Best Rank Four Xyz Homework

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Exams, Homeworks, and Grading

Students are responsible for understanding all of the policies on this page.  These policies are subject to change at any time by the instructor.

Assumptions and Intent in Exam Problems
Grading and Exams
    Grading Philosophy
    Graded Items
    Exam Scores
"Fair Game" Material
Rough Cutoffs
Grading Comment Codes
Reading Quizzes
In-Class Quizzes
    Excused Absences
"Log Confidence" Exam Problems
Improving Your Performance
Additional Comments

Assumptions and Intent in Exam Problems

Students should be very careful about what formulas and techniques are used in the solution to a problem.  Points will not be awarded if students simply cite formulas that were not intended to be used in the solution to the problem.

For example, consider the problem of finding the antiderivative of the function (1-4x^2)^(-1/2) with respect to x.  There are multiple expectations that an instructor might have for solutions to this problem:

   1. Some instructors might allow students to cite formulas from integration tables.  In that case, an acceptable solution to the problem would consist only of plugging in to a memorized formula.
   2. Other instructors might allow students to cite the formula for the derivative of the arcsine function, in which case an acceptable solution to this problem would require the student to perform a substitution.
   3. Another instructor might allow neither of the above, and want the student to do a trig substitution -- thus assuming even less and showing more work in the derivation of the solution.

This choice is entirely at the discretion of the instructor.  If it is not explicitly clear in the statement of the problem or from the course materials or lectures, the student should clarify before making an assumption that might well turn out to be wrong.  Misunderstandings on this do not excuse inappropriate solutions.

For most Duke math classes, students can rule out certain possibilities by some simple reasoning.  In the example above, for instance, note that the first option allows the student to come to a correct answer by having memorized a formula, without any demonstration of understanding of significant ideas or techniques.  This is not typical of math classes at Duke, and so students should not assume that such a solution would receive credit.  Both the second and third solutions above though do demonstrate understanding of important techniques, and either could reasonably be what the instructor expects; students should get clarification from the instructor.

Other conclusions can be drawn by thinking about the motivation for a given exam problem.  Suppose for example that a problem asks you to compute the derivative of the composition of two functions, h(x) = g(f(x)), where f and g are given.  The clear intent of the problem is for students to demonstrate their understanding of the chain rule by using it to derive the solution.  As this is the clear intent, students should suspect that they are supposed to do the problem that way.  Now, a correct final answer can be computed by first computing h(x) explicitly and then computing its derivative directly, with the usual shortcut rules -- but, as this method circumvents the clear purpose of the problem, students should not expect to receive credit for this.

Remember, your solution is not graded on the correctness of the final answer, but on the extent to which you have demonstrated comprehension of the corresponding techniques and concepts.  So you should make sure to identify what those are as you work any problem.  Again, in any situation where you are unsure, you should make sure to clarify before making what might turn out to be an importantly wrong assumption.

Of course sometimes there are multiple ways to solve a problem, several of which are substantial and appropriate.  In those cases, it is not expected that the student will necessarily make the same selection as the instructor.

Grading and Exams

Grading Philosophy

At the end of the term, every course grade assigned in this course is determined by the instructor's thoughtful and impartial assessment of the performance of the student in the course, in the context of the grading standards of the Department of Mathematics.  The instructor will consider the performance of each student, from multiple perspectives. 

There is no single formula by which the course grade is computed.  Any computations done by the instructor are simply tools contributing to one or another point of view on these considerations. 

As is emphasized elsewhere in this website, the "rough cutoff" grades have no bearing on the determination of letter grades for the course.  You do not "have" any sort of a letter grade at any point in the course, or "going into the final exam"; any possible such suggestion is intended only as an estimate.  It is not possible or reasonable to expect that it can be stated in advance of the final exam what sort of performance on the final would yield a particular letter grade for the course. 

The course grade assigned by the instructor will be made in consideration of all of the graded items, including very substantial consideration of the final exam.  Any statement, comparison, or any other sort of assessment made by the instructor before the availability of final exam scores is therefore based on incomplete information; such an assessment must therefore be interpreted only as a guess, and does not obligate the instructor in any way.

All students in this course must understand and agree to all of the terms above concerning the determination of course grades.



Critically, note that it is not feasible for the instructor to predict accurately your future course performance or course grade based on one or even two midterm scores.  So very much can happen to your performance in the course at the end of the term, and the last midterm exam and final exam are extremely important in the determination of your grade for the course.  The instructor has no control over your performance on these exams -- only you do -- so it is entirely your responsibility to make sure that you are able to perform to your desired level on these exams.

