Mathematics Feed

Cellular Automata Caterpillar

My son found this caterpillar in our yard Friday. I don't know what species it is. (Anyone know?) I took its picture because of the cellular automata-style patterns on its back. I sent it to the folks at Wolfram, who I'm sure could tell me what pattern that is.

(Regarding the ichor it's sitting in, we didn't know if it was injured or whether this was some other kind of secretion.)

Future Shock & Third Grade Math

DonpostmutantI'm a bit unsure what I shold have said this morning when my son's new third grade teacher told me, in the context of a discussion of the math cirriculum, that in the third grade "we don't teach memorization; we teach concepts." Perhaps she was a little too candid.

Good thing that here in the Future we can all work out multiplication problems from first principles whenever we need to multiply, isn't it?


WolframtonesAND NOW FOR SOMETHING COMPLETELY DIFFERENT: In the midst of all this mess, I got an email from one of the folks at Wolfram telling me about their lovely new Internet widget WolframTones, which is essentially the aesthetics of A New Kind of Science rendered as sound. One of the potentially revolutionary strengths of Mathematica is it's ability to render mathematics as sound, allowing us to gain greater understanding of math using our faculties for appreciating and understanding music. I've been looking into this myself, reading up on the neurology of math. One interesting book on this subject I have in hand is Functional Melodies: Finding Mathematical Relationships in Music by Scott Beall.

But what is special about the Wolfram version, and sets it apart from other attempts to integrate mathematics and music, is that it takes on the gnarly natural mathematics derived from Wolfram's attempt to parse the complexity of the geometry of nature. The piano selections remind me of Philip Glass's "Closing," which I think of as the best Thinking Music I have in my iTunes.

Teaching Perfect Squares with Legos

Lego_pyramidOne of my summer projects is to teach Peter his multiplication tables before school starts. Our exceptionally fine school district had an extremely difficult time teaching him his basic addition and subtraction facts in the first and second grade, and I have no reason to believe that they will have more success in the third grade with multiplication. Peter has David's amazing associative memory, and while associative memory is great for learning about, say, red efts, since calling to mind all the information you have about surrounding concepts such as salamanders and newts gives you context and allows you to interpolate information you don't know. But for recalling information about, say, the number seven it is a disaster (as my Google search link handily illustrates: nearly a billion results for the numeral 7; only a hundred and sixteen million for the word seven spelled out).

I have arrived at this formulation: Memory is something I do; memory is something that happens to David and Peter. So in order to get Peter over certain academic hurdles, I need to teach him how to work at memory. Simple recitation does not do it for numbers. The public schools have a slightly more complex technique that boils down to repetition which has failed us utterly, so far.

So I have been looking for alternatives. One of the alternatives, has been assigning multiplication problems to particular places in a classically organized "memory palace" structured around the pool area and grounds of the hotel where the International Conference on the Fantastic in the Arts is held. Each memory place is a place where he found a memorable creature. (The iguana he spotted by the whirlpool is given the spot 7 X 7 and is names Fortunine to invoke 49; the place he found a favorite caterpillar is designated 8 X 8, and the caterpillar is named "Sticky Boy" to invoke 64.) This was succeeding up to a point, but lacked a structure that could be extrapolated upon.

So this afternoon, I hit upon the idea of building a Great Pyramid, complete with a Lego Pharaoh, to illustrate the concept of perfect squares in a way that could be generalized to other multiplication problems, and would also allow us to deduce the existence of prime numbers.

The hardest part was sorting his vast and diverse collection of Legos for the collection of 200-odd square Legos with four bumps on them. This allowed us the make the first 8 layers of the pyramid. Starting with one, I had him tell me what the product of each number was when multiplied by itself; then we collected the right number of square Legos,; then we built the next layer of the pyramid. (Because of the tightness of the fit needed for the Legos near the middle, I did the middle parts, and he did the perimeters.) Having verified that the square of each number indeed yielded a square, we moved on to rectangles; and then we demonstrated experimentally that there are some numbers of blocks that can't be made into rectangles (the example we tried was 19). Then I explained about prime numbers.

I am pleased with this Lego activity, but also think that it would not have worked if I had not first helped him memorize the perfect squares using the pool-side memory palace. I was taking things he had memorized as arbitrary concepts and giving them a more conceptually based architectural structure which can be extrapolated from.

PS: I must say I'm rather hardpressed to understand the intended mneumonics of the "Fact Triangles" in the Everyday Mathematics curriculum that Peter's school uses.

