There's a related thought I have with this whole idea, and it's that these types of tools mainly end up used for "circus math". Math that once may have been useful, just as slide rules were useful, but in modern times have much less direct value and seem mainly to be for show. When you're revisiting calculus in order to accomplish some goal, are you doing everything by hand or are you leveraging software like Maple, Octave, Wolfram Alpha, Python, Julia, or a plain TI-89? Do you derive or memorize or keep a handy table of integration/derivative formulas? Do you ever do integration by parts and show all your work? Where there are nth order DEs, Laplace Transforms can be very useful (and lead to the tremendously useful Fourier Transform), but do you do partial fractions by hand so you can get an expression to something you can easily invert by inspection (with a handy table reference maybe)?
I wonder if there's not some way to skip a lot of the tedium of algebraic manipulation that is forced upon students, such that students can learn how to use algebra as a tool to solve problems, rather than as an interesting written dance where each step is shown that they must perform for points. These sorts of games may make the tedium go by quicker, and there is something to be said that understanding can come through rote, but once a student grasps the meaning of these things, I think we should immediately encourage that student to avoid as much tedium as possible and move on to higher subjects instead of more and more worksheets testing knowledge of process rather than knowledge of usefulness.
I occasionally link back to this text (ignoring the controversial remarks on violent video games): http://www.theodoregray.com/BrainRot/ In short, if you think of the brain as a limited resource, then all these numerical and analytical methods that were needed before computers have a cost -- one which our intelligent ancestors paid for out of necessity, and it's foolish to suppose these things don't require significant amounts of brainpower or cognitive resources. Is this cost still worth it for most of them, is the amount of brainpower in fact trivial despite our ancestors' struggles, were they just stupider back then? Do our children have enough resources that they can learn all they knew, at least until the final exam, and then all we've found out about higher levels of math and about automated computation this last generation? I don't think so.
This is a point of view that I'm hearing a lot now, mostly from technically capable people who know that computer algebra systems exist and are more reliable than doing everything by hand. And there is merit in the argument, but I've always felt uncomfortable about it, as if something was missing.
More recently I think I've identified what it is, and I included a little rant about it in my blog post about the birthday problem[0].
In particular, you've said:
> I wonder if there's not some way
> to skip a lot of the tedium of
> algebraic manipulation that is
> forced upon students,
I'd like to compare this with the idea of missing all the tedium of practising the cross-court forehand drive in table tennis. And the answer in that case is no, not if you want to be a top flight player. You need your body to recognise the shot automatically and play it without thinking, so your brain is released to do the higher-order stuff necessary to work on the problem, not the detail.
But more than that, sometimes it's the hours of practice in algebra (or similar) that means that when something turns up in disguise then you still recognise it, and still know how to torture the equations to twist them into the standard form.
It's really hard to explain. Sometime I'll have another go at it, try to put into words the meta-intuition I've developed over the past 40 years. In the meantime, the side-box with the rant is the best I've managed.
I wonder if there's not some way to skip a lot of the tedium of algebraic manipulation that is forced upon students, such that students can learn how to use algebra as a tool to solve problems, rather than as an interesting written dance where each step is shown that they must perform for points. These sorts of games may make the tedium go by quicker, and there is something to be said that understanding can come through rote, but once a student grasps the meaning of these things, I think we should immediately encourage that student to avoid as much tedium as possible and move on to higher subjects instead of more and more worksheets testing knowledge of process rather than knowledge of usefulness.
I occasionally link back to this text (ignoring the controversial remarks on violent video games): http://www.theodoregray.com/BrainRot/ In short, if you think of the brain as a limited resource, then all these numerical and analytical methods that were needed before computers have a cost -- one which our intelligent ancestors paid for out of necessity, and it's foolish to suppose these things don't require significant amounts of brainpower or cognitive resources. Is this cost still worth it for most of them, is the amount of brainpower in fact trivial despite our ancestors' struggles, were they just stupider back then? Do our children have enough resources that they can learn all they knew, at least until the final exam, and then all we've found out about higher levels of math and about automated computation this last generation? I don't think so.