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:
  1. 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.
  2. 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.
  3. 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. 
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.