Midterm grades, if submitted, are not to be interpreted as a promise of any sort, and do not in any way contribute to the determination of course grades at the end of the term.  They are nothing more than fully disclaimed estimates designed to help some students recognize as early as possible that they might need to change their approach to the course.

Here though are some notes about how your perfomance on the various exams might be interpreted.  (Students in my summer courses, which usually have only two midterm exams instead of three, should interpret their second midterm exam by the comments in both items 2 and 3 below.)
  1. The first exam will be considered significantly in the determination of your final grade, but it is certainly also true that at this early stage no score is so low that it dooms you to a bad grade in the course.  The last two midterm exams and the final exam together carry much more weight, and strong scores on those exams would still result in a reasonable course grade. 

    More important than the direct contribution to the course grade is the fact that a low score on the first exam is strongly suggestive that there is a significant problem with the way the student is approaching the course; and if not corrected, this would be expected to lead to low scores on the remaining exams.  It is critical that these sorts of problems be identified and corrected as soon as possible.  See "Improving Your Performance" below for discussion on this.

    Also of serious concern from a low score on the first exam is that a weak understanding of the material at the start of the course will make it particularly difficult to understand the rest of the material for the course.  This material should be reviewed thoroughly and understood at a deeper level so that the effects of such a deficiency can be minimized.

    If you take the necessary time to review the past material and achieve an appropriate level of proficiency, and make necessary changes to how you approach the course, it is certainly still possible to earn higher scores on the remaining exams and achieve a desirable grade for the course.  Note however that this will require lots of additional time and work.

    In the interests of good planning this would be a good time to check the official university calendar to be certain of the withdrawal date for the course.  You should also have a plan in place for how you will evaluate your continuing performance in the course and, if necessary, how you will decide on the possibility of withdrawing before that deadline.

  2. The same interpretations can be made from another low score on the second exam; but by this point the accumulated affects of two bad exam scores are, unavoidably, more significant in their direct influence on course grade at the end of the term.  Due to the later stage of the term, there would also be more past material that would require review and it could very well be harder to make significant changes in course approach.  So, with two low exam scores, the situation is urgent.  Recovery and the achievement of higher performance worthy of a desirable grade in the course will require dramatic changes, almost certainly including much more time being dedicated to the course.

    Of course you must make your own assessment of your own situation, and there are many factors that might have contributed and which might be interpreted in different ways in terms of how the course will turn out for you.  Be sure to be very honest with yourself in making these assessments.

  3. After the third exam, it is too late to speculate about making changes to your course approach, and it is also too late to withdraw (with a possible exception in the case of a serious medical condition, which you should discuss immediately with your academic dean).  Depending on how far you believe you are from your desired course grade, and the possible needs of your other courses, you should simply dedicate as much time as possible to your most effective methods of studying.

  4. Of course, after the final exam, the course is over and there is nothing more you can do to influence your course grade.  See Additional Comment 10 below for more discussion on this.


Graded Items

Typically there will be three midterms and one final exam.  There will also be homeworks assignments and attendance quizzes.  There might be reading quizzes or pop quizzes, at the discretion of the instructor. 

The scores on each of these items will be reported in "points", but these points should not be assumed all to be of equal significance in the determination of final grades for the course. 


Exam Scores

The student should be prepared for the fact that, on an exam in this course, the grading system is NOT the same as the one most students became accustomed to in high school. There are two main properties in particular of the high school system that will not be used in this class :

1) In most high school grading systems, there are fixed, arbitrary numbers that determine the cutoffs between different letter grades -- these cutoffs were invariant, and independent of the exam. The problem with this that it forces the instructor to create exams that are always the same difficulty; in other words, the instructor must make sure that all exams will yield the same mean score. Furthermore, it requires that the distribution of scores also be roughly constant. Achieving both of these goals is not only difficult, but impossible to perform perfectly.

This system ties the instructor's hands severely, and is totally unnecessary! Of course it is important that final letter grades for a class follow a prescribed plan, so that those letter grades have some meaning outside of the context of that class. Ensuring that is actually easier if the instructor chooses the cutoff numbers after having seen the distribution of student scores. The cutoffs can then be chosen while incorporating important considerations such as the difficulty of the exam, or any other points about the exam that may be relevant.