A Few Shiny Pebbles on the Infobeach

I wrote to Alice Flaherty, expert on the neurology of writing, for help with references on the neurology of math. She suggested some places to look and some search terms, so I've been playing with PubMed and discovering interesting things such as that a lot more seems to be known about the neurology of metaphor than about the neurology of math. I came across a couple of articles with interesting descriptions which I though I'd share:

Research into the origins and characteristics of unicorns: mental illness as the unicorn. Abstract:

Ethical Hum Sci Serv. 2000 Fall-Winter;2(3):181-92.
Research into the origins and characteristics of unicorns: mental illness as the unicorn.
Simon L.
Kingsborough Community College, City University of New York, USA.

    Basic research, particularly into the psychological and neurological underpinnings of schizophrenia and other "mental illnesses," is flawed because of its adherence to the ideology that unwanted, hard-to-understand behavior constitutes true medical illness. It is argued here that psychiatric diagnostic terms represent moral judgments rather than medical entities. By reducing experimental subjects to a moral label, and assuming that neurological differences associated with unwanted behavior are brain diseases, researchers fail to take into account the conscious experience, organization of self and self-image, patterns of motivation, history and social contexts of their patients. The failure to consider the psychology of their subjects renders the results of these studies ambiguous and irrelevant for any uses except bolstering the biomedical model of psychiatry.

    PMID: 15278984 [PubMed - indexed for MEDLINE]

(I had recently noticed that the literature associated with various conditions affecting the social skills is often contaminated by the researchers' dislike of the research subjects.)

And Neural activity associated with metaphor comprehension: spatial analysis.

Neurosci Lett. 2005 Jan 3;373(1):5-9.
Neural activity associated with metaphor comprehension: spatial analysis.
Sotillo M, Carretie L, Hinojosa JA, Tapia M, Mercado F, Lopez-Martin S, Albert J.
Departamento de Psicologia Basica, Facultad de Psicologia, Universidad Autonoma de Madrid, 28049 Madrid, Spain.

    Though neuropsychological data indicate that the right hemisphere (RH) plays a major role in metaphor processing, other studies suggest that, at least during some phases of this processing, a RH advantage may not exist. The present study explores, through a temporally agile neural signal--the event-related potentials (ERPs)--, and through source-localization algorithms applied to ERP recordings, whether the crucial phase of metaphor comprehension presents or not a RH advantage. Participants (n=24) were submitted to a S1-S2 experimental paradigm. S1 consisted of visually presented metaphoric sentences (e.g., "Green lung of the city"), followed by S2, which consisted of words that could (i.e., "Park") or could not (i.e., "Semaphore") be defined by S1. ERPs elicited by S2 were analyzed using temporal principal component analysis (tPCA) and source-localization algorithms. These analyses revealed that metaphorically related S2 words showed significantly higher N400 amplitudes than non-related S2 words. Source-localization algorithms showed differential activity between the two S2 conditions in the right middle/superior temporal areas. These results support the existence of an important RH contribution to (at least) one phase of metaphor processing and, furthermore, implicate the temporal cortex with respect to that contribution.

    PMID: 15555767 [PubMed - indexed for MEDLINE]

and Neural correlates of metaphor processing.

Brain Res Cogn Brain Res. 2004 Aug;20(3):395-402.
Neural correlates of metaphor processing.
Rapp AM, Leube DT, Erb M, Grodd W, Kircher TT.
Department of Psychiatry, University of Tuebingen, Osianderstrasse 24, D-72076 Tuebingen, Germany.

    Metaphoric language is used to express meaning that is otherwise difficult to conceptualize elegantly. Beyond semantic analysis, understanding the figurative meaning of a metaphor requires mental linkage of different category domains normally not related to each other. We investigated processing of metaphoric sentences using event-related functional magnetic resonance imaging (fMRI). Stimuli consisted of 60 novel short German sentence pairs with either metaphoric or literal meaning. The pairs differed only in their last one to three words and were matched for syntax structure, word frequency, connotation and tense. Fifteen healthy subjects (six female, nine male, 19-51 years) read these sentences silently and judged by pressing one of two buttons whether they had a positive or negative connotation. Reading metaphors in contrast to literal sentences revealed signal changes in the left lateral inferior frontal (BA 45/47), inferior temporal (BA 20) and posterior middle/inferior temporal (BA 37) gyri. The activation in the left inferior frontal gyrus may reflect semantic inferencing processes during the understanding of a metaphor. This is in line with the results from other functional imaging studies showing an involvement of the left inferior frontal gyrus in integrating word and sentence meanings. Previous results of a right hemispheric involvement in metaphor processing might reflect understanding of complex sentences.

    PMID: 15268917 [PubMed - indexed for MEDLINE]

Pokémon Cards & Folk-Mathematics

Through most of my career as a mother, I have made it a point of aligning my interests with my children's interests. This has taken me to many interesting places, taught me many interesting things, and even gotten me published in the science magazine Nature (reprint on Fantastic Metropolis).

