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Your quest uses tags of 'mathematics' so I found you this >>>>
Kill Math
Bret Victor / April 11, 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.
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.
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. Unfortunately, most people are not comfortable with bundling up meaning into abstract symbols and making them dance. Thus, the power of math beyond arithmetic is generally reserved for a clergy of scientists and engineers (many of whom struggle with symbolic abstractions more than they'll actually admit).
We are no longer constrained by pencil and paper. The symbolic shuffle should no longer be taken for granted as the fundamental mechanism for understanding quantity and change. Math needs a new interface.
Project
Kill Math is my umbrella project for techniques that enable people to model and solve meaningful problems of quantity using concrete representations and intuition-guided exploration. In the long term, I hope to develop a widely-usable, insight-generating alternative to symbolic math.
Someday there will be an introductory essay on this page, and it will move you to tears. That essay is not yet written -- it will take a lot more thinking, and a lot of examples, before I understand what I'm trying to do well enough.
Here's what I have for you so far:
Scrubbing Calculator demonstrates a tool for exploring practical algebraic problems without symbolic variables. Instead of x's and y's, you connect concrete numbers and adjust them interactively.
Interactive Exploration of a Dynamical System demonstrates a tool for manipulating differential equations where every variable is shown as a plot, and every parameter has a knob that can be adjusted in realtime. This helps the user develop a sense for how the parameters of the system influence its behavior.
Up And Down the Ladder of Abstraction is an interactive essay about using visualization in a systematic way to design and understand systems.
Simulation As A Practical Tool is its early precursor, where I started working out the ideas behind this effort.
Below is a collection of blog-quality ramblings on the topic, which I suppose are intended more to attract like-minded people than to convince the skeptical. (The skeptical should refuse to be convinced until they see more examples.)
My plan is to collect a number of meaningful problems across different application areas and areas of mathematics, and for each one, design a means of solving it that is line with the philosophy here, and compare the benefits of this solution to the benefits of a conventional solution. The techniques and design patterns that emerge during this process will, hopefully, inform a more general framework in the long term.
As always, if you're playing with ideas along similar lines, I'd love to see what you've come up with.
Some additional thoughts were published in this Fast Company article.
A Cobbled-Together Assortment of Unconnected Notes from Various Times and Places
Language and Visceral Interpretation (1)
The ability to understand and predict the quantities of the world is a source of great power. Currently, that power is restricted to the tiny subset of people comfortable with manipulating abstract symbols.
By comparison, consider literacy. The ability to receive thoughts from a person who is not at the same place or time is a similarly great power. The dramatic social consequences of the rise of literacy are well known.
Linguistic literacy has enjoyed much more popular success than mathematical literacy. Almost all "educated" people can read; most can write at some level of competence. But most educated people have no useful mathematical skill beyond arithmetic.
Writing and math are both symbol-based systems. But I speculate that written language is less artificial because its symbols map directly to words or phonemes, for which humans are hard-wired. I would guess that reading ties into the same mental machinery as hearing speech or seeing sign language.
I don't believe we have the same innate ability for processing mathematical symbols.* * Papert might disagree, and claim that a child raised in "Mathland", an immersive interactive mathematical environment that "is to math what France is to French", would become as fluent in symbolic math as in language. With regard to symbolic math, I might respond that a child raised in Antarctica would be quite tolerant of the cold, but maybe people shouldn't need that sort of tolerance. Instead, we tend to reply on implicit physical metaphors, both for the mechanics of symbol manipulation (e.g., "moving" a term to the other side of the equation, "canceling out" two terms, etc.) and for the semantic interpretation of the symbols (e.g., exponential "blow-up", or the "smallness" of a neglibible term). To a certain extent, a person's mathematical skill is tied to their ability to "feel" the symbols through these physical metaphors, and thereby make the abstract more concrete.
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. And, more contentiously, I believe that a person should not have to imagine the interpretation of abstract symbols. Instead, dynamic graphs, diagrams, visual models, and visual effects should provide visceral representations. Relationships between values, exponential blow-ups and negligible terms, should be plainly seen, not imagined.
Language and Visceral Interpretation (2)
Humans are built for language -- we're symbol-processing machines -- so I can't say "symbols bad". But I feel that there are things that we need to see or experience in order to truly understand. And there are things that are easy to draw or build, but impossible to describe (without years of practice in arcane specialized languages).