2) The class average on exams in most high schools was usually expected to be somewhere in the mid-eighties. While this is reasonable considering the nature of high school, it is not always appropriate for a college setting.

In this class, certainly, there are expectations for the student that are much more demanding than those of most high schools. We expect that the student will achieve much more than the mere ability to reproduce what he or she has seen in class. In particular, we expect that the student will achieve an understanding of the ideas that are at the foundation of the methods -- and thereby gain the ability to apply those ideas to situations that he or she has not already been exposed to.

Since the expectations of this class are more difficult than those of high school, it stands to reason that the exams, designed to test the students mastery of these more lofty goals, must involve more difficult questions; and therefore, the exams must be more difficult. Clearly this will result in class averages that are lower than what one would expect if the exams were more like those of high school. It will also tend to result in score distributions that are more broad, since the students responses can be expected to be more varied.

For example, it is typical for a midterm exam in this course to have a median score of about 60, and a broad distribution.  Sometimes, if the exam is particularly challenging, the median score might be below 50; and of course if the exam is more routine the median could approach 80.  If the median is 60, note that this means that about half of the class will have scored below 60 -- these scores should not all be interpreted as failing scores!  Certainly scores below the median suggest significant gaps in understanding of the material and much room for improvement, but these gaps can be filled and that improvement can be achieved. 

(Scores at the very bottom of the distribution (this changes with every exam) might very well be indicative of failing performance, requiring dramatic changes in approach to the course.  If your score is near the bottom of the distribution, please consider seriously that you might need to make such a significant change in your approach to this course.  See for some discussion on this, and feel free also to come talk to me about it.)

The student should be fully aware of these points before taking an exam in this class.

"Fair Game" Material

This class, as with most math classes, is largely "vertical", in the sense that the material covered later in the course usually depends on the material from earlier in the course.  Because of this, it is not possible to make an exam that is entirely non-cumulative.  Questions on topics from later in the course unavoidably will require proficiency in topics from earlier in the course.

Nevertheless, we can still note that most problems have a primary focus, which can usually be associated to a topic covered at a specific point in the course.  Each midterm exam will have an associated range of topics from the syllabus that are declared to be "fair game" as a primary focus for problems on that exam.  These fair game ranges will be specified by the instructor, and will not overlap; that is, if a topic is fair game as a primary focus for the first midterm, it will not be for the second midterm.  (Note, these ranges refer to material from all aspects of the course -- the book, the lectures, the homework problems, the recitation sections (for courses with recitation sections), and any other official part of the course.)

This makes the midterm exam as non-cumulative as possible.  Still, of course, students must be sure to retain proficiency in topics that were fair game for earlier midterms, because even though those skills will not be the primary focus of problems on later midterms, they still should be considered as necessary skills in solving such problems completely.

The term "fair game" is used as an acknowledgement that for most midterms, there is not enough time to allow for testing all of the skills and concepts relevant to that exam.  Midterm exams will cover a representative sample of the relevant material, but not all of it.  Calling a topic "fair game" means that the instructor might choose to create an exam problem testing a particular topic, but makes no explicit promise.

To summarize then:  On each midterm exam, the primary focus of each problem on that midterm will be from the fair game material for the exam, which includes all of the material in the course up to a certain point in the syllabus (to be specified by the instructor) that was not declared as fair game for a previous exam in that same term. 

Unlike the midterm exams, the final exam is cumulative.  All of the material from the course will be fair game.

Rough Cutoffs

It is very dangerous to associate letter grades with performances on individual exams, because it is very difficult to predict how the distributions for those exams will interact when any sort of total score distribution is formed. Therefore, the class will usually be informed only of the class median and mean for a given exam -- letter grades will not be assigned.

On the other hand, I believe that it is desirable to give students some idea of their position in the course.  To this end, I will sometimes assign "rough cutoffs" for current total scores.  Students might find them helpful for roughly interpreting their performance, and they also might find them useful in communicating with others, perhaps parents, about their position in the course.  Usually I will not make such rough cutoffs after the last midterm exam, since it is so close to the end of the semester.

Given the "roughness", these numbers will not account for incidentals.  So, if the rough cutoffs are given as 173 for an A, 141 for a B, 107 for a C, one might interpret a total score of 170 as a B+; and likewise, one should probably view a 145 as a B-, and a 111 as a C-. 

I will usually not give a rough cutoff for the grade of D.  Any score below the rough cutoff for a C should be interpreted as unacceptable, and precariously close to failing.


