Math 102, Clark Bray

Values of Sine and Cosine at Standard Angles -- All students are strongly advised to know the values of the six trig functions at the standard angles (all multiples of pi/4 and pi/6); this short commentary and the linked diagram will help you both remember these values, and understand the fundamental relationship between trigonometry and the unit circle.

The Remainder Theorem and the Factor Theorem -- These are fundamental facts about polynomial algebra, and they are simply stated and easy to prove; but for some reason they are not covered in most high school curricula any more. I strongly encourage students to look at this brief description and ensure an understanding of these theorems and their proofs.

Some comments about Single Variable Integrals -- This is a document that I wrote in the Fall of 2004 after teaching the Math 103 section about integrals of parametric functions. When we get to the integrals portion of this course, the first part of this could be useful to you.

My other Course Websites -- Students can find here all of the websites for every course I have ever taught.

Java Applets -- This is a site that was created by a colleague at Stanford; several of the tools there could be useful to you for visualizing multivariable functions.

Exam Solutions

2008-2009 Spring:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions
Exam 3
Exam 3 Solutions -- (There are arithmetic errors in the solution to number 5.)

2008-2009 Fall:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions
Exam 3
Exam 3 Solutions

2007-2008 Spring:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions
Exam 3
Exam 3 Solutions -- (On problem 4 there is an error in the analysis of edges E1 and E2. The easier observation is simply that the function is identically zero there.)

2007-2008 Fall:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions (Note, there is a computational error in the bottom left element of the solution matrix in #2a.)
Exam 3
Exam 3 Solutions

2006-07 Spring:
Exam 1
Exam 2
Exam 3
Exam 1 Solutions
Exam 2 Solutions (Note, there are computational errors in the solution to problem 1a)
Exam 3 Solutions (Note, there is an arithmetic error on number 5; the answer should be 11, not 7.)

Homework

Homework problems are assigned for every lecture, and students should ideally complete each assignment on the day of the lecture. The assigned problems for each lesson will be listed on the syllabus. (Note, we might find ourselves behind or ahead of the posted schedule; if so, you should do the problems 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, and the section number(s) for those problems.

In order to give flexibility to students, the assignments for the previous week will be picked up in section every Tuesday, and will be graded and returned as soon as possible.

No late homework will be accepted without the Short Term Illness Notification.

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. Also, it is inadvisable to skip a homework unless absolutely necessary, since only one homework will be dropped.

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!

Here is the procedure we will use this semester for 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 section, when your TA is collecting the next week's homework.
(4) The grader will receive your note and original paper, will give it fair consideration, will consult with me or your TA if necessary, and then will make a change to the score if that is deemed appropriate. He 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.

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 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 fair and reasonable consideration of your request might yield no additional credit.

Grading and Exams

Final grades for the class will be determined by the total number of points earned in the class. These points are given based on performance on the items below, with the following maximum possible scores:

Homework average:   50 possible points (the lowest homework score is dropped)
Tests:                           300 possible points (3 exams x 100 points each)
Final Exam:               200 possible points
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Total:                            550 possible points

The student should be prepared for the fact that the grading system for these exams 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.

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

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 the 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.

I will periodically distribute a list of all student total scores in the class (with names removed), so that students can see exactly how their own total scores compare to the rest of the class.

Calculators

We will not be using calculators in any aspect of this course. You may use a calculator on a homework problem if you feel that it will help you understand the concepts, but you may not make any reference to the use of a calculator on the homework you turn in, except in the case of a problem that requires substantial and nontrivial arithmetic.

Calculators will NOT be allowed on the final exam or on the in-class exams. I also generally discourage their use on the homeworks, since students may develop a dependence that will hurt them on the exams.

You are encouraged though to take advantage of the online resources linked to from this page, including the Java Applets (there are many tools there that can help you visualize and develop a geometric intuition for some of the ideas in this course).

Getting help

There are several resources that students should be aware of; make sure to read the Sources of Help for First-Year Students.

I'd also like to emphasize that classmates can be an excellent resource as well. I refer you above to my comments on this in the homework section.

Be sure to realize that you are encouraged to use these resources for more than just help on the homework... Ask questions about general ideas you are having trouble with, specific parts of the lectures that you did not understand... Of course you should also seek help with homework if you find yourself stuck on a problem for an extended period of time.

Duke Community Standard

The Duke Community Standard is taken very seriously on Duke campus, and you are all reminded to make certain you are familiar with it.

In this course some collaboration is allowed and encouraged, but of course your work must all be your own. Here are some specific comments about the graded items in this class:

Homeworks -- You are encouraged to work in groups to exchange ideas and help each other understand how to approach problems, but the student's work must be his or her own. Copying and dictating are not allowed.

Exams and Quizzes -- Students are not allowed to have any outside help during exams or quizzes.

