view, and it will not lead to the
desired credit on the exam.
Instead, think of your writing on homework and exam problems as
being documentation of your thought process.
You are communicating to another person, namely the grader, and your
goal is to communicate that you understand the tools needed to solve
the given problem and that you know how to implement those
tools. Of course in communicating that to the grader, you will
also write down the necessary algebra to allow you to arrive at the
final answer.
One of the great tragedies of the way math is taught in many high
schools is that, for whatever reason, these courses often end up
being "symbolic manipulation" courses. Along those same lines,
students often simply memorize algorithms that are applied to given
problems, allowing for the computation of the correct final
answer.
Of course, as we have discussed previously, this is certainly not a
workable strategy in this course. The goal of this course is
for students to have comprehension of ideas, to a level that allows
the student to use those ideas for the creative solution of
problems, and also to be able to communicate that comprehension in
writing.
There is an analogy that can help students understand the propriety
and necessity of these goals. Consider instead the history
courses that one might have had in high school. Certainly
there are many history courses in which the student is expected
simply to memorize large amounts of data -- roughly boiling down to
names and dates.
Of course in a college setting, history courses are much more than
that. Students are expected to know names and dates,
certainly, but the point of the course is much more than that.
Students are expected to know how and why things have changed over
time -- and to be sufficiently familiar with the facts and prevalent
theories that they can form their own ideas about how different
aspects of society have interacted to cause these changes.
Analogously, in this course and the rest of the math courses at
Duke, students are expected to understand at a higher level.
Inadvertent errors cannot be completely avoided. Certainly it
is entirely understandable that students will make algebraic and
even arithmetic mistakes while working problems on the exams and the
homework; and usually the presence of these types of errors, in the
absence of conceptual or important procedural problems, can still
allow for most of the credit on the problem to be awarded.
However, there are some types of algebraic and arithmetic errors
that are so outrageous that they should almost never happen; and
when they do, they are usually a sign of serious problems with
algebra, as opposed to careless oversight. These types of
errors very often will greatly reduce the number of points awarded
on the problem.
For example, in simplifying the expression "-3(x-y)", one might
inadvertently lose a negative and mistakely arrive at
"-3x-3y". This is an understandable error, and if this is the
only error on the problem the student would probably be awarded most
of the points for the problem.
However, in simplifying the expression "sqrt(a+b)", one should never
make the mistake of rewriting this as "sqrt(a) + sqrt(b)".
This is not an inadvertent error, but rather evidence that the
student does not thoroughly understand the algebraic skills taught
in early high school algebra. This lack of understanding is a
significant impediment to the ability of the student to be able to
work the problem, and so very likely the points awarded on the
problem would be low.
In this course we assume that students are thoroughly competent with
the entire standard math curriculum of high school, including
algebra, geometry, and trigonometry. Students are also
expected to be thoroughly proficient and refreshed on material in
all of the prerequisite courses for the courses they are taking.
In some settings, very often including high school math courses, the
grading scheme is set up so that students do not lose significant
points, if any, for errors relating to mathematical content from
previous courses. For example, in such grading schemes,
algebra errors would not cause points to be lost in a calculus
course.
As discussed previously this is not the case in this course.
Unfortunately these schemes cause problems that are not limited to
the courses in which they are applied. Specifically, students
in such courses might notice that they can increase the number of
points they are awarded by intentionally making certain types of
errors.
For example, suppose a student in such a calculus course is faced
with finding the antiderivative of the function "f(x) =
sqrt(1-x^2)". The correct solution involves making a trig
substitution and using a half-angle formula; and if the student is
unable to do this, other solutions are not available and few points
would likely be awarded.
But, by intentionally making a "mistake" in recopying, the student
can turn this into "sqrt(1-x)", which can then be antidifferentiated
by simpler methods; similarly, the student could intentionally make
the algebra "mistake" of rewriting this as "sqrt(1)-sqrt(x^2) =
1-x", which is also very easy to antidifferentiate. Students
who intentionally make either of the above algebraic "mistakes" will
be able to come down to an answer, and might hope for strong partial
credit based on the fact that none of their errors involved
calculus.