Students should also be aware of the statistical fact that positions in any sort of rankings do not behave as might be expected when distributions are added.  Certainly, rankings do not "average".  Even more, it is not hard to create a scenario showing that a student can score in the top half of the class (above the median) on all of the exams, and yet still be in the bottom half of the class for total scores.  This phenomenon is not even particularly rare. 

A closely related point is that very often a student might be above the rough cutoff for a B- after the second exam, score approximately the median on the third exam and the final exam, and yet still end up with a C+ for the course.  (And likewise near other cutoffs.)  Of course this goes the other way equally often, but students are usually far less inquisitive in those cases.

One point of view that can be taken on this is that a student is mostly in competition with only a small fraction of the students in the class, and much less with the entire class.  Specifically, a student is mostly in competition with the other students that are nearby in the distribution of current totals.  If on an exam you score above most of the students that are near you in the distribution, then you would pass them in the totals and would expect to move up in the rankings; and likewise if most of those students score above you, then they will pass you in the totals and you would expect to move down in the rankings.  On the other hand, scoring higher on an exam than some student that is way above you in the totals would probably not move you up or down in the rankings.

As a result, it is impossible to draw conclusions about movement in the rankings based on how your score compares to the mean or the median. 

While this difficulty in predictions might seem undesirable, the more important point is the simple reality that grades in this course are based on a complete picture of performance over the entire course; and an incomplete picture simply does not allow for accurate predictions.  The difficulty of making predictions should be interpreted as part of the unreliability of the rough cutoffs, and not as a flaw in the grading system.

Grading Comment Codes

Graders in this class might choose to use the shorthands listed on the Grading Comment Codes webpage.  If you see a code on your graded exam paper that is not self evident, check this webpage and see if it is one of those listed shorthands.


Homework exercises are assigned for every lecture, and students should ideally complete each assignment on the day of the lecture.  The homework exercises are part of the learning process, so working on them that same day will help you be more fully prepared for the next lecture, and thus help you get the most out of the lectures and the course.

The assigned exercises for each lesson are listed on the syllabus.  (Note, we might find ourselves behind or ahead of the posted schedule; if so, you should do the exercises as we actually finish the sections.)


Make sure you staple your homeworks!  We cannot give credit to students for work that was lost as a result of not being stapled.  Also, make sure to put at the top of the front page your name, the section number(s) for those exercises, and the course information (Math XYZ, Clark Bray)

In order to give flexibility to students, the assignments for about one week of lectures will be picked up about once per week, usually on Fridays, and will be graded and returned as soon as possible.  Note however that students are still strongly encouraged to do the homework exercises throughout the week as soon as the corresponding sections are completed.  Procrastinating them all to the day before they are due creates a heavy workload, and makes it difficult to seek help on specific exercises if needed.

You will turn in the homeworks to your recitation TA; talk to your TA for those details.

The exercises due on a given due date will be those from sections on the syllabus that we finished talking about in lecture at least two days prior to the due date.  For example, for a course in which the homeworks are due in Thursday recitations, then the exercises to be collected will be those from sections we finished talking about in lecture on or before that Monday.  For a course in which the homeworks are due on Fridays, the exercises to be collected will be those from sections we finished talking about in lecture on or before that Wednesday.

No late homework will be accepted without filling out the Short-Term Illness Notification form, or obtaining a Dean's Excuse.

In calculating homework grades, the lowest of your homework scores will be dropped.  The purpose of this policy is to handle exceptional circumstances.  Please do not request to have late homework accepted without filling out the Short-Term Illness Notification form.  Also, it is inadvisable to skip a homework unless absolutely necessary, since only one homework will be dropped.


In previous terms a collection of solutions written by a previous instructor has been used.  This previous instructor taught the course from a different point of view, and the resulting inconsistencies with how this course is being taught has proved to be very problematic.  These solutions are no longer being supplied to students by this course.

It is suggested that you NOT seek out this document on your own for use in this course.  The instructor and TA's for this course are not responsible for the content of that document.


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.

Reading Quizzes

Students are required to read the corresponding sections of the book before they come to class.  This gives students a rough idea of what will be covered in class, allowing for a more thorough treatment during class time.

Sometimes, reading quizzes will be set up on Sakai to test that students have done this reading on time.  These quizzes will be short -- typically only one or two questions, typically worth one point apiece -- and should be easy questions if the student has read the sections in question.  Students will be informed of the creation of these quizzes by email.  Make sure that you take the quiz AFTER doing the reading and BEFORE class -- because these are timed tests, and furthermore are available only before class time.