I have made an exception for annoying fads, especially the Pokémon thing. (See my May 18th, 2003 post, "Pokémon Infestations and Other Matters.")

I realized in the middle of the night, night before last, that there was something big I had been missing about the whole phenomenon. Here is an out-take from what I wrote about it:

One puzzling phenomenon I've observed watching 2nd graders is how kids, who are only just getting basic addition and subtraction of multidigit numbers by the tail, can spend literally hours trading Pokemon cards (by which I mean 2 or 3 hours at a time). The decisions of whether or not to trade are based on multiple factors, some of which are linear functions like how many hit points does a given card have (or is the sum of the hit points of the two cards you are offering me equal to or greater than the hit points of the card of mine you want), and some of which are binary (is it a "shiny", i.e. a holographic card).
. . .
I spot-checked Peter's sense of the relative value of cards back in February. I had him show me what he thought of as his three best cards. I priced them on The cheapest of them came in at $47.00. I then had him show me three of his cards that he thought of as "not-so-good." priced those between 75 cents and $3.00.

Given what I know of the scholastically measurable of the math skills of the kids in question, there has to be some kind of pre-verbal calculation going on. They seem to me to be carrying out complex calculations involving multiple variables of different types, and arriving at basically correct conclusions via some kind of folk-math.
. . .
One other implication of this phenomenon, it seems to me, is that the equals sign, as a piece of mathematical notation, is highly socially embedded. I remember something about a second grade playground bead market at Ravenna during recess that spontaneously emerged and then spread until teachers banned it after a few weeks. It may be that there is a developmental phase around 7 or 8 in which the social embedding of trade is explored.

I would be interested in your anecdotes about young kids and card trading. I've decided to investigate further.

I should also say that this realization was inspired partly by Munir Fasheh's essay "Can We Eradicate Illiteracy Without Eradicating Illiterates?", an expansion on a paper given at a UNESCO meeting in Paris, on 9-10 September, 2002, to celebrate the International Literacy Day. The meeting was entitled "Literacy as Freedom."

In it, he dscribes his realization of his illiterate mother's mathematical sophistication:

My 'discovery' of my illiterate mother's mathematics, and how my mathematics and knowledge could neither detect nor comprehend her mathematics and knowledge, mark the biggest turning point in my life, and have had the greatest impact on my perception of knowledge, language, and their relationship to reality. Later, I realized that the invisibility of my mother's mathematics was not an isolated matter but a reflection of a wide phenomenon related to the dominant Western worldview. In this sense, the challenge facing communities everywhere, is to reclaim and revalue the diverse ways of learning, teaching, knowing, relating, doing, and expressing. This reclaiming has been the pivotal theme of my thinking and work for the last two decades.

My concern is not about statistical measures - for example, how many learn the alphabet - but about our perception of the learner and what happens to her/him in the process of learning the alphabet. My concern is to make sure that the learner does not lose what s/he already has; that literacy does not replace other forms of learning, knowing, and expressing; that literacy is not considered superior to other forms; and that the learner uses the alphabet rather than be used by it. My concern is to make sure that in the process of eradicating illiteracy, we do not crush illiterates.

In the 1970s, while I was working in schools and universities in the West Bank region in Palestine and trying to make sense out of mathematics, science and knowledge, I discovered that what I was looking for has been next to me, in my own home: my mother's mathematics and knowledge. She was a seamstress. Women would bring to her rectangular pieces of cloth in the morning; she would take few measures with colored chalk; by noon each rectangular piece is cut into 30 small pieces; and by the evening these scattered pieces are connected to form a new and beautiful whole. If this is not mathematics, I do not know what mathematics is. The fact that I could not see it for 35 years made me realize the power of language in what we see and what we do not see.

Her knowledge was embedded in life, like salt in food, in a way that made it invisible to me as an educated and literate person. I was trained to see things through official language and professional categories. In a very true sense, I discovered that my mother was illiterate in relation to my type of knowledge, but I was illiterate in terms of her type of understanding and knowledge. Thus, to describe her as illiterate and me as literate, in some absolute sense, reflects a narrow and distorted view of the real world and of reality. A division, which I find more significant than literate and illiterate, would be between people whose words are rooted in the cultural-social soil in which they live - like real flowers - and people who use words that may look bright and shiny but without roots - just like plastic flowers.

(It's a neat essay. Read the whole thing.)

Rudy Rucker, Live, in Santa Cruz

Rudy Rucker announces an interesting event that I'd love to attend, except that I'm on the wrong coast. But if you live near Santa Cruz, go and then tell me about it. He says:

I’ll be performing with Phil Curtis at a small-scale event called ELSA, that is, ELectron SAlon #11, on Friday, June 3rd , starting at 8 PM .

