This is not an answer, but I think it's too important to leave as a comment. I strongly urge you to rethink your goals for this course. Your goals should be to prepare your students to teach ...

First of all, you should test them on remembering the definitions. Second, there are probably a significant number of your students who do not understand the definitions. Suppose you gave them an ...

The Association for Computing Machinery (which is the professional association for computer scientists) puts out a curriculum guideline for computer science. While the guideline is for what an entire ...

It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.

The standards in math are way higher because of supply and demand. Lawyers and doctors directly help people. We need lots of them. Mathematicians do some difficult to understand work that might ...

"For all" statements are hard to grasp because they don't really exist outside of mathematics (or philosophy or other abstract domains). In any vaguely empiricist epistemology, you can have "there ...

The main reasons both of these are called the same thing are historical. Group theory was more or less invented by Galois to study when one could solve polynomial equations by radicals, and ring ...

This is a text for an "introduction to proofs" course. It might not be well-known outside mathematical circles, because mathematics educators don't like to advertise this fact, but, outside ...

You might know (or not) enough computer science to know there are such things as functional programming languages. These are programming languages (the most popular are probably Scheme, ML, and ...

I think it matters what you mean by flub. There are a couple specific exceptions, but I think any of my colleagues in my department could teach any of our undergraduate courses in an emergency. ...

The set of solutions to a nonhomogeneous linear differential equation form an affine space. (The underlying vector space is the set of solutions to the associated homogeneous equation.) Every time I ...

I think this is one of those places where teaching a detailed strategy is a form of "teaching to the test" that is counterproductive for the students' intellectual development. It's ...

From a practical point of view, your friend is right. One just has to be careful to test enough numbers. Engineers don't believe calculus because it has proofs based on analysis which is based on ...

I think Lay's and Poole's books are easily the best of the common ones on the market. However, if you have stronger students than we do, Strang or Bretscher might be more appropriate. (Not having a ...

If you are teaching to an audience that will largely be computer science majors (which is often the case), then you should look at the ACM guidelines for an undergraduate computer science curriculum, ...

WolframAlpha is not your problem. It really isn't. The students who are cheating by and large aren't good enough to figure out WolframAlpha's syntax and interpret its answers. Your problem is that ...

Analysts would be horrified, but I'm a combinatorialist, so when I teach calculus, I take the viewpoint that a function is first and foremost a table of data. We have a bunch of $x$'s, and for those $...

I have heard of people teaching Calculus II with the theme of approximation. Integrals and Riemann sums (as well as other numerical integration techniques) are approximations of each other, and ...

I don't know whether this is a comment or an answer. There is a bigger picture to keep in mind, which is that, beyond developing the students' understanding of the epsilon-delta definition and their ...

The sad fact is that your mathematics preparation may very well be inadequate for graduate school. My department admits with some frequency people from small regional public universities or unknown ...

One of the important meta-lessons you will learn in differential equations is how to deal with problems you cannot solve completely. For almost all differential equations, no one knows how to write a ...