I think that quantity and measure fall into that category. Reading "1m" and "1mm", versus actually observing those two measures -- one is just numbers on a page, the other hits you viscerally. Do you think most people understood, really felt, the difference between a $1B and a $1T bailout? Three orders of magnitude hidden inside a symbol.
The point is that you need that visceral sense, that gut feel, to reason about a problem by intuition. Good circuit designers can "feel" how a circuit behaves. They look at a schematic and in their mind's eye, they see the voltage going down over here and pushing the voltage up over there, as if they were looking at a see-saw or water pump. It requires years of practice to develop this sense, this ability to look at symbols (in some domain) and feel what they represent.
Likewise, people used to think that reading and making sense of huge tables of numbers was an essential skill for working with data. But then William Playfair came along and invented line graphs, and suddenly everyone could feel data through their eyes. Their plain old monkey-eyes!
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.
It's the responsibility of our tools to adapt inaccessible things to our human limitations, to translate into forms we can feel. Microscopes adapt tiny things so they can be seen with our plain old eyes. Tweezers adapt tiny things so they can be manipulated with our plain old fingers. Calculators adapt huge numbers so they can be manipulated with our plain old brain. And I'm imagining a tool that adapts complex situations so they can be seen, experienced, and reasoned about with our plain old brain.
Kitchen Math
In The Children's Machine, Papert describes "kitchen math". A certain recipe serves 3, but the cook is only cooking for 2, so she needs to 2/3 all of the ingredients. The recipe calls for 3/4 cup of flour. The cook measures out 3/4 cup of flour, spreads it into a circle on the counter, takes a 1/3 piece out of the circle and puts it back into the bag. That's 2/3 of 3/4.
Some people would be horrified that this person can't multiply fractions, but I find the solution delightful. It's concrete, visual, tangible, direct. As opposed to the conventional approach of "canceling out the 3 on the top and bottom", which has no physical meaning whatsoever in this case.
I want to create an environment for turbo-charged kitchen math.
Mathematical Arts
This project is not an attack on practicing math for its own sake. I have no problem with mathematics for recreation, or as an art form. All my life I've studied math out of personal interest; I play with math all the time. I resonate deeply with Lockhart's lament, and I'm amused by the work of Vi Hart, Mike Keith, and so on. There's beauty in patterns and rules; there's challenge in discovering it; that's all fine. My problem is when mindless tradition and lack of imagination compel us to use this art form, with all of its archaic restrictions, as a practical tool.
Consider martial arts, another art form that evolved out of immediately practical needs. Like math, people might practice martial arts for exercise (physical or mental), for the challenge and reward of mastering a skill, for its elegance and beauty, or as a social activity. Unlike math, we recognize that the martial arts are no longer suitable for their original practical purpose, now that technological progress has yielded more wonderously effective ways of smashing people.
(Also unlike math, we don't force-feed twelve years of lessons to every child on the planet, and those who are unskilled at the art aren't made to feel ashamed and vaguely inferior.)
A Possibly Embarrassing Personal Anecdote
When I was in high school, I would go down to the local college a few times a week to learn about differential equations. One day, after the instructor solved a second-order equation, say:
he threw out an offhand question: "Why do you think the solution has two arbitrary constants?"
I was confused by the question. It does because it does, I thought. I could see how the solution would have two degrees of freedom, that made sense to me, but it never occurred to me that there was some deeper cause.
The instructor continued, "Because you're integrating twice." And then moved on to some other subject while my young brain twisted into a knot.
I had never considered solving a differential equation to be integration. It didn't feel like integration. I knew what integration felt like -- it was adding things up, a little tank filling up with water:
And I knew what an equation felt like -- it was a balancing act, a little scale coming to rest:
I went on to college, and grad school, and an engineering career, and I must have solved, what, hundreds of differential equations? Thousands? Obviously I understood the formal relationship between differential equations and integration. But I don't know that I ever felt it.
Then, one day, I was reading (for fun) Strogatz's phenomenal book, Nonlinear Dynamics and Chaos. And he asked, how do you solve this differential equation:
And he said, well, you don't. You can't. It's nonlinear. Our symbol-pushing tricks don't work here. But what you can do is decompose it into a system ....... http://worrydream.com/KillMath/
| 10 years ago. Rating: 2 | |
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