I call this the "technique of intentional errors". Because
this practice of making intentional errors is effectively rewarded
in these systems, very often students leave the class with the idea
that this is a respectable practice, and try to make use of it in
other classes.
I feel that this technique of intentional errors is directly
anti-intellectual, and therefore entirely opposed to the
philosophical direction of this class. Let me be
clear that in this class, this technique of intentional errors is
not condoned, nor is it advantageous to the student in this class in
any sense.
For one thing, as I have described previously, in this class
outrageous algebraic errors in the solution to a problem will not
allow for the awarding of more than minor partial credit.
Hopefully this fact alone will stop most of this sort of thing.
Furthermore, students who make common use of the technique of
intentional errors should note that the grading system in this
course does NOT make it a strategically useful technique
anyway. Specifically, in this class points are awarded, not
taken off -- so the mere lack of errors in the calculus does not
imply that no points will be "taken off". Rather, on any given
question the student is expected to demonstrate understanding of
particular skills, and points are awarded based on the extent to
which students have achieved that demonstration. In the
example problem above, the student is supposed to be demonstrating
the ability to use trig substitutions and trig identities to compute
an antiderivative; by instead transforming the problem into
something entirely different and significantly simpler, the student
simply has not demonstrated the required skills. So very few
points would be awarded.
Finally, I note that this technique of intentional errors is
intrinsically dishonest -- that is, the practitioner presents as
inadvertent a mistake that is actually intentional, and therefore is
making a deliberate misrepresentation in the hopes of personal
benefit. And of course, academic dishonesty is certainly
inconsistent with the Duke Community
Standard.
If you find yourself unable to work a question on an exam in this
course, the most effective way to get partial credit on the question
is to make honest, legitimate attempts with whatever techniques you
feel might have a chance of working. I will award partial
credit based on progress made toward a solution even if the work is
not finished. I will not award partial credit for a wrong
answer if the method is intrinsically flawed and never would have
had a chance to result in a valid solution.
Students in this course no doubt have been exposed to much
discussion of the Duke Community Standard. I remind all
students that cheating, in any of its forms, will not be
tolerated in this course.
Note further that students who do cheat are much more likely
to be caught than they probably realize. Students
who have been found responsible for cheating naturally do not
advertise that fact, and organization on campus that have the
information are bound by law not to reveal this private
information.
For obvious reasons I will not disclose all of the methods that we
can use for this, but let me say simply that this is something that
I view as very serious, and I take an active role in ensuring that
students abide by the rules.
Students should note carefully that the atmosphere and attitude
about cheating on Duke campus is significantly more serious than in
most high schools, and many other universities. Duke does not
"look the other way", or simply dismiss incidents as "no big
deal".
If you think that there is any reasonable chance that you are not
clear on exactly what this means, I urge you to read about the Duke
Community Standard and talk with someone at the Office of Student
Conduct.
Write neatly!
The all-too-common attitude in high school mathematics courses is
that the answer is the only thing that matters; if the grader can
read the answer, then the clarity of the rest of the work is not
relevant.
As per the above comments, this is not the case in this
course. Everything that is
written on the exam is going to be read and considered for the
contribution it makes to demonstrating comprehension of the ideas.
So it must all be legible.
The same standards of neatness and legibility appropriate to, say, a
history class should be applied in this mathematics class.
Because of the types of notation used we do not expect that the
student will type the solutions to homework exercises but, we expect
that the solutions will be written neatly and legibly, the papers
should not be crumpled or stained, and there should not be large
areas of the paper scratched out. (If you are not sure how to
work a problem and if you are going to put its solution on the same
page as that of another problem, you should do your scratch work on
another piece of paper, and then write out the solution neatly on
the paper you will turn in.)
Furthermore, note that the flow of
ideas over the page should be reasonable. Ideally, it
should be "top down", or perhaps two or three column if the student
prefers. Either way, the grader should be able to look at the
page and effortlessly identify the location of the beginning of the
argument, and easily follow each successive step until arriving at
the conclusion.