Attendance

Attendance at all lectures and sections is required . If you miss a lecture, it is your responsibility to catch up on the topics that you missed. You should keep in mind that in this course, the material builds on itself; if you miss some of the material, subsequent lectures will seem much more difficult to you.

Absences from exams will be excused only for reasons such as serious illness or appropriate official university activities. In either case, a written notification from the dean is required. In the case of illness, this must be done with the Short-term Illness Notification form. In the case that an absence from an exam is excused, the grade will be determined based on your performance on the final exam for the course, and relative to the performance of the rest of the class on that exam.

Students should note that use of the Short-term Illness Notification form is subject to the Duke Community Standard, discussed and linked above. In particular, it is expected every student will take reasonable responsibility for his/her own health at least to the extent that such health is needed to be able to participate in classes. Here are some examples:

-- If on the day of an exam in this course you have a debilitating headache caused by a virus, then it would be appropriate to use the short term illness form. However, if you have a debilitating headache caused by a hangover, it would not be appropriate to use the short term illness form.

-- If on the day of an exam in this course you have a severe cold caught two days earlier while camping outside for basketball tickets, then it would be appropriate to use the short term illness form. However, if you have had the cold for two weeks and are still camping outside for basketball tickets, it would not be appropriate to use the short term illness form.

Additional Comments

As was discussed in the section on Grading and Exams, the goals of this course are higher than those of high school math courses, and even higher than most single variable calculus courses. Specifically, students in this course will be expected not only to be able to perform computations, but also to be able to demonstrate comprehension of the ideas behind those computations.

We expect students to demonstrate this comprehension by showing the process used to perform the computation, with sufficient commentary to allow the reader to conclude the thought process the student was using in working that problem.

The correct numerical answer does not guarantee that the student will get full or even majority credit for the problem. Rather, students will be graded on the validity of the method of computation, the extent to which the computation and commentary demonstrate the student understands the underlying ideas, and the clarity of those explanations.

It is very likely that this system of evaluation is very different from the ones the students might have become accustomed to in high school.

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 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 attitude, 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.

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 problems; 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, and ideally no two that are nearly identical. Obviously this makes it very difficult to read and interpret while grading.

Here is an incomplete list of some common such problems:

- "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

- 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.

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.

There are many resources available to students in this class, and we expect students to avail themselves of all of them.

Yourself -- By this I mean that you are ultimately the one responsible for ensuring that you learn the material. You cannot wait for help to come to you, but rather you must seek it out independently and aggressively. You must make sure that you are learning the material to a sufficiently deep level, and you must be responsible for finding the ways to improve your understanding if you are not certain that your understanding is sufficient. Think creatively about the different ways that you can learn!
Book/notes -- I strongly recommend that you read the appropriate sections in the book before class! Of course with a first reading it is not expected that you will understand everything, but a rough idea of what the topic is and how the problems will be approached is extremely valuable when you come to lecture; it will allow you to spend more time thinking about the higher level ideas that we discuss there, and to minimize the distractions of thinking about where things are going, and tedious minutia that you could have picked up by reading.

I also strongly recommend reading the appropriate sections of the book after class! This will allow you to solidify the things you learned in class.

Note also that in class, we will NOT cover all of the material in the book for which you are responsible. There are far too many things in this course to be able to spend sufficient class time on all of them. So, in class I will discuss the ideas that I think are the most important, and also the ideas that I think are most difficult to get from the book. I will also do as many examples as time permits, but the priority must be on the concepts.
Instructor -- Of course this is a given, but the point here is that you must come to class in order to get any use from me as the instructor. It is a tragic error to skip classes, thinking that your time is better spent doing other things. You simply never know what will come up in class, and if you miss the presentation of an idea that you are not familiar with, it will be very difficult to make up for that loss.

Note, in this course and most other math courses at Duke, the level at which the material is presented is dramatically higher than that in even the strongest high schools. It is at that higher level that you will be tested... So, while you may have done very well in your calculus courses in high school (note, almost every student in this class has already had a course in calculus in high school!), that is no guarantee that you will do well in this course.
TA -- Your TA for this course is a highly qualified graduate student in mathematics, and is another very valuable resource for asking questions and such. You should make sure to go to the recitation section with an idea of questions that you would like to hear the answers to, and if no one else asks those questions (this is fairly likely actually) you should be prepared to ask them yourself. The questions you can ask can be either specific ("I don't follow the argument in the book on page 275") or general ("I don't understand why elementary matrices are important").

Of course, as with class, you have to attend section!
Classmates -- As we discussed already in the homework section, you are encouraged to work in groups on the homework, to share ideas and exchange assistance on problems you are having trouble with. In general I think that your classmates can be very useful resources for you. Again, don't limit your use of this resource merely to specific problems. Try to initiate broad conversations on topics you are not sure about, and share your different points of view. If you find that neither you nor any of your classmates understands a particular point, then seek out help either from the TA or from me.

