Learning Expectations

Student Learning Objectives
Block Calculus Courses
Department of Mathematics

In these courses, students should:

1. Learn the definitions of the mathematical concepts of calculus enumerated on the syllabus.
2. Understand analytic and geometric perspectives on these concepts.
3. Attain proficiency in the techniques of working with these concepts algebraically, graphically, and numerically.
4. Use these tools for applications in the natural and social sciences.
5. Develop skills in creating and working with mathematical models to solve problems.
6. Learn to formulate and communicate mathematical arguments effectively in writing.

• Students in this course will be expected to learn the material at different levels:
  • Execution/memorization -- As with any math class, of course you will be expected to know formulas and be capable of executing algorithms relevent to the course.
     
  • Understanding -- Very importantly, the above will not be sufficient.  On the exams we will be designing questions that go to gauging the extent to which students actually understand the formulas and algorithms...  Where do they come from?  What do these objects mean?  Why do the algorithms work?
     
  • Applications to unfamiliar situations -- On exams, you should expect also to see questions that require applying the ideas of the course to applications that were not specifically discussed in class or on the homework, but which are susceptible to the same tools and ideas that were presented.  Students that have a clear understanding of the ideas underneath the formulas and algorithms should be able to identify how those ideas can be applied to the question, adapted as needed, to solve the problem. 
     
• Why is it not sufficient just to memorize formulas and rehearse algorithms?  Keep in mind that computer algebra systems these days are extremely powerful; Wolfram Alpha, for example, allows a user to type in a math question, in casual language without any significant syntax requirements, and will return almost instantly with the answer.  (For example, type in something like, "What is the integral of x times the sine of x squared?" and look at the result it gives!)
  
  So if all you can do is execute algorithms, then you can't offer future employers or research programs anything that can't be done more quickly, probably more reliably, and certainly more inexpensively by this free website!  (This is sort of a more modern version of what happened with the rise of computers and calculators...  before such tools were available, someone could be useful even if all they were capable of was arithmetic/evaluations; but once calculators became commonplace such a person became much less useful.)
  
  On the other hand, computer algebra systems can NOT look at a real world problem, identify key features, recognize mathematical tools that might apply, and create a mathematical model for understanding the problem.  These require real understanding, and even the emerging AI tools cannot do this reliably and creatively.  That understanding is what you can offer in your future, and that is a large part of the reason we focus on understanding so much in our courses.
   
• Why can't we just identify exactly the skills you will need and the problems you will face in your future work, and show you the math you need to do those things?  Because, very importantly, we don't know what those skills are!  They will come up in the context of the world of the future, and of course no one knows what that world will look like. 
  
  We will show you some examples and applications, of course, largely because this helps with developing a real understanding of the ideas.  But it is the ideas, abstractions, and general strategies that will help you solve future problems once you face them.  This is one of the big reasons that this course will focus on ideas and generalities, and why even students primarily interested in applications should be interested in them. 
   
• With these previous points in mind, be prepared that the questions on the exams will not be just like those on the homeworks.  Students are often surprised by this, perhaps because in their high school math courses maybe the exam questions were just like the homeworks.  In this context though, exam questions that are just like homework questions don't allow us to distinguish students who understand the important ideas from those who simply memorized.  Said differently, in order to test true understanding, we really have to put unfamiliar questions on the exams.
   
• Similarly, keep in mind our interest in testing understanding when you write on the exams.  Be sure to show all of your work and explain your reasoning as you work toward the answer.  We will be looking at the work and explanations far more than the final answer when thinking about scoring your work! 
  
  Of course you don't have to write a paragraph explaining every routine step of algebra that you use.  But if you make a step in your solution on the basis of the geometry of the problem, draw a figure!  If you use a theorem, indicate so and be sure to check any necessary conditions on that theorem!  Your explanations can be terse, even very terse; but in order for us to give you points for your work, we need to be able to see clearly and with confidence, from what is written on your paper, what you were thinking in your solution.  Don't presume that we will assume you understood something relevant to your work.
   
• Keep in mind that comprehension is not a binary thing, but instead is a spectrum.  When you get to a point that you feel you understand a certain topic, it may be that you have simply raised your understanding up to a familiar "bar", defined by previous experience (in the past when you have gotten understanding up to that level your exams have gone well).
  
  Very importantly, the "bar" is set higher in higher-level math courses, and all math courses at Duke set their "bar" higher than almost all high school courses (even at prestigious high schools).  So in order to be ready for this higher level of expectation, you have to recalibrate your internal expectations and standards accordingly.  In other words, you must raise your internal bar.
  
  How do you know how high to raise your internal bar?  This is a hard question to discuss, as there are no conventional metrics we can refer to.  My old midterm exams and solutions (available at links above) give you a great number of such reference points though.  Note in those exams the kinds of questions there are, the great variety of questions there that are not like any other; and also pay special attention in the solutions to the way the work is explained with words to accompany equations as needed to make the necessary ideas clear.
  
  You might also consider how confident you might be in giving a talk on a given subject.  For example, suppose you had to give a 15 minute talk to a group of classmates on a given topic (integration by parts, perhaps), addressing why it works, showing examples, and being prepared to answer questions that your classmates might have...  how confident might you be in your ability to do this?  If you feel super confident with the prospect of giving such a talk, then maybe you are okay with that topic.  But if considering this makes you nervous, or if you find yourself worrying about certain topics you hope that no one would ask you, then these might be signs that your understanding is not at the level that it should be. 
   
• It is common knowledge at Duke that this is a very challenging course.  Make sure that you have enough time in your schedule to dedicate to this course, and expect to spend significantly more time in this course than most others.  As a reference point, I think on average semester students should expect to need to spend about 15 hours per week on this course outside of class, on homework and general studying; so because of the faster pace in summer terms, summer students should view this course as a full-time commitment!  See the above recommendations on schedule for more specifics.  And note that this is an average; some might do okay with less, but some will need more!)
  
  But also be prepared for the fact that this course accelerates though the term.  So the amount of time you will need to digest the ideas at the start of the term is likely less, and at the end of the term you should expect to need more.  If you make time management decisions near the start of the term on the basis of your experience with the first couple of weeks of the course, you will likely need to re-evaluate those decisions later.