Box Orbits
Andy writes this morning:
A fun online game, and a great example of how computer games can be educational. I think I may understand something about box orbits at last!
http://www.bigideafun.com/penguins/arcade/spaced_penguin/default.htm
Cheers!
A
If you’ve been obsessed in recent years by your failure to completely grasp box orbits, here’s an explanation for nonmathematicians. The second illustration amounts to a defnition.
On the other hand, if you like to keep track of what you don’t know about the analysis of box orbits in a triaxial galaxy, here’s a fine start.
If you’d like to look at the math of orbits, with some handsome diagrams, these are courtesy of Prof.dr. M. Franx of Leiden University.
For myself, I think I’ll just sling Kevin a few more times and get back to work.
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There’s something profoundly useful about math education methods that tie mathematics to physical intuition – especially in ways that are interactive, that let us poke, prod, and experiment to discover the behavior of a system. Computers open up tremendous possibilities, which I think have largely gone unused. People make math games that use the game aspect to make fundamentally boring stuff flashy and attention-grabbing, which seems not very useful to me. Kevin the Penguin definitely fits in that better “poke, prod, and experiment” category.
But I’m with Grow With the Flow on this one: we shouldn’t think for a minute that only intentionally “educational” games are actually educational. So yeah, slinging Kevin around is a good time!
Man, I was all excited because I got a score over 300,000, then I see the high scores over 2 billion….
Paul,
I’ve been meaning to do a post sometime on Lakoff and Nunez, Where Mathematics Comes From. Lakoff, of course, is interested in the metaphors that “ground” math; but those metaphors are based in physical realities like walking some-place or filling baskets. Reading your comment, it struck me that poking and prodding is just right – the physical intuitions a game like this engenders are the base in the senses of the metaphors that led the human brain to imagine math.
Incidentally, I showed the game to a younger member of the species, who is 1/10 of my age, and he blew past my score without even looking around…
I had an interesting conversation about math learning with the inimitable Ms. Paige, a speech pathology student who has to take a math-heavy acoustics class. She was a bit surprised (and encouraged!) to learn that even those of us who have a lot of background in math find new math extremely confusing at first.
I think it is not the initial confusion, but rather the long-term building of intuition that distinguishes those who do well in math. Every mathmetical concept, no matter how abstract, eventually takes on a kind of intuitive familiarity that is very much akin to basic physical intuitions like “when I push this, it will tip.”
On intuition in math: There’s software nowadays that can find mathematical proofs automatically. I remember reading that one of the things researchers used it for, when the software was first invented, was to check over various proofs that they found in scholarly journals. They found that human mathematicians tended to come up with correct conclusions significantly more often than they had correct proofs to justify them. That seems to me to fit with what Paul is saying, about learning math being in large part about building intuitive models of the ideas…
Also, in case anyone missed it, there’s a sequel to Spaced Penguin which features basically the same gameplay with duct tape balls.
Paul says “There’s something profoundly useful about math education methods that tie mathematics to physical intuition.” When we play Spaced Penguin, we set a vector for Kevin with fingertip adjustments of a mouse. Is there “physical intuition” without the feel of the sling pulling back against us, of a real mass (Not a penguin!), of our eyes, arms and legs moving to set a direction?
When I sling Kevin in an introspective mood (my mood, not his – I intuit his mood is one of perpetual e-terror), there is no sense of the physicality of aiming him – the process seems entirely analytical. But when I release him, there’s a strong physical sense of his trajectory – my body tries to influence his route as a crowd tries to make a baseball go fair or foul. Is that similar for those of you who grew up with computer games, or do you transcend the abstraction and experience the “feel” of the sling when you pull back the mouse?
If something is lost in the abstration of the mouse, I think something is gained: the representation and management of real-world events via screens, computers, and interface tools is important now and stands to become essential. Perhaps we need to be able to work intuitively both with the real events and with the computerized abstraction of them? Perhaps math classes need to give kids both real slings, and abstrated models of them?
Joel notes that Theorem Prover software “found that human mathematicians tended to come up with correct conclusions significantly more often than they had correct proofs to justify them. That seems to me to fit with what Paul is saying, about learning math being in large part about building intuitive models of the ideas…”
Agreed. But what’s the base of these intuitions? I take an intuition (for example, that an unproved mathematical assertion is correct) to be the output of an algorithm that we aren’t aware we possess. What’s the nature of that algorithm?
Lakoff and Núñez propose four “grounding metaphors,” based in our physical experiences in the world, on which arithmetic is built. For example, we can use the metaphor, Arithmetic Is Motion Along a Path, because we have the experience of walking through a space. That makes sense: The act of moving along a path allows us to intuit addition as walking a distance, and then walking another distance, such that our new location is a sum. Because we have neural networks that we built by walking along paths, we can use that grounded and neurally registered reality to think about addition. Without such metaphors, they say, we couldn’t conceive or do math. The book’s subtitle says it: “How the Embodied Mind Brings Mathematics into Being.”
They then extend their method to abstract math (from algebra on “up") by linking metaphors, so that “…branches of mathematics that have direct grounding are extended to branches that have only indirect grounding.” (p. 102) When I first read the book, the ground began to look rather far off to my mathematically untrained mind, and the scaffolding a bit shaky. I talked to three mathematicians and an English professor of a philosophical bent about this method of explaining what we do in higher math. All had their doubts about the extension.
If mathematicians see their way to correct conclusions without proofs, what does it feel like to them? What do they think they’re doing? Do they smell metaphor at the bottom – does the highly abstract eventually link to the grounded? Is some sort of verbal reasoning without talking involved? Is the intuition visual / nonverbal, as I think Einstein suggested?
Or is high-level mathematical intuition an exclusively neural event at the start, in which understandings that have been laid down as neural networks fire together so as to create new linkages? ("What fires together, wires together.") Ultimately, whatever happens, it must register as changes in the brain; that is necessarily true for any cognitive understanding. But perhaps the intuition is laid down as a neural event, which language or metaphor or visual centers access when the intuition emerges. This would fit with Paul’s sense that new ideas are difficult for everyone, and that for some, “Every mathmetical concept, no matter how abstract, eventually takes on a kind of intuitive familiarity that is very much akin to basic physical intuitions like ‘when I push this, it will tip.’”
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Joel added a footnote:
Also, in case anyone missed it, there’s a sequel to Spaced Penguin which features basically the same gameplay with duct tape balls.
The second version I take as yet more proof that duct tape and The Force are one and the same.
I like the idea of mathematics as being based on metaphors, however I think it goes beyond just physical metaphors.
Certainly we all have experiences of navigating a hallway or rocky path. But also we have interactions with other beings - over time we have experiences making other people happy or sad, and that becomes a part of us as surely as the memory of a sudden slip on a stair. Although I have not ventured far down the path of serious mathematics from programming I know my inclination of metaphor in an anthropomorphic form - I think of code as being confused, or wanting to be a certain way to ease pressures on itself. I often think of code as being a (sometimes malevolent) being. Anthropomorphizing things is such a common habit I can’t help but think it has some strong relationship to intuition.
So it seems like a combination of not just physical, but also emotional metaphors would be at work in intuiting mathematics or anything else. Indeed to me the action of intuiting something about anything reminds me of the show “Connections” where one invention in history had an interesting effect on another development elsewhere. Intuiting something about code or other things seems to be about seeing as many connections affecting a thing as possible, both the straightforward (physical) and surprising (emotional) - then you have a rough idea of how a thing will react when you poke it.