As previously discussed, the expectations in this class are very high.

Memorization/Execution -- Students in this course were almost certainly made very familiar in high school with the idea that they should be able to remember needed formulas and techniques, and work a problem out to its conclusion. That is, that they should be able to execute skills that are discussed in class. Students in this class are generally very good at this sort of thing, because of that familiarity and the predominance of that in high school classes.

However, because most high schools emphasize ONLY this skill, many students in this class expect that this is the only expectation that will be made of them. This is most certainly NOT the case.
Understanding -- Very often a student can be sufficiently familiar with an algorithm that he or she can execute the algorithm, and manipulate symbols in a way that yields an answer; but still this student might not understand either why the algorithm works, or the meaning and interpretation behind each individual step in the algorithm.

Students in this course will be expected to move beyond this level. That is, they will be expected to understand what is really happening with the algorithm, when it can be applied and why, and all of the appropriate interpretations.

Problems will be constructed for the exams that will test these sorts of understanding. Students will also be expected to demonstrate such understanding by their written annotations and explanations of their work on their exam papers. Solutions that do not have sufficient or communicative explanations will not receive full credit, even if the final numerical answer is correct.
Creative Application -- If a student truly understands the tools that we discuss and how and why they work, then he or she should be able to apply those tools and ideas to new situations. That is, students should be able to use those tools TO SOLVE TYPES OF PROBLEMS THAT THEY HAVE NOT SEEN BEFORE, EITHER IN CLASS, THE BOOK, OR THE HOMEWORK.

This will seem difficult at first, as students adjust to the needed higher level of understanding that they must demand of themselves in this course. But with sufficient practice, students will become comfortable with this level of expectation.

To prepare for these sorts of problems, students can pick a particular idea or tool from the course, and then try to think themselves of a new sort of problem or circumstance in which the tool might apply. Then the student should work out those details.

Of course students can also look at all of the exams I have given in previous semesters in this course.

Students in this course will need to "raise the bar" on their personal expectations on understanding, in order to do well in this course.

Note, understanding is not a binary thing. That is, it is not the case that someone either understands something, or they don't. Rather, there is a continuous spectrum of comprehension, and study continuously increases the level at which someone understands an idea. (One might compare this scale to those for skills in a sport, say, basketball -- certainly one cannot say that a player either does or does not have a certain skill in the game; but rather, for each skill, there is a continuous spectrum of levels of ability.)

In high school, there is a "bar" that is implicitly set on this spectrum by the types and difficulties of the problems that students are expected there to be able to solve. This bar is set at a level that, while it might be appropriate in a high school setting, is much lower than is appropriate at Duke.

This presents a kind of psychological challenge to students in this course. That is, students must find a way to motivate themselves to understand things at a level higher than they are familiar with, and furthermore they must then also find ways to gauge their understanding.

This is a difficult task, but one that every student here must undertake.

I recommend the following exercise to help gauge your understanding on any particular topic. That is, ask yourself,

"Am I prepared to give a 15-minute presentation to a group of other students explaining what the big idea is here, how it works, why it works, and then to answer questions from the group?"

This hypothetical question has two things going for it. First, note that the question forces a student to consider his or her ability to communicate on a particular topic. My experience has been that in order to communicate effectively on a topic, the most fundamental requirement is a clear and comfortable understanding of the topic in question. Furthermore, the process of preparing to communicate on a given topic forces the mind to organize the ideas in a way that helps understanding.

Second, public speaking is something that most people are naturally apprehensive about. Often this apprehension is based in some sort of fear of the possibility that one might be exposed for not understanding something. By considering a public speaking scenario, the possible resulting feelings of apprehension might be a clue that you do not actually understand things as well as you might have thought.

Note, some students take this to an even higher level by actually giving such presentations, in groups that they form themselves. I think that this is a wonderful idea, and both presenting and listening can be useful learning tools for participating students in such groups.

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.

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.

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.

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 actual incompetence in 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 Algebra 1, often in middle school. 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.

At Duke, we assume in all math courses that students are thoroughly competent in with the entire standard math curriculum of high school, including algebra, geometry, and trigonometry. Students are also expected to be thoroughly competent 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 problem 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. And of course, academic dishonesty is certainly inconsistent with the Duke Community Standard.

If you find yourself unable to work a problem on an exam in this course, the most effective way to get partial credit on the problem 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 problem 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 arrive at the correct answer.

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. 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 very seriously, 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. We do 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 Judicial Affairs.

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.

- 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 this?

(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.

Much better would be to write the following:

96
---------------
8 3 2

3 * 2 * 8 * 2
= ------------------------
8 3 2

For many more examples of these sorts of things, see "The Most Common Errors In Undergraduate Mathematics".

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?

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.

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 as a clear and intentional violation of the Duke Community Standard, and I will notify the Office of Judicial Affairs.

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.

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.

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.