In-Class Quizzes


In Math 216, watching and studying the lecture recordings before coming to class meetings is required.  At the start of each class meeting, there will be a short in-class quiz on the material from the relevant lecture recordings.  The quiz will require the student to have viewed the recording and studied the content to achieve basic understanding.  These quizzes will contribute to your course grade in a way that will be determined by the instructor.

The quiz question(s) will be displayed on the projector screen in the classroom.  Students will submit their answers through Sakai, in a quiz listed (by date) in the "Tests & Quizzes" section of the course site.  It is the responsibility of the student to be sure to have access to Sakai during all class meeting periods; laptops are recommended, but are not required if the student is certain to be able to complete the quizzes with another device such as a phone or a tablet.  (Helpful hint:  If a Sakai page shows an error, this might simply be a sign that you need to clear your browser cache.  Be sure that you know how to clear your browser cache, reliably and quickly!)

Students who arrive late after the quiz has begun will not receive extra time; and students who are not present for the quiz, and unexcused, will not be able to make up the quiz.  Students may take the quiz ONLY in the section of Math 216 for which they are registered.



Any form of cheating on the in-class quizzes will be reported to the Office of Student Conduct, and will be considered as equivalent to cheating on an exam.  This includes (but is not limited to):
  • any communications, giving or receiving, direct or electronic, with anyone on any topic that is relevant to the quiz
  • consulting any materials that might reasonably be expected to contain information relevant to the quiz
  • looking at the laptop screen of another student
  • submitting responses to the quiz while you are not physically present in the classroom
While in-class quizzes are in process, the following rules will be in effect.  These rules (much like the rules during an exam) apply to all students, independent of whether they have submitted their response(s) for the quiz:
  • students may not talk, even in whispers
  • students may not leave the classroom
  • students may not consult any course materials (lecture notes, lecture recordings,...) or other references with similar content
  • students may not use their laptops, tablets, or phones for purposes other than submitting responses to the quiz
Importantly, students also are explicitly disallowed from communicating about the quiz with anyone until after the quiz has been given IN BOTH SECTIONS of Math 216.  If a student in the earlier section of Math 216 gives information about the quiz to a student in the later section of Math 216, both students will be reported to the Office of Student Conduct and the cases will be treated as equivalent to cheating on an exam.

Students should be aware that the potential sanctions for cases of academic dishonesty are extremely serious, almost always involving suspension (for at least one semester, and usually two semesters), even in cases involving only a minor impact on the student's grade for the course.

If you hear discussions that trivialize the gravity of respecting the integrity of the in-class quizzes, please understand that this naive perspective is based on the very limited experience of a small set of students.  Keep in mind that cases in which students have been caught and sanctioned for these sorts of violations are usually kept very quiet, according to the rights of those students; while this is necessary, an unfortunate side effect is a lack of appropriate perception of the very real risks of being caught, which can result in an inappropriately (and naively) casual attitude in some students.

If you have information about other students who are not following the rules regarding the in-class quizzes, please make a record of the evidence if you can and let me know, as per the "obligation to act" component of the Duke Community Standard.


Excused Absences

If you are unable to attend class due to an illness, you must submit a Short-Term Illness Notification form in order to be excused from the corresponding in-class quiz or quizzes.  Submit an STI form for each lecture that you miss due to illness -- or, for an illness that covers several days, you can submit a single STI form and indicate clearly in the comments section exactly which lecture dates you missed for that illness.  Absences due to university representation and religious observance will also be excused (athletes missing class due to athletic events should submit a NOVAP form). 

Other absences will be excused only in extreme circumstances, and at the discretion of the instructor.  If you anticipate a possible absence and are not sure it will be excused, you should communicate with the instructor in advance of the class in question.  Note, absences due to holiday travel are voluntary and will not be excused; students must make their own decisions about choosing to travel before the official beginning of the university holiday in question.

In the case of an excused absence, I will make a note of the exception in my own records.  The excusal for that in-class quiz will not show up on Sakai -- but it will be considered in the final tally on my gradesheet at the end of the term.


Here is the procedure we will use this semester for homework regrades:

    (1) Write a clear and complete description of why you feel your paper deserves more points than you originally received.
    (2) Attach that description to your homework paper.
    (3) Put that paper into the pile in the following week, when the next week's homework is being collected.
    (4) The grader will receive your note and original paper, will give it fair consideration, will consult with me (or the TA) if necessary, and then will make a change to the score if that is deemed appropriate.  He or she will then also make the change on the homework gradesheet.
    (5) The grader will put the paper back in the pile and it will be returned to you along with those other homeworks.

