The idea of "geometrization" of mathematical education as opposed to "algebraization" has been discussed widely in the recent decade at least, maybe more. In Russian school of mathematics, Vladimir Arnold (http://en.wikipedia.org/wiki/Vladimir_Arnold), was a very strong proponent of the idea. You can read his talk of the subject here http://pauli.uni-muenster.de/~munsteg/arnold.html .
Octave is Matlab clone, in fact Octave developers openly say that except for some special cases, any difference between Octave and Matlab is a bug.
The biggest difference between Matlab and Octave is JIT compiler in Matlab, which does incredibly good job at vectorizing simple (or sometimes even not-so simple) loops.
I think it's fair to say that Octave performance is very close to a Matlab in a pre-JIT time.
There's also a huge difference in toolboxes, profiling, sparse matrix operations, parallel computing and many-many more. In these areas I'm afraid Octave is light-years behind Matlab.
However, you still can do a lot of useful simple stuff with Octave and it's free! Matlab-like syntax is really, really cool then it comes to vectorized operations. So probably these two reasons determined Andrew Ng's choice of Octave as a main environment for ml-class. Huge win for Octave I guess. This might spur some interest in the development, attract new people to the product. I think it's a well-deserved success for John W Eaton and other people who develop(ed) Octave all these years.
Unfortunately, it's almost impossible to work with a very large datasets in R, because of the speed limitations. Many researchers I know use Matlab because of this.
There's a long tradition in ex-USSR math schools and math department of "teaching by doing", or by solving to be more precise. And I think it gives superior results given talented and motivated students.
have you ever seen this book?
