Kill Math
July 24, 2011 6:47 AM Subscribe
Bret Victor on WorryDream The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots.
I believe that both of these forms of mental contortion are artifacts of pencil-and-paper technology. A person should not be manually shuffling symbols. That should be done, at best, entirely by software, and at least, by interactively guiding the software, like playing a sliding puzzle game.
Isn't this kind of like saying that, instead of learning to read, people should just use screen readers? I mean, that's almost like being literate...
Honestly, I think most people aren't very good at math because a) it's not all that well taught (maybe I had a bad childhood, but I had one teacher K-12 who got me really excited about math, and I was predisposed to like the stuff), b) people don't think about it all that much -- the daily uses of math are under their radar because they do it more by instinct than design. Since people don't think about it in any formal way, they don't value it.
Note that people don't think about reading and writing in any formal way, either, but minor errors in language still allow reasonable communication. Minor errors in math lead to modern accounting practices.
posted by GenjiandProust at 7:51 AM on July 24, 2011 [3 favorites]
Isn't this kind of like saying that, instead of learning to read, people should just use screen readers? I mean, that's almost like being literate...
Honestly, I think most people aren't very good at math because a) it's not all that well taught (maybe I had a bad childhood, but I had one teacher K-12 who got me really excited about math, and I was predisposed to like the stuff), b) people don't think about it all that much -- the daily uses of math are under their radar because they do it more by instinct than design. Since people don't think about it in any formal way, they don't value it.
Note that people don't think about reading and writing in any formal way, either, but minor errors in language still allow reasonable communication. Minor errors in math lead to modern accounting practices.
posted by GenjiandProust at 7:51 AM on July 24, 2011 [3 favorites]
This seems like an exceptional amount of effort to get around a little bit of thinking. I bet it will catch on like gangbusters.
posted by Dmenet at 7:52 AM on July 24, 2011
posted by Dmenet at 7:52 AM on July 24, 2011
I kind of see where he's going, I think.
As a child of the graphing calculator era I've gotten used to being able to just plot things and fiddle around with parameters and see how they change (this bas basically how math was taught in my high school). For all that there was much hand-wringing over the fact that my class' algebra skills were the shit, later on in life I've found the ability to visualize and sketch out problems like I was taught in HS has made more advanced math a hell of a lot easier.
It is tough to learn symbol manipulation for the sake of symbol manipulation, however if you've already learned some math then it isn't so bad.
posted by selenized at 8:08 AM on July 24, 2011
As a child of the graphing calculator era I've gotten used to being able to just plot things and fiddle around with parameters and see how they change (this bas basically how math was taught in my high school). For all that there was much hand-wringing over the fact that my class' algebra skills were the shit, later on in life I've found the ability to visualize and sketch out problems like I was taught in HS has made more advanced math a hell of a lot easier.
It is tough to learn symbol manipulation for the sake of symbol manipulation, however if you've already learned some math then it isn't so bad.
posted by selenized at 8:08 AM on July 24, 2011
The catch with his system seems, to me, that to work the software you have to already understand how to set up the algebra. By hiding the variable, you're obscuring some of the meaning, and making the solving just seem like wizardry.
He's also falling into the trap of presenting real-world things as being harder than they are. In the Scrubbing Calculator section he talks about setting up the graph to fit a 768 pixel display. If you're using algebra to find this you're overthinking it - just start with the height, subtract the header and footer height, and divide by 9. What's that? A decimal? Round down. Done. Algebra is useful when you're trying to generalize. In specific cases, arithmetic will suffice.
I'd still like to play with the software, if it ever materializes.
posted by Wulfhere at 8:11 AM on July 24, 2011
He's also falling into the trap of presenting real-world things as being harder than they are. In the Scrubbing Calculator section he talks about setting up the graph to fit a 768 pixel display. If you're using algebra to find this you're overthinking it - just start with the height, subtract the header and footer height, and divide by 9. What's that? A decimal? Round down. Done. Algebra is useful when you're trying to generalize. In specific cases, arithmetic will suffice.
I'd still like to play with the software, if it ever materializes.
posted by Wulfhere at 8:11 AM on July 24, 2011
OK, now try doing this for something where your intuition is not adequate. This point is different for everyone, but once you get there I just don't see how you do it without the symbols and the small, verifiable steps.