For exam regrades, the policy is similar except that you will give the exam paper to the TA instead of putting it into the homework pile.  Also, exam regrade requests will be accepted NO EARLIER than 24 hours after you receive your paper in class (or discussion section) to ensure that students have compared their papers to the posted solutions before requesting regrades; and NO LATER than one week after papers are returned in class (or discussion section).  If you are not in attendance when exam papers are returned, it is your responsibility to come collect it; note, the TA (or I) will probably not bring it to the next class or discussion simply on the chance that you might be there, so you will need to make an arrangement to come to pick it up from the appropriate office.  You will still be required to wait 24 hours before making a regrade request, but the deadline remains as one week after the papers were returned to the rest of the class.  No requests for regrades of midterms will be accepted after the final exam.

For the final exam, regrades are made only in exceptional circumstances.  Note that it is Trinity policy that grade changes for the course can be made only in the case of "an error in calculation or an error in transcription"; I interpret this as excluding subjective choices on grading of particular problems.  Students may view their final exams after the grading is completed and grades are in, but should be aware of this Trinity policy on grade changes.

Here are a few thoughts to keep in mind about regrades:

(a) It is entirely possible and reasonable that the grader might have misread your paper, and with your explanation realize that you do indeed deserve more points.  In such a case, he or she will be very happy to award more points.

(b) It is also very common for a student to feel simply that too many points were taken off for a given error.  In these cases, the student should be prepared for the likely conclusion that no additional points will be awarded.  The point here is that this is a subjective situation, and a choice has to be made.  The grader makes the decision based on his feeling about the importance of a given aspect of the problem, and the grader's opinion on this question is the standard.

Common examples of these types of disagreements involve the amount of explanation that should be given, and the relative importance of different parts of the problem.  These are highly subjective questions, and reasonable people will come to different conclusions.

Remember that this is a curved class.  So, when it comes to questions about too many or too few points being taken off, it is far more important that the grader's scheme be applied consistently across the board for all students than that it be something other people might or might not agree with.

(c) When you submit your paper for a regrade, the grader might possibly come to the conclusion that too many points were awarded in the first place.  In such a circumstance, your score could go down.  Of course the grader will always make such decisions dispassionately and fairly, but certainly you should only submit for a regrade in a situation where you feel you have a comfortably strong claim.

(d) The grader is a very reasonable and intelligent person, and absolutely deserving of being addressed politely and treated with respect.  Make sure to phrase your requests calmly and reasonably.  And of course, always be prepared for the possibility that the grader might have a different point of view than you on a given question, and that his or her fair and reasonable consideration of your request might yield no additional credit.


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 previous discussion of the expectations in this course.)

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 before the next lecture, that next lecture 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.

"Log Confidence" Exam Problems

In some of my classes, I will employ a type of exam problem that I call a "log confidence" problem.  If you are not certain whether these types of problems will be used in the class you are taking, please ask me.

Log confidence problems are similar to multiple choice problems, except that instead of choosing only one of the possible responses, you can indicate next to each one your confidence that this is the correct response.  These problems allow students to remove the randomness normally associated with multiple choice problems.  You can read more about these types of problems on my Log Confidence Problems webpage. 

Note, there is much to say about this type of exam problem.  If the class you are taking from me will employ this type of exam problem, you should make sure to read all of the contents of the Log Confidence Problems webpage to make sure that you are fully prepared and aware of how the problem works and how to respond to your own advantage, so that you will not have to spend valuable time during the exam doing this.  If after reading this webpage you still have questions about how best to respond to this type of question, make sure to ask me well before the exam during office hours.  (Email is probably not a good choice for this type of conversation.)

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 well 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, as per the earlier comment on this page.  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 activites.

  • 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 comprehension, 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 was discussed earlier on this page -- that is, for some given topic, you can consider 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 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.  Some of those old exams might 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:
    1. Was this problem too hard or too easy?
    2. Does the problem focus on important ideas from the course, or is it tangential and/or irrelevant?
    3. What important idea does this problem allow the tester to gauge in the student?
    4. What other ways might this problem be rephrased, while still being basically the same problem?
    5. 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 his or her 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 exam scores.  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.

Additional Comments

1.  For a course in which the exams are graded objectively, based merely on the correctness of the final answer, it is possible to make a grading system that can be advertised in advance, serving as evidence of the objectivity and complete impartiality of the system.

