> Its hard to communicate a fraction to a computer
> "Explain whether 4/3 or 3/4 is closer to 1, and how you know."
You just did that. Twice.
Square root ? "(3/4)^(1/2)" or maybe "sqrt(3/4)". There's no complexity in parsing that. I do agree it is not as natural as on paper but maybe tablets will find a way to improve that. Thats what innovation is here for after all.
>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."
I am not familiar with the domain, but dont we have some automatic theorem-proving tools? Validating the answer to such a question would look like a perfect use case to me.[edit : clarified]
> [the description example]
Im not sure about this one. On the one hand i used to have a project in college about reconstructing a picture from an incomplete description - and its hard. On the other hand we are expecting a perfect description. Hence it would be pretty much isomorphic to the code of a program used to draw the picture. Matching the two images is also doable. Pseudocode would actually be the best way to transmit this image .
I have no problem using this machine to say:
abs(4/3 - 1) = abs(3/3 + 1/3 - 1) = abs(1/3)
abs(3/4 - 1) = abs(4/4 - 1/4 - 1) = abs(-1/4)
abs(1/3) > abs (-1/4), so 3/4 is closer to 1 than 4/3.
Try something harder - at least, say, the quadratic formula. Or, if you really want to appreciate what we're getting at, try a chewy example, like Maxwell's equations:
These are expressible on the calculator command line, and first-year grad students with Mathematica licenses type them out fairly often, but you're going to discover that Knuth spent a decade of his life inventing TeX for a very good reason. Without proper notation it is hard to reason about math.
You can, of course, use the better class of notation editors to get your computer to display math in proper notation, just as the Wikipedia authors did, but it's a bunch of fiddly work, rather more work than writing out the math by hand. And then you find that you can't make very faint tickmarks or cross-outs or circles on your math. You can't easily draw arrows connecting one line to another.
A useful side-effect of pencil-written mathematics is that the intermediate steps are there in front of you. Watch a professor talk you through a derivation on a chalkboard. Observe that, despite the fact that chalkboards can be erased, using technology that has been available since prehistory, the professor rarely transforms equations by erasing the old ones and replacing them with new ones in-place. That's because recopying the equation after every one or two transformational steps leaves behind a changelog that represents your train of thought. When the problem is almost done but you're trying to track down the sign error you'll be grateful for that.
And, yes, computers have undos, so you can rewind and fast-forward your math. But that's inferior technology for thinking about problems. Ask Tufte: The secret to reasoning about data or logic is to spread it out in front of you in as flat a manner as possible, so that you can move from step to step using nothing but your eye muscles, or defocus your eyes a bit and view the whole problem space in the abstract.
Giant desk-sized iPads may one day render pencils and paper obsolete for serious math, but I'm not convinced that will happen in my lifetime. The hardware is expensive, the software is far more expensive, and paper is cheap, and scanners for digitizing paper are cheap.
I don't doubt that one can, and that we will, build a computer system for manipulating higher mathematics that's so much better than a stack of paper, a pencil, and a decent eraser that you won't even own the paper. What I doubt is that it's done yet. But I haven't exactly been looking for it, so maybe I'm wrong. Certainly, once it's done there won't be any problem selling it to me (except for the sad fact that I no longer manipulate equations on an everyday basis).
There is plenty of room for automation in math education. But in a really good math education, the automated tools need to be balanced with more socially-oriented approaches to education. Students need to talk to each other and to good teachers about their work. Students need to see each other's approaches and hear each other's ideas, and have face-to-face conversations about math.
I think we presume a great deal in suggesting that a simple flat array of characters and operators is somehow less understandable than a nicely typeset equation (especially when you've never written one before!).
One advantage would be that you could try invalid syntax and operations (i.e. x/0) and see the errors that result in real time as opposed to an hour or a day later after the teacher marks up your test. Then you're more likely to stick with it until you get it right, which in turn means the answer is more likely to stick with you.
"Pseudocode would actually be the best way to transmit this image ."
I wholeheartedly agree. A graphics routine or a LOGO program would do a great job of describing that.
The author seems to have an irrational dislike of people trying to use computers in this fashion, which I find strange. I certainly agree that something like a geometric proof (in absence of a good modeling language) is difficult to automatically check, but at the same time I question whether or not the human element would be any more useful here. Math teachers, especially at lower levels, are not infallible.
I would almost venture that a better test, one that examines both critical thinking and ability to logic about a problem, would be a battery of small programming problems to solve some kind of geometric or graphical challenge. It's a bit all-or-nothing, but it would show that the student can both interpret a problem and also describe the steps to solving it.
>>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."
>I am not familiar with the domain, but dont we have some automatic theorem-proving tools? This would look like a perfect use case to me.
Theorem proving tools would work if the students wrote their answers in a format that the tool would work in. In this case, it would be a natural language proof instead of a formal proof, which simply isn't possible to parse right now. Perhaps it will be in the future.
They will learn the process if the tool is only used behind the scenes to validate their answer
Theorem proving tools would work if you were writing formal proofs. At the level he's talking about, students are not writing formal proofs -- they're writing explanations.
Perhaps a partial solution to this problem can be found in... wait for it... programming! That is, it is really hard to write a program to find general answers to problems unless you understand the basic idea. This definitely doesn't work for everything, but I wonder how big the domain is where this is a really good way to mechanically assess understanding, and whether the language that'd be required would itself become a bigger component of the measure than what you're trying to assess -- mathematical reasoning.
Or, heck, just think of how annoyed some developers get when asked to write code on a whiteboard during an interview. "That's not how we code! If you want me to write code, give me a computer."
A human instructor asks for the explanation to explore a bit if the answer was just a guess or if it was reasoned out, and whether the reasoning was correct. An automated evaluation system would have an easier time factoring in results from previous evaluations (it can perfectly remember an arbitrary number of tests across an arbitrary number of students...) and could check understanding by presenting several questions (it won't get sick of looking at the answers).
I guess if the proposition is that human instruction can be replaced by automated systems that is crazy. Really, I expect most people are looking to supplement it and to make it more effective.