On the other hand, non-symbolic methods (visual methods) are probably good for developing intuition, making them a good thing.
posted by jepler at 8:42 AM on July 24, 2011 [3 favorites]
On the other hand, non-symbolic methods (visual methods) are probably good for developing intuition, making them a good thing.
posted by jepler at 8:42 AM on July 24, 2011 [3 favorites]
I think what he's doing is pretty neat and would probably be a better way to develop intuition at the high-school level. Why does everyone need to learn double integration just for the sake of finding instantaneous change?
But for students who need to take higher maths, I don't know how this would develop logical reasoning skills used in say, proofs by induction/contradiction or group theory.
Honestly I have no intuition for a n-dimensional matrix but I do understand the individual steps in being able to precisely express it on paper and being able to manipulate and decompose it to the form I want. And sometimes it's after I do all that symbol manipulation, then I develop the intuition for the grand overview. I guess YMMV.
posted by tksh at 9:32 AM on July 24, 2011
But for students who need to take higher maths, I don't know how this would develop logical reasoning skills used in say, proofs by induction/contradiction or group theory.
Honestly I have no intuition for a n-dimensional matrix but I do understand the individual steps in being able to precisely express it on paper and being able to manipulate and decompose it to the form I want. And sometimes it's after I do all that symbol manipulation, then I develop the intuition for the grand overview. I guess YMMV.
posted by tksh at 9:32 AM on July 24, 2011
I like the approach but it has it's limits. They're going beyond pencil and paper but only to the computer screen, picturing multidimensional relationships will still be a challenge. The 2D representation of a hypercube still looks the same, perhaps drawn somewhat better than a quick sketch on a chalkboard and still has to be imagined.
I think it's best to look at this as an alternative or additive way to look at certain types of problems interactively, but it's still a symbolic approach. To the degree that "mathematics" only exists in the mind as a descriptive modeling language. Things and events in the real world may be mathematically predictable or measurable but they are not mathematical objects in and of themselves. Replacing alpha numeric symbolism with graphical symbolism would be like throwing out written language once someone invents a voice recorder. They both have uses that are not mutually exclusive and can be mutually beneficial.
posted by doctor_negative at 9:47 AM on July 24, 2011
I think it's best to look at this as an alternative or additive way to look at certain types of problems interactively, but it's still a symbolic approach. To the degree that "mathematics" only exists in the mind as a descriptive modeling language. Things and events in the real world may be mathematically predictable or measurable but they are not mathematical objects in and of themselves. Replacing alpha numeric symbolism with graphical symbolism would be like throwing out written language once someone invents a voice recorder. They both have uses that are not mutually exclusive and can be mutually beneficial.
posted by doctor_negative at 9:47 AM on July 24, 2011
jepler, I think the situation is even worse than what you describe. You won't necessarily know when the limits of your intuition have been passed, even with simple plane geometry. For example, there's the paradoxical dissection.
And personally, I've used high-powered software that can do algebraic manipulations for me, but I can understand problems a lot better when I'm playing with the symbols myself on a sheet of paper.
posted by Hither at 10:40 AM on July 24, 2011 [2 favorites]
And personally, I've used high-powered software that can do algebraic manipulations for me, but I can understand problems a lot better when I'm playing with the symbols myself on a sheet of paper.
posted by Hither at 10:40 AM on July 24, 2011 [2 favorites]
This sounds like fun, especially for visual types who really want to be able to see the changes happening in front of them. I would love to play around with this software.
Would this be a great tool for learning to visualize equations and relationships between variables for some people? Yes, probably. Is it some vastly new method of teaching that will do away with the need for understanding the basic rules and symbols? No, I don't think so. Some people think better in those symbols.
They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions.
*ugh* Yes, but lets take a moment to read more into this analogy and maybe take a few hints from it. The moment I want to do something that the designer of my "better" interface hasn't anticipated, I'm stuck unless I know more than just how to use that interface. He has a fine idea, but how about we hold off on replacing or "killing" anything just yet, okay?
posted by Avelwood at 10:51 AM on July 24, 2011 [1 favorite]
Would this be a great tool for learning to visualize equations and relationships between variables for some people? Yes, probably. Is it some vastly new method of teaching that will do away with the need for understanding the basic rules and symbols? No, I don't think so. Some people think better in those symbols.