However, in a course such as this one, this is simply not the case.  Grading on any individual problem is intrinsically subjective, based on the view of the grader as to how well the student communicated in the written solution his or her clear understanding of the method, theory and technique relevant to solving that problem. 

Of course, fairness is still critical.  In order to ensure as much fairness as possible, the grading on any given problem will always be done by the same grader for each student in the class.  If the grader is generous, then this generosity will affect all students equally in expectation value, and then because of the curve it effectively does not have any systematic influence on grades at all.  Similarly, if a grader is harsh, but applies the same harsh grading system to all students on that problem, then again after the curve the effect is that there should be no systematic influence on the grades.

Because of this subjectivity it is likely that the student might have his or her own opinion as to whether the grading on a given problem is too harsh or too lenient.  Certainly students have their rights to their own opinions on this question.  But when it comes to regrades, in preserving the fairness discussed above, it is essential that regrades be based on that grader's consistent view of the grading.  Thus, requests for regrades based on an assertion simply that it is your opinion that too many points were taken off for the acknowledged error will generally not be granted.  (Similarly unlikely to yield extra points is an appeal based on your claim of what your high school math teacher used to do.)

Of course students are always welcome to submit their papers for regrades, but in those instances that boil down to a simple difference of opinion, further argument will not yield any benefit.  In such a circumstance, the student will be far better off trying to understand the grader's perspective, so that necessary adjustments can be made that will avoid such problems in future exams.

2.  Very often, a grader will establish a system for grading in order to aid in the consistency of evaluation over large numbers of students.  For example, the grader might decide that one particular part of the problem is worth some number of points, or certain steps (or errors) are worth some number of points.  Similarly, a system is in place regarding the accumulation of points in this course and the process by which final letter grades will be determined. 

These sorts of systems are useful tools for graders.

It should be emphasized however that these systems are decided on by the grader voluntarily, and for the purpose of assisting the grader.  It is not to be assumed from the existence of such a system that the grader abdicates his or her right to form any opinion about the quality of a student's work.

For example, note that on a given problem (on an exam, for instance), there might be multiple ways to work the problem; and even worse, there are countless ways that a student can make mistakes.  A given system might allow the grader conveniently to determine grades for most papers, but for another the system might not have been set up to account for the pecularities in that particular paper.  In such a case, the grader is entirely within his or her rights, and in fact obligated, to award points based on his or her true opinion of the work, and not based on the system.

The grades on homeworks, exams, and the letter grade for the course will be determined entirely by the corresponding grader's considered opinion as to the quality of the work done by the student.  Systems are useful tools to help the grader achieve that goal, but ultimately it is only the opinion of the appropriate grader that determines the grade awarded.

4.  During an exam, if you have a question you may come up and ask me.  However, I will only answer questions that concern a clarification of what the question is asking -- I will not give any information that will help you formulate a solution to the problem.

For example, suppose the question says, "Bob is pushing a 20-pound box up a ten foot ramp angled at 30 degrees.  How much work does it take to get the box to the top?"  Questions that I can answer include:

- Should we ignore friction?
- Is ten feet the length of the surface of the ramp, or is it the height of the ramp?
- Is it 30 degrees from horizontal or from vertical?

However, I cannot answer questions such as:

- What is the sine of 30 degrees?
- What is the formula for work?

5.  If you ask me a question during an exam -- MAKE SURE THAT YOU WHISPER YOUR QUESTIONS!  When you speak in a normal speaking voice in a virtually silent room, everyone in the room can hear you.  So, for one thing, speaking above a whisper is a distraction to all of the other students.

More importantly though, if your phrasing of your question itself contains any content, and if another student overhears you, then you may have communicated assistance to that student!  For example, if you ask, "Is this where we use the formula about force times distance?" in a voice that can be overheard by another student, then you have communicated to that student some assistance in the solution to that problem. 

This is entirely avoidable of course, and I expect that all students will take simple and obvious precautions in order to avoid this sort of thing.  Failure to do so might be viewed as a violation of the Duke Community Standard.

6.  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, scan, and post, and lots of other little 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. 

7.  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.

8.  You may NOT bring in scratch paper to use during the exam.  All of your responses must be written on the exam paper itself, ideally in the space provided.  You are welcome to use the backs of those pages also if needed, but if you do this please make sure to indicate very clearly in the intended space for the problem exactly where the remainder of your solution is located.  All together this should be more than enough space to solve each of the problems.

