Math and programming are a bit different though - to my knowledge in math you're not dealing with malicious users trying to give you incorrect input in hopes of exploiting bugs in your algorithm.
Edit: but to be honest if you needed to give the result Z for a set of inputs X and Y without worrying about invalid/malicious input, then a Stack Exchange copy/paste is totally fine by me.
> Math and programming are a bit different though - to my knowledge in math you're not dealing with malicious users trying to give you incorrect input in hopes of exploiting bugs in your algorithm.
But in fact you are! Actually it's even worse; in programming, you are just up against the ingenuity of actual human (or at least humanly programmed) users, whereas, in mathematics, you are up against the whole of 'reality' (in a Platonic sense). A whole tower of consequences will be built upon one mathematician's work, and, even if no human can spot the flaw in that work, if it is there then 'reality' will find it, and somewhere in the chain of consequences there will be an error that will bring the whole thing crashing down.
The thing is, if no humans can spot the flaw, then what difference does it make if they cheated in school? Either way they'll do flawed calculations if we assume your logic.
However, I wasn't really talking about people wanting to become actual mathematicians - those probably wouldn't use Wolfram just because they actually love crunching those numbers manually. The people who we're talking about here just see math as a roadblock they need to get through to do whatever they really want to do (programming, etc), and in this case this "cheating" is totally fine by me.
> The thing is, if no humans can spot the flaw, then what difference does it make if they cheated in school? Either way they'll do flawed calculations if we assume your logic.
I don't think that my logic allows us to conclude that. The idea of mathematics is that it is possible for humans to create and apply a system whose correct application makes the genesis of errors, howsoever subtle or undetectable, impossible.
Given this, and the likelihood that a mathematical error (of the conceptual type "any convergent sequences of continuous functions converges to a continuous function", rather than of the computational type "1 + 1 = 1") will not be found, it is especially important that practitioners of mathematics know how to apply their tools correctly, which they probably will not have learned by cheating in school; and, if they are able to apply those tools correctly, then they will not create errors.
(I grant that the weasel word 'correct' and its derivatives risks making this argument circular. I grant that human mathematicians collectively make an awful lot of errors, although I hope that we make fewer professional errors than many other professionals.)
> However, I wasn't really talking about people wanting to become actual mathematicians - those probably wouldn't use Wolfram just because they actually love crunching those numbers manually.
This comment seems to suggest to me the source of our disagreement in the first paragraph. As a mathematician, I don't crunch numbers professionally, and, when I have to do so outside of my profession, don't love crunching them manually. I suspect most mathematicians are in the same boat.
I definitely agree that someone using tools should know how to use them right - however in this case maybe the curriculum should be tweaked to point out mistakes when using a tool? Ie, instead of assuming that someone would do it by hand, assume they'd do everything they can to cheat their way out of doing the work and trick them as much as possible so the tools would only work if you use them right. Instead of focusing on teaching them how to do it by hand (which they would never do in the real-world given the time constraints), teach them which tools to use and how to use them properly.
My point about crunching numbers manually or not was more about the fact that a lot of people taking those math tests do so because it's required by X policy and not because they are genuinely interested in math, and IMO that's fine - not everyone aims for a job that involves mission-critical math. Some for example might just want to develop games, where a screw-up could at worst result in a graphical glitch.
> I definitely agree that someone using tools should know how to use them right - however in this case maybe the curriculum should be tweaked to point out mistakes when using a tool?
I totally agree! I structure my classes to point out both common classes of mistakes that everyone makes, and uncommon classes of mistakes that are subtle and difficult to catch. I even have a special way of presenting it (I switch to a colour I only use for discussing mistakes).
Students hate it. One of the two comments that I consistently get on my evaluations is "stop telling us how not to do it." (The other is that my tests are too hard, precisely because they don't involve just rote computations.) I've been told by classroom observers that many students literally ignore it, ceasing to take notes while I discuss mistakes and resuming only afterwards.
I keep doing it anyway, and I make a point of why I'm doing it, but it can't all be me; some of the onus has to be on the students to be willing to think about understanding failure modes as being as important as success.
Edit: but to be honest if you needed to give the result Z for a set of inputs X and Y without worrying about invalid/malicious input, then a Stack Exchange copy/paste is totally fine by me.