Step-by-Step Math(blog.wolframalpha.com)
blog.wolframalpha.com
Step-by-Step Math
http://blog.wolframalpha.com/2009/12/01/step-by-step-math/
33 comments
Priceless:
Suffice it to say, my students love this. Graded homework is now obsolete.
Of course, the department says graded homework must still be turned in...
Of course, the department says graded homework must still be turned in...
Give homework where the most interesting part is not something that Wolfram Alpha can solve.
I remember university teacher of mine had a collection of a few problems which if you typed it into Mathematica would give an answer on a very different form than what you would be likely to get if you solved by hand the 'normal' way. Handy for giving you a heads up on who might be 'cheating'.
Same is true here. Look at the solution to the quadratic in the beginning. I suspect most students would use the quadratic formula and not complete the square by hand.
Assume learning by doing, and Alpha's ability to solve any introductory problem, and you are screwed.
There have always been ways to cheat at basic school work, if nothing else by grabbing someone who did know how to do it and copying theirs. This does add one more way, but it can also be constructive if struggling to learn something.
You'll be able to tell who those kids are because they'll die on the exam.
I've been offering these kind of calculators for a while to students... but the department doesn't require a grad their homework. Is this college? What college department requires that sort of thing?
Cheating is now easier. That's good, for two reasons:
1. The steps are small and have clear documentation. Copying a wall of numbers does nothing. Copying delimited steps while being told exactly what they are helps. Actually, it's a lot like typical classroom instruction!
2. Cheating from someone "smarter" allows rationalization. The class is hard, you suck at math, they're smart. But here a dumb process spits out answers. The typical response to this sort of narrow AI is "well, it's not that difficult after all". This suggests that you don't need an abundance of ingenuity to do rote math of this sort. And that's motivating, if you're feeling daft.
The mediocre borrow and the great steal; if a novice learns from brute copying then we shouldn't give a damn.
1. The steps are small and have clear documentation. Copying a wall of numbers does nothing. Copying delimited steps while being told exactly what they are helps. Actually, it's a lot like typical classroom instruction!
2. Cheating from someone "smarter" allows rationalization. The class is hard, you suck at math, they're smart. But here a dumb process spits out answers. The typical response to this sort of narrow AI is "well, it's not that difficult after all". This suggests that you don't need an abundance of ingenuity to do rote math of this sort. And that's motivating, if you're feeling daft.
The mediocre borrow and the great steal; if a novice learns from brute copying then we shouldn't give a damn.
Aww, after I saw the integration examples I got my hopes up for a Show Steps for this...
http://www.wolframalpha.com/input/?i=integrate%5Bexp%28-t%28...
http://www.wolframalpha.com/input/?i=integrate%5Bexp%28-t%28...
NoScript filters that link as a potential XSS attack. I guess it doesn't like brackets in URLs. That could be a problem for Wolfram if the noscript author can't find a safe way to whitelist wolfram alpha links.
on a limb: I would say the overlap of no-script to wolframalpha users is minimal / WolframAlpha has other bigger problems to fix first.
How is that noscript strips URLs with brackets is WolframAlpha's problem?
Mathematica (and thus Wolfram Alpha) is unusual in it's use of brackets to enclose function arguments. Every other computer algebra system I've used has been content with parentheses. And NoScript is pretty much the leading edge for browser security, so if it thinks brackets are dangerous, they probably are and other browsers could follow at any time. Wolfram Alpha's direct use of Mathematica's notation in URLs seems a bit naive, and it wouldn't be the first time Wolfram embarrassingly misunderstood the way the web works.
how would brackets be dangerous? just because the author of noscript thinks they are, doesn't make them so. he's definitely not infallible.
my point exactly / sorry for confusion.
Hmm, can I get Wolfram Alpha to compute a closed form solution of a recurrent sequence - like the Fibonacci numbers?
It would be also be nice if the list of steps wasn't displayed all at once, but more like a lazy evaluator (Show-Me-Next-Step?). That way you can let Mathematica lead you for a bit, but then still be able to proceed on your own without too much guidance.
Then again, knowing the answer ahead of time doesn't really hurt since understanding the steps in between is what matters.
Then again, knowing the answer ahead of time doesn't really hurt since understanding the steps in between is what matters.
I have been looking at sage (sagemath.org) and sympy to see if I can use open source math packages to do something like this for http://www.mathmaster.org but I don't have any successes so far. Have you seen something like this besides commercial software? They have a nice API but looking at the licensing cost make me read no further.
Aren't they spreading themselves a little thin a little too early? Google can afford to explore different areas, but I think these guys should first work on an impressive engine for pulling together information on the web in a meaningful way before looking at more niche things like resolving math. Unless it somehow calls on the same AI base and was very little work to develop.
I am so glad this didn't exist when I was taking calculus. I would have never learned anything :)
Why oh why did you not exist when I was still in school! I was born too early...
How do you program something like that? I'm impressed.
Let's treat this as an interview question:
First, I'll assume alpha works similar to a compiler. It parses the text, and generates an AST for the input expression. Your goal is to replace the tree with the 'best' possible equivalent AST. Best here is a fitness function, that provides a score for a given AST. At first glance, we'll say the AST with the smallest number of nodes is optimal.
Then, create a list of legal operations on a subtree. Try an operation, and see if the AST gets better. Repeat.
First, I'll assume alpha works similar to a compiler. It parses the text, and generates an AST for the input expression. Your goal is to replace the tree with the 'best' possible equivalent AST. Best here is a fitness function, that provides a score for a given AST. At first glance, we'll say the AST with the smallest number of nodes is optimal.
Then, create a list of legal operations on a subtree. Try an operation, and see if the AST gets better. Repeat.
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As a personal exercise, I've been working on a basic math engine of my own recently. It's not very difficult to break problems down into computer-friendly steps, but if you approach them that way it's another challenge to choose the steps you should pull out to show in human-friendly output.
Here's a screenshot of the output from a very simple example in the current version: http://imgur.com/hevar.png. It uses LaTeX for the equations and Flot for the graphs. It's a fun project to work on; it can be surprising how far a program can get by just repeatedly applying some simple rules. :)
Here's a screenshot of the output from a very simple example in the current version: http://imgur.com/hevar.png. It uses LaTeX for the equations and Flot for the graphs. It's a fun project to work on; it can be surprising how far a program can get by just repeatedly applying some simple rules. :)
This'll be perfect for precalc next semester