They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions.
*ugh* Yes, but lets take a moment to read more into this analogy and maybe take a few hints from it. The moment I want to do something that the designer of my "better" interface hasn't anticipated, I'm stuck unless I know more than just how to use that interface. He has a fine idea, but how about we hold off on replacing or "killing" anything just yet, okay?
posted by Avelwood at 10:51 AM on July 24, 2011 [1 favorite]
I like aspects of this this. I've used tools that seem aligned somewhat with this philosophy in the classroom, and I was trained by a scientists who believed that a good plot that really gets to the kernel of the physical phenomenon at work was the most valuable output of any scientific endeavor.
As a way to build intuition, supplementing analytical math, excellent. As a way to give students who would otherwise never use analytical math a window into quantitative relationships, thumbs up.
I worry, though, at the point where these tools are good enough to serve as a replacement for good old fashioned analytical math, because so often students (and, um, sometimes *cough* myself) accept whatever garbage is barfed out of Mathematica, Excel, and (bane of my existence as an instructor) Wolfram Alpha uncritically. These tools can fail in unexpected and insidious ways, and if nobody in the room knows how to confirm the results with pencil and paper, the results could be disastrous.
This seems like an exceptional amount of effort to get around a little bit of thinking. I bet it will catch on like gangbusters.
Yep.
posted by BrashTech at 11:01 AM on July 24, 2011
As a way to build intuition, supplementing analytical math, excellent. As a way to give students who would otherwise never use analytical math a window into quantitative relationships, thumbs up.
I worry, though, at the point where these tools are good enough to serve as a replacement for good old fashioned analytical math, because so often students (and, um, sometimes *cough* myself) accept whatever garbage is barfed out of Mathematica, Excel, and (bane of my existence as an instructor) Wolfram Alpha uncritically. These tools can fail in unexpected and insidious ways, and if nobody in the room knows how to confirm the results with pencil and paper, the results could be disastrous.
This seems like an exceptional amount of effort to get around a little bit of thinking. I bet it will catch on like gangbusters.
Yep.
posted by BrashTech at 11:01 AM on July 24, 2011
The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
I stop with the first sentence. People who are very good at math aren't good because they "freakishy manipulate abstract symbols." not there aren't plenty of mathematicians (and others) who are good at symbolic manipulation, but really good mathematicians are good at realizing that abstract symbol games can be mapped into something they already understand i.e. something intuitive (to them) that they can manipulate. this isn't necessarily apparent from the outside. there isn't one way to make something intuitive, it's different for different people and notation is really just a common ground so people can communicate.
but reading on...
This mechanism of math evolved for a reason: it was the most efficient means of modeling quantitative systems given the constraints of pencil and paper.
Ignoring the vast history of mathematics before pen and paper were cheap.
Complex numbers provide a similar example. Being able to work with complex numbers (as abstract values) is seen as an essential skill in many scientific fields. Then David Hestenes came along and said, "Hey, you know all your complex numbers and quaternions and Pauli matrices and other abstract funny stuff? If you were working in the right Clifford algebra, all of that would have a concrete geometric interpretation, and you could see it and feel it and taste it." Taste it with your monkey-mouth! Nobody actually believed him, but I do, and I love it.
LOL... Clifford algebras are less abstract than complex numbers?
And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface.
OS X Math with.... lickable buttons?
I'm actually sympathetic to his plight: trying to manipulate symbols you have no intuition about is a frustrating and painful... but this is kind of painful. in the end, he's trying to blame his own problems on the notation rather than the simple fact that, like barbie says, math *is* hard.
posted by ennui.bz at 11:34 AM on July 24, 2011
I stop with the first sentence. People who are very good at math aren't good because they "freakishy manipulate abstract symbols." not there aren't plenty of mathematicians (and others) who are good at symbolic manipulation, but really good mathematicians are good at realizing that abstract symbol games can be mapped into something they already understand i.e. something intuitive (to them) that they can manipulate. this isn't necessarily apparent from the outside. there isn't one way to make something intuitive, it's different for different people and notation is really just a common ground so people can communicate.
but reading on...