Note, explanations that bounce around the page in a seemingly random
pattern will be more difficult for the grader to follow, which
invariably leaves the grader finding less clarity. This can
lead to the awarding of fewer points.
Make sure also to use only standard notations, or notations that
have been used in class or in the book. If there is a notation
that was used in a previous course that you are likely to use in
your work, and if you are not sure if it is acceptable notation, do
feel free to ask me about it at any time.
There are countless different styles of handwriting; some people
make certain letters one way, some another, some use cursive, some
print,...
It is perfectly okay to have your own style of handwriting.
The only requirement that I have is that, however it is that you
write, make sure that there are no
two characters in your handwriting set that look identical or
nearly identical. Obviously this makes it very
difficult to read and interpret while grading.
Here is an incomplete list of some common such problems, and some
suggestions for how to deal with them:
- "2" and "z" very often look similar. Putting a horizontal
bar through the "z" is a common way to solve this problem.
- Capital "I", lower case "l", and the number "1" each are commonly
written with merely a vertical line! A simple solution to this
problem is to put horizontal bars on the top and the bottom of the
capital "I" as is done in many fonts, and to write the lower case
"l" in script. Alternatively, one could also put the usual
angled tip on top and horizontal bar on the bottom of the number
"1", although for many people this ends up looking too much like a
"2".
- Some people make their "x" and "z" in a script style that makes
them look very similar. If this is the case for your writing,
please make an adjustment to at least one of these so that there is
no confusion.
- The print lower case "t" can look very similar to a "+".
When you are using "t" as a variable in an algebraic expression, you
can distinguish it from "+" by writing it in script.
- The print lower case "u" and "v" can look practically identical
when written in a hurry. One way you can help with this is to
write the "u" in script (with the tails on both sides), and leave
the "v" in print.
Students in this course should make sure to obey the standard
rules of "mathematical grammar". The most
important point on this is that notations have precise meanings; and
the student should use them in such a way as to communicate
precisely that which he or she is intending.
For example, to write the derivative of x^2, it is acceptable to
write:
(x^2)', or
d/dx (x^2), or
df/dx, where f(x) = x^2
It is NOT acceptable to write:
x^2 (d/dx), or
x^2 dy/dx, or
d/dx f(x)=x^2
These latter expressions have either different meanings or no
meanings at all. Students may be accustomed to this sort of
sloppiness, perhaps having successfully argued with previous
instructors that while the notation was not actually correct, the
meaning was still communicated, and that that was what really
mattered.
In this course we will not accept this sort of sloppiness
as being unimportant. Mathematics is about the communication
of precise ideas, and so the precision in the communication itself
is critical.
Here are a few other issues of mathematical grammar:
- Of course students should all be aware of the standard order of
operations.
- Generally it is preferred to place numbers before variables in a
product. For example, one should write "2x" instead of
"x2".
- One should think about other instances of order in a product, and
how choices on this relate to clarity. For example if one
writes "x cos y", the meaning is clear. On the other hand,
"cos y x" might have been intended to mean the same thing, but in
this form it is not clear if the intent is "(cos y) x" or "cos (y
x)". Given this confusion, it is preferred to write "x cos y";
"(cos y) x" is acceptable but not optimal.
- Students must be careful to use parentheses when needed to clarify
possible confusion with notation. The previous example
illustrates how this can be done.
- Students should not use the equal sign as some sort of
multi-purpose punctuation. For example, when asked to find the
derivative of the function f(x)=x^2, some students will write:
f(x) = x^2 = 2x = f'(x)
While it may be the case that the grader can figure out from what is
written here that the student did understand the answer to the
question, the critical fact remains that what is written is entirely
false. This sort of sloppiness is unacceptable and points will
be counted off for this sort of thing in this course.
Likewise, students should not use arrows as a similar sort of
multipurpose notation, nor as an alternative to an "=".
- Many students, when simplifying, will cross out cancelling factors
as a visual aid to themselves while working. It is okay to do
this sort of thing for your own convenience. But, be sure to
realize that very often this does not communicate the process to the
reader; in fact, sometimes what remains on the page is very
difficult to interpret.
For example, what is supposed to be represented by the following?
