Aleks sends along this (by Robert Ghrist). It reminds me a lot of my class notes from high school.

I took a look at the book, starting on page 1 with the description of functions, and it made me think that maybe a more historical approach would be useful. The idea presented in this book, of a function as an arbitrary mapping from inputs to outputs, is a relatively modern idea (I associate it with Fourier in the 1800s, but my remembered history might be wrong here), and it might be helpful to build up to it from simpler ideas.

Similarly, instead of starting with definitions of limits etc., I suspect it would be better to start with the theorems and then motivate the definitions as the things you need to prove the theorems rigorously. That’s the historical order, right? Theorem, then proof, then defining the conditions of the theorem.

I sent the above paragraphs to Aleks, who commented:

Agreed strongly. But you’re assuming that theorems are intrinsically interesting, or that proving theorems is something one might want to do. History shows that all these tools have been developed to solve real problems in engineering – the same way Fisher developed his tools to solve real problems in agricultural experimentation and the same way probability was developed to make sense of gambling.

Yup. But then that just adds a couple more steps to the beginning of the process: first the real problem, then the series of possible solutions, then the theorem, etc. Definitions still come at the end.

That said, this attitude of mine is not new, yet it remains standard to teach math in this definition-theorem-proof style. So there must be some good arguments on the other side. To me, the Ghrist book looks like an excellent effort but from a backwards perspective. But I’m sure Ghrist could provide some good reasons why I’m wrong.