This mechanism of math evolved for a reason: it was the most efficient means of modeling quantitative systems given the constraints of pencil and paper.
Ignoring the vast history of mathematics before pen and paper were cheap.
Complex numbers provide a similar example. Being able to work with complex numbers (as abstract values) is seen as an essential skill in many scientific fields. Then David Hestenes came along and said, "Hey, you know all your complex numbers and quaternions and Pauli matrices and other abstract funny stuff? If you were working in the right Clifford algebra, all of that would have a concrete geometric interpretation, and you could see it and feel it and taste it." Taste it with your monkey-mouth! Nobody actually believed him, but I do, and I love it.
LOL... Clifford algebras are less abstract than complex numbers?
And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface.
OS X Math with.... lickable buttons?
I'm actually sympathetic to his plight: trying to manipulate symbols you have no intuition about is a frustrating and painful... but this is kind of painful. in the end, he's trying to blame his own problems on the notation rather than the simple fact that, like barbie says, math *is* hard.
posted by ennui.bz at 11:34 AM on July 24, 2011
I think he's on to something with the monkey eyes bit, but I’m uncomfortable with the idea of replacing something-you-know with something-you-have, especially when it’s a $500+ flavor of the decade piece of hardware. Abstract reasoning is table stakes for modern life, and every developed country in the world spends substantial energy trying to teach this to its youngest members. To characterize math as the domain of a “clergy of scientists and engineers” is pretty ugly and disingenuous.
posted by migurski at 11:50 AM on July 24, 2011 [2 favorites]
posted by migurski at 11:50 AM on July 24, 2011 [2 favorites]
I'm torn about this. He's certainly right that our ability to reason mathematically is enhanced when we can find ways to bring mathematical objects within the grasp of our native intution (which is great at space and motion, less great at symbolic manipulation.) The complex numbers are a great example, except the "killer app" here is not Clifford algebras but the idea (due to Argand, I think?) of thinking of complex numbers as points on the plane, thereby bringing the full force of our geometric intuition to bear on them.
That said: we don't use symbolic methods just for fun. We use them as an immensely powerful cognitive prosthesis -- a kind of "force multiplier" for our intuition. Without them we would be able to do less stuff. When it's time to find a local maximum of a function on 10,000-dimensional space, you want to supplement those monkey eyes with something a little more up-to-date.
posted by escabeche at 2:45 PM on July 24, 2011 [2 favorites]
That said: we don't use symbolic methods just for fun. We use them as an immensely powerful cognitive prosthesis -- a kind of "force multiplier" for our intuition. Without them we would be able to do less stuff. When it's time to find a local maximum of a function on 10,000-dimensional space, you want to supplement those monkey eyes with something a little more up-to-date.
posted by escabeche at 2:45 PM on July 24, 2011 [2 favorites]
but nobody is suggesting we should teach everyone to use the command line.
I will go ahead and suggest that we should teach everyone to use the command line.
(Actually, I'll go ahead and suggest that most people who use a modern computer already uses a CLI or three, albeit in stealth forms like the browser location bar or a search box.)
posted by brennen at 4:13 PM on July 24, 2011
I will go ahead and suggest that we should teach everyone to use the command line.
(Actually, I'll go ahead and suggest that most people who use a modern computer already uses a CLI or three, albeit in stealth forms like the browser location bar or a search box.)
posted by brennen at 4:13 PM on July 24, 2011
Ignoring the vast history of mathematics before pen and paper were cheap.
Wax tablets were cheap, and in common use in the classical era. Slate and chalk are cheap, and. handheld slates were standard issue for enlightenment-era schoolchildren, and earlier, for pretty much anyone who needed to calculate a few problems, most notably military sappers and engineers from the dark ages.
So, yeah, we've been doing things the same for a really, really long time... at least a thousand years for our modern numeral system.
posted by Slap*Happy at 5:30 PM on July 24, 2011
Wax tablets were cheap, and in common use in the classical era. Slate and chalk are cheap, and. handheld slates were standard issue for enlightenment-era schoolchildren, and earlier, for pretty much anyone who needed to calculate a few problems, most notably military sappers and engineers from the dark ages.
So, yeah, we've been doing things the same for a really, really long time... at least a thousand years for our modern numeral system.
posted by Slap*Happy at 5:30 PM on July 24, 2011
This is against everything I stand for in math.