DO NOT tear the pages out of the staple or remove the staple.  All of these pages must remain attached to ensure that pages are not lost.

9.  This is a curved class, in that the determination of your letter grade at the end of the course is based on your performance relative to the rest of the class -- not based on arbitrary cutoffs determined beforehand.  Specifically then, your grade in this course depends in some part on the scores of the other students in the class.

Because of this, it is particularly important in this class that, during each exam, all students must have the same amount of time to work on the problems.  If one student should somehow have more time to work, the extra points that student gets in that extra time will negatively affect the relative performance of the other students; and clearly this is not acceptable.

Making sure that all students get the same amount of time on the exam is accomplished by two steps -- starting everyone at the same time, and ending everyone at the same time.  At the beginning of the exam I will pass out the exams and tell students not to turn over the cover page until everyone else has a copy of the exam and I say "begin".  At the end of the exam, when I say "stop", students should immediately put down their pens/pencils, and bring their papers up to me.

Note the following, from the Duke Community Standard In Practice: A Guide for Undergraduates, specifically in the section on Academic Dishonesty, among the list of forms of cheating:

•   working on any examination, test, quiz or assignment outside of the time constraints imposed;

Note also the following from the same document, specifically in the section on Failure to Comply:

A student or group may be held accountable for failure to comply with:

Some students seem to feel that they can get away with continuing to write for a minute or two after the official end of the exam -- perhaps because they feel they will be unnoticed in the hustle and bustle of other students getting up and turning in their papers, or perhaps because this sort of thing was condoned or even accepted in their high school math courses.

Students should be very clear that it is critically important to the fairness of the course that they do indeed stop when instructed to do so.  I will allow students a few seconds to finish a thought, but after that I strictly require that no more writing on the exam paper should take place.  If any writing on the exam paper should take place significantly after I have very clearly called the end of the exam, I will consider this to be a clear and deliberate violation of both of the above cited aspects of the Duke Community Standard, and I will notify the Office of Student Conduct.

10.  Performance at the end of the semester is reported in the form of a letter grade with incidentals.  Of course these letter grades are determined by the instructor based on numerical scores on the graded items from the course.  There are several facts about this conversion that students should be aware of.
  • The spectrum of total scores for the course is hundreds of points wide, filled with a very large number of students, and yet there are typically only 11 or 12 different letter grades (counting incidentals) awarded -- and fewer that are awarded commonly.  As a result, it is unavoidable that sometimes students with somewhat different total scores will receive the same letter grade for the course.  It is also unavoidable that sometimes students with very similar total scores will receive different letter grades.

    Students should be prepared for these possibilities, and aware that such occurences are not reasonable grounds for requesting reconsideration of letter grades. 

  • It is the responsibility of the student to make sure that scores on graded items for the course have been recorded correctly.  Please make sure that you have confirmed these records as they are made throughout the term.  Corrections such as this must be made before the final exam -- after grades for the course have been submitted, it is too late to make such corrections.

  • As discussed previously in the above section on rough cutoffs, students should be aware that those rough cutoffs have absolutely no role in the determination of letter grades at the end of the course.  Please make sure to read carefully the discussion of this in the section on rough cutoffs.

  • Since the third midterm is usually extremely close to the end of the semester, I will usually not report any rough cutoffs after that exam.  At that stage of the course, a student's only concern should be studying as much as possible and doing as well as possible on the final exam.

  • Duke University policy ( clearly indicates that, with only a very few exceptions, letter grades are final after they have been reported.  Please do not request reconsideration regarding letter grades based only on previous expectations, appeals based on effort exerted, appeals based on the personal consequences of your letter grade, or subjective interpretations of how letter grades for the course should have been determined.

  • Final exams are not returned to students in this course.  It is Department policy that all final exams remain secured.  You may come and look at your final exam in your instructor or TA's office, but you may not remove it from that office, even temporarily.  Because of the Duke University policy linked above, regrades for the final exam are considered only in the instance of an extreme error of grading; students should be aware that this is very rare, and should not come to look at the final exam motivated by the hope for a regrade.

11.  Sometimes there might be material that is fair game for an exam, but for which the homework problems are not due until after the exam due to the way the schedule and due dates are set up.  Make sure to consider this possibility when you are studying for the exam, and if this should be the case you are strongly encouraged to do the corresponding homework exercises before the exam in question, so that you can get the needed practice for the exam.


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