(96)(32)(16)(2)
----------------------------
8 3 2
At first glance it might appear that there was initially a large
fraction, and then some cancellations were done; but a closer
inspection shows that what might have appeared to be cancellations
could not be valid, because 96*32*16 is certainly not equal to
8*3*2.
After some consideration, one might make the following plausible
guess as to what happened. The initial expression was
96
----------------
8 3 2
The student noticed that 96 has a factor of three, so he cancelled
the 3, and cancelled the 96 but replaced it with 32. Then he
noticed that 32 has a factor of 2, cancelled the 2, and cancelled
the 32 but replaced it with 16. Finally he noticed that 16 has
a factor of 8, cancelled the 8, and cancelled the 16 but replaced it
with a 2.
The point here is that while this process might have been convenient
for the student in the first place, it is very uncommunicative to
the grader as to what actually happened; and even worse, as we saw
with the above example, it can even communicate the wrong
thing. NB, cases like this that are either wrong or ambiguous
will be interpreted literally and thus the student will not receive
credit.
Much better would be to write the following:
96
---------------
8 3 2
8 * 3 * 2 * 2
= ------------------------ = 2
8 3 2
For many more examples of these sorts of things, see "The
Most Common Errors In Undergraduate Mathematics".
In many high school math classes, and even on some of the
standardized exams such as the AP, students are not required to
simplify arithmetic expressions. The rationale for this that
has been presented to me is that the final arithmetic simplification
is not the component of the problem being tested, and so students
should not be liable for having to do it.
For example, in such contexts, if the student has reduced the answer
to a question down to the expression "3*32 - 12", it is considered
acceptable to leave the answer in that form, and not to perform the
final simplification to the final answer "84".
As I wrote in a comment earlier on this page, I do not believe in
absolving students of all responsibilities for being competent in
prerequisite material for the course. Rather, the solutions to
most questions involve techniques both from the course and the
prerequisites, solving such questions requires being proficient in
both, and students are responsible for providing a full solution to
the question.
Along these lines, students should
do all of the arithmetic in the problem, and simplify their
answers completely. Answers such as "3*32 - 12" will
be considered incomplete. Similarly, students should simplify
the values of standard functions -- "sin(pi/6)" should be simplified
to "1/2", "ln(8)/ln(2)" should be simplified to "3",... The lack of such simplifications will
be interpreted as an indication that the student does not have the
ability to execute such simplifications correctly, and the
work will be graded accordingly.
In some instances though, the arithmetic for the question might be
particularly inconvenient. Usually I will announce these
circumstances at the beginning of the exam period, and indicate that
students do not have to perform the final arithmetic for that
question. If a student believes that a question requires
particularly inconvenient arithmetic and thinks that I might simply
have forgotten to make the announcement, they should ask me during
the exam period before making the decision to skip that arithmetic.
Students should think very
carefully about how to make the most effective use of the lecture
notes in class.
For example, having a printed copy of the lecture notes obviates the
need to spend valuable class time transcribing what is on the
board. Since classes like this include so much verbal
discussion of ideas, one would expect that having such a printed
copy would then allow you to focus more on these
discussions, and thus get more benefit from your time in
class.
Of course it is also to be expected that you would want to make
notes to yourself about things that are talked about in class.
So you should be prepared to write your own comments on the printed
notes. You might find it helpful to write with a
colored pen, so that your own comments are more
prominent on the page; so, you should consider coming to class with
a colored pen.
Since it is often the case that some students might not understand
everything in the lecture the first time through, you should expect
to need to make some use of the lecture recordings. You might
want to rewatch entire lectures that you found difficult, or you
might prefer simply to watch specific parts you found confusing.
In order to help keep track of things you want to watch again,
consider making time stamps on the printed lecture notes.
For example, you might simply indicate next to each important or
confusing topic the date and time it was discussed. You could
also make such indications at the top of each page, or next to every
example and/or theorem. You could also make a notation to
yourself to review specific portions of the lecture in the future --
then, later that day or whenever you were doing your studying, you
would have a convenient indication of where to find that specific
portion of the lecture.