He rejects the whole idea of translating a problem into another language: physical quantitative problems should have only "intuitive", physical solutions, in which all steps must have "physical meaning". Never mind if it is vastly easier to reason correctly about discrete objects like symbols. Never mind how wondrous and delightful it is that it is even possible to translate problems into different yet equivalent forms, in which operations you can't even conceive in the original setting become trivial — and for god's sake let's not waste time thinking about what these possibilities tell us about the true nature of our questions! We have "meaningful problems of quantity" to solve!
And what a narrow view of math that phrase embodies. Perhaps that narrowness explains how he is able to completely overlook the thing that actually makes math hard, namely thinking logically. Synthetic geometry, for example, doesn't have the symbol-pushing he is so opposed to, and it's all about visualizable objects — but it's still hard, because logic is hard, because rigour is hard. But there is no place in this guy's notion of math for logic, just some kind of intuitive monkeying with quantity, which means he's not really talking about math at all.
posted by stebulus at 5:35 PM on July 24, 2011 [6 favorites]
He rejects the whole idea of translating a problem into another language: physical quantitative problems should have only "intuitive", physical solutions, in which all steps must have "physical meaning". Never mind if it is vastly easier to reason correctly about discrete objects like symbols. Never mind how wondrous and delightful it is that it is even possible to translate problems into different yet equivalent forms, in which operations you can't even conceive in the original setting become trivial — and for god's sake let's not waste time thinking about what these possibilities tell us about the true nature of our questions! We have "meaningful problems of quantity" to solve!
And what a narrow view of math that phrase embodies. Perhaps that narrowness explains how he is able to completely overlook the thing that actually makes math hard, namely thinking logically. Synthetic geometry, for example, doesn't have the symbol-pushing he is so opposed to, and it's all about visualizable objects — but it's still hard, because logic is hard, because rigour is hard. But there is no place in this guy's notion of math for logic, just some kind of intuitive monkeying with quantity, which means he's not really talking about math at all.
posted by stebulus at 5:35 PM on July 24, 2011 [6 favorites]
I've seen this guy posted before and I have the same reaction - he is right about the problem and his is an interesting solution among many possibilities.
I took the two advanced math classes at high school and I immediately grew weary of learning solving methodologies. I struggled through the symbology of maths even though conceptually it was very interesting to me. I just didn't like the aesthetics of maths.
I knew there had to be other ways to represent mathematics which were more more suited to the way I thought so I brought this up with my teacher. He looked at me as if I was insane - it was 1991.
we should teach everyone to use the command line
That might have even suited me. It would make intersting to see if learning programming abstractions first and then applying those to solving problems via computation would suit those who aren't drawn directly to maths. Maths IS translated at some point into computation anyway. Yes I know the limits of computation - they are worth learning.
posted by vicx at 9:30 PM on July 24, 2011
I took the two advanced math classes at high school and I immediately grew weary of learning solving methodologies. I struggled through the symbology of maths even though conceptually it was very interesting to me. I just didn't like the aesthetics of maths.
I knew there had to be other ways to represent mathematics which were more more suited to the way I thought so I brought this up with my teacher. He looked at me as if I was insane - it was 1991.
we should teach everyone to use the command line
That might have even suited me. It would make intersting to see if learning programming abstractions first and then applying those to solving problems via computation would suit those who aren't drawn directly to maths. Maths IS translated at some point into computation anyway. Yes I know the limits of computation - they are worth learning.
posted by vicx at 9:30 PM on July 24, 2011
And another thing. According to him, math is hard because symbol manipulation is not natural, and people who are comfortable with symbol manipulation are therefore not really human (they have a "freakish knack"!), and have unfairly used their inhuman abilities to gain power over you and me. If you ever had trouble understanding math, he implies, it's because those Others want you to "feel ashamed and vaguely inferior".
This is bigotry and demagoguery. It's foul.
posted by stebulus at 9:41 AM on July 25, 2011 [1 favorite]
This is bigotry and demagoguery. It's foul.
posted by stebulus at 9:41 AM on July 25, 2011 [1 favorite]
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This is good thinkin'.
posted by otherthings_ at 7:35 AM on July 24, 2011