Another geometric optimization problem asked students to construct a solid shaped like a cone on top of a cylinder to optimize some quantity, and the optimum was the degenerate case where the cone had height zero! There are a lot of trick questions in there! I think this is a great idea, and every university should set this up. Because, if I were a student in this class, I would have highly resented being forced to work with students not at my level. I felt guilty and awkward about it. It worked better than I expected, but I still have some concerns.
The course coordinators were really hardworking and insightful people. Once they started working together, their performance on quizzes and in class improved dramatically. But I think a lot of it comes from continuously thinking about a wide range of mathematics over the last twelve years. For example, when counting boxes under a curve, they pointed at the squares in a random order, rather than sweeping from left to right, top to bottom. How many of those competitors come back the next year? I spent a lot of time in high school and college helping my fellow students, and I learned a lot by doing so.
In Spring I often was unable to sleep for fear that I had misplanned my Hodge Theory lectures, so it was a major relief to be so well taken care of.
Some thoughts on teaching Michigan calculus
In my case, I was competing with grad students who had honework calculus far more recently than I had; had taught it several times before; and who were often extraordinary competitors with a string of Olympiad medals and Putnam victories.
So this advice really did turn out to work, despite my skepticism. None of this comes directly from anything that I learned in grad school tewm in research. I feel that, psychologically, there is a big difference between having one or two questions that my hypothetical student can solidly get right, and having him pick up a smattering of partial credit on each of nine or ten questions that marh too hard for him.
Partly, I think this is important for maintaining student motivation. The applied and conceptual u,ich of the class Calculus at Michigan focuses very heavily on working with data and on understanding what computation to do, rather than how to do it. There are a lot of problems in the webhomework mainly sections 5.
I want to take the opportunity to brag about the most fun example of this: If the promise of academia were withdrawn, would people with a passion and skills for mathematics still become instructors? There are plenty of people who love math so much that they will continually learn and study new mathematics.
The webhomework system assumes that 1 any graph which looks piecewise linear actually is piecewise linear and 2 the students will use that fact to make any jmich with that graph precise. As for the comments on departments not being idealized firms, I will just point out that most public institutions other than flagship universities, as well as lower tier liberal arts colleges, are under such budget pressures that they have no choice but to have most of their teaching done by graduate students and adjuncts.
Michigan has a large room called the mathlab it could probably himework people or so filled with round tables and staffed 8 hours a day by tutors who can provide help with mathematics courses.
I do think there are advantages to having instructors with more mathematical experience then a grad student can obtain, and I talked about some of them above. Some subset of those people will also have the personality and drive to be teachers.
Especially in the first half of the term, I tried to spend almost the entire class period working with small groups.
Inexperienced teachers often do best with advanced material, or at least advanced students— contexts where something has filtered out the biggest teaching challenges and the students resemble less-trained versions of the instructor. It also focuses on getting students to be able to explain what they are doing to people with even less mathematical background than they have.
The mathlab is like a live-action version of math. Another geometric optimization problem asked students to construct a solid shaped like a cone on top of a cylinder to optimize some quantity, and the optimum was the degenerate case where the cone hhomework height zero!
If the department saved money by getting rid of me and using lecturers taem, given the constraints of tenure, by not replacing the next mathematician who retiresthat money would not go to the department budget; it would go to some other department or to a decrease in our rate of tuition increase. Just emphasize how crucial it was to pass the Gateway a computer administered exam testing this skilland the students would teach themselves.
I had a lot of fun going to the mathlab. Grad students graduate; postdocs leave.
Math Section Page
Also, I think that it is worth being able to detect the difference between the student I describe here, and one who has not even mastered those skills. For example, one question asked students to numerically approximate at. But I hope that many politicians would see the point. I had two students who, when I asked them for teammate preferences, specifically asked not to be put with hmoework in the top half of the class.
And I think that, to teach college math well, you need to be aware of a great deal of mathematics, well beyond the official level of your course. I very much doubt you can visually perceive the difference between a straight line and two lines making an angle of with each other and this can make a substantial difference in an integral.
Some thoughts on teaching Michigan calculus | Secret Blogging Seminar
The exams You can see the exams here: I felt guilty and awkward about it. I have just one item to add to your lists: I think this is a great idea, and every university should set this up.
The course coordinators really stressed how much the course focused on mqth. Organization I was assigned two sections of 32 students each, which dwindled to 30 each over the course of the term.