I’ll read my story “Ain’t Paint” which appears in my forthcoming The Lifebox, the Seashell, and the Soul.  Phil will create some appropriate heuristic electronic music on the spot, and for video, we’ll use live demos of nine of my CAPOW software Zhabotinsky scrolls; the guys shown below. . . .

Phil and I will go first, so if you want to see us, you actually have to be there at 8. Usually at ELSA events there’s some free wine and food. It’s almost like a party.

Set 1: Rudy Rucker and Phil Curtis
Set 2: Run Return, an electronica duo with Kevin Dineen and Tommy Fugelsang
Set 3: The inimitable DJess and mixmaster, Ms Pinky, a. k. a. P. Minsky, with friends.

The venue is Next Door, 1207 Soquel Avenue, Santa Cruz, CA 95062, 831-429-1596. Next Door is next to the Rio Theater, see map.

Real Life Hard SF

Yesterday, I came across a really charming anecdote that reads like hard SF, but is in fact non-fiction. It is from the question and answer session following astronaut Michael Foale's keynote address at the 10th Anniversary Mathematica Conference, Friday, June 19, 1998, published in The Mathematica Journal in 1999. Foale took has own laptop with Mathematica on it with him to the Mir:

I had Mathematica with me; I owned it personally. It wasn't even a copy that NASA had bought for me. And I had intended to work on tensor calculus in all that free time that I was going to have. And I had it along with my music CDs in my CD pack that NASA nicely made for me, in the Spektr module. I also had it on the hard drive, installed on a laptop in the Spektr module.

But there was an accident. An experiment in which the crew was to try to dock a Progress convoy vehicle to the station didn't work out and caused severe damage to the Mir.

A simple TV image was used to measure the rate at which we were closing in. That's "black ground rush" to a parachutist. As you come in closer, the image gets bigger, and you can try to use that to calculate what the speed is while at the same time deriving a closing rate. Then you figure out the docking, using a little joystick to fire the thrusters.

As you know from the media, this was a terrible mistake. It left the station not mortally, but severely, wounded. The Progress basically impacted, we think now, on this part of the solar array on the Spektr module, and then it bounced and slowly floated away along the base block.

The Progress weighs seven tons. We think it collided at about three meters a second. I was in the base block; I didn't see it at all. Sasha Lazutkin saw it; he told me, in all haste, to go straight to the Soyuz escape craft, and as I was passing into the node region of the Mir, I heard a big thump. . . .

It had hit the Spektr module. If we'd been strapped in, we'd have all been shaken around. This is just the opposite of being on Earth, where you're in a car and you're always supposed to strap in. Bash the space station, and nothing happens to you because you're not in contact with the station -- an interesting backwards twist.

Like any good hard SF protagonist, Foale set out to do a bunch of calculations aimed at solving the problems encountered by the crew of the crippled space station. (For those who want to know all about the calculations, the keynote speech discusses them in detail.) He whipped out his trusty slide rule. Well, no, it was a little more complicated than that.

First, the problem he was trying to solve:

The task was this: when you lose attitude control on the station, what happens? The station, low-powered, starts to tumble; then the solar arrays are no longer pointed toward the sun; and then slowly the batteries of the station start to deplete, because the solar arrays aren't charging the batteries. And then in about two or three hours you have no power on the station.

Gyrodynes, the momentum wheels, are always acting, spinning at different rates to change the orientation of the station very slightly. Once the station's lost all its power, or the guidance and control system has failed, the gyrodynes start to spin down, and that momentum gets transferred back into the station. It spins in the opposite direction to the gyrodynes.

Lo and behold, because you have twelve gyrodynes all spinning and working really well to do a nice job at holding the station in attitude, as the space station loses control of those gyrodynes and the gyrodynes spin down, then the space station picks up all of the angular momentum that was in the gyrodynes and starts to spin in the opposite way -- and in an unpredictable way.

So my whole task was to basically try to figure out what the rotation was, null it, establish an orientation, and then spin. But the problem with the station is that it has unequal moments of inertia.

So, hard SF readers. You're in a damaged space station and you need to do some calculations on your computer. But the power keeps going out. Your install disks for the crucial program flew out will the escaping air when the station was damaged. People on the ground are trying their darnedest  to help. What else can possibly go wrong. Read it and find out! Actually solving the equtions seems to have been the least of the problems.

(Did you know that the IBM Thinkpad warranty does not cover exposing your laptop to hard vacuum?)

PS: Further to the subject of math, check out the amusing flame war in the reader review section of the Amazon page on Stephen Wolfram's A New Kind of Science; the book, published 3 years ago, has 318 reader reviews so far.