Versus Maple, there's really no contest in my opinion, but that's mostly related to the rather poor quality of Maple. I remember even five years ago, when Sage

Versus Maple, there's really no contest in my opinion, but that's mostly related to the rather poor quality of Maple. I remember even five years ago, when Sage was considerably less mature, Sage was able to outperform Maple on first-year calculus problems (i.e., there were symbolic integrations that Maple choked on but Sage did perfectly). Sage is not only better than Maple at symbolics, but also at numerics, and the gap only widens as the mathematics level increases. As an amusing anecdote, I had to use Maple for a course once, and compared the speed of a simple algorithm to find the order of an element of Z/nZ (by taking x, and adding it to itself in a loop until you get to 0). The Maple code ran approximately 100x slower than the Sage code, even though the Maple code used machine integers but Sage uses bigints by default. With Mathematica, the general impression that I get is that for making interactive graphics, Mathematica is slightly better, but otherwise they're mostly approximately even for the first-year case, with a slight edge to Mathematica. Sage's friendly user interface is a web-based notebook, whereas Mathematica has a native application, but it's pretty good. Sage has very good documentation for common use-cases, but I haven't used the Sage notebook in a while, so I can't really say how discoverable it is. There are some resources on using Sage in an undergrad curriculum here, if you're interested: http://www.sagemath.org/doc/prep/index.html http://www.sagemath.org/doc/prep/index.html though the examples presume you're actually running a Sage notebook, so they don't show what happens, for instance, when you add an @interact decorator to make an interactive thing, like so: http://wiki.sagemath.org/interact/ http://wiki.sagemath.org/interact/ which has a nice example of an interactive Taylor-series approximation.