"Babies who look longer at certain objects are counting, not just looking at new shapes and textures, according to a study by Yale University researchers."

Like their predecessors in similar studies elsewhere, these five-month-olds looked longer at novel stimuli, this time at multiple groups of objects with different numbers of groups than scenes they'd looked at earlier. The assumption here is that their increased attention means the babies were aware of the different numbers of groups (all other variables were strictly controlled), and so were counting.

I'm never really convinced that experiments like this measure what it's claimed they do. Were the babies counting or merely perceiving? It is known that humans and even some animals and birds can "subitize", that is they can instantly recognize small numbers of items. It seems much more likely that the babies merely recognized the different numbers of groups (no more than four) perceptually rather than by counting "one, two, three, four, ..." Counting is a significantly more sophisticated process than basic perception, and there doesn't appear to be any real evidence here that the babies were doing that.

The idea that basic arithmetical ability is somehow innate is a theme of several recent books, e.g. The Number Sense: How the Mind Creates Mathematics (Dehaene), Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (Lakoff and Núñez) and What Counts: How Every Brain Is Hardwired for Math (Butterworth). I don't find any of these really convincing. My own guess is that mathematics comes from a mixture of human perception of the external world and our need to order it somehow. In the unlikely event I ever understand the subject more clearly than that, I'll write my own book.

Stolen from Martin's Rants.

You find all sorts of stuff on the web. For those bibliophiles among you with a taste for the macabre, here's a piece about books bound in human skin - no, the practice didn't start with Nazi Germany.

Snow Crystals. "This site is all about snow crystals and snowflakes, and how these remarkably complex and beautiful structures appear, quite literally, out of thin air."

Interesting not only for the exquisite photos, but also for the description of how snowflakes form and how scientists create them in the laboratory. Almost makes me forget the hours spent shovelling my car out of the white stuff before I moved to milder climes. Almost.

I promised myself when I started this blog that I would refrain from writing "me too" posts, but this article by Andy Oram hits the nail too squarely on the head to resist. A one sentence summary: existing concepts of the semantic web - complex information tagged for organization and searching - are problematic; active collaborative filtering - exchanging information among individuals of like interests - is potentially more workable and more useful.

I have a visceral empathy with this: though I'm a trained mathematician, I instinctively resist attempts to formalize systems beyond their inherent natural structures. I was once involved in a project using OpenMath, a complicated system for organizing mathematics into "content dictionaries" to enable mathematical software packages to communicate with each other (amongst other things). The OpenMath structures grew more abstract with each iteration and it was never clear who was expected to write the necessarily huge number of dictionaries to make it all work. OpenMath appears to have descended into near oblivion.

My guess is that this fate will eventually befall the various initiatives to classify and archive learning objects. The schemes I've looked a bit at (CanCore, SCORM) don't appear to capture enough of the semantic content of LOs to make them worth the tagging effort. Nevertheless, LO proponents are trying to enforce adherence to such standards for anyone wanting to distribute online learning commercially. Where that leaves us non-commercial academic types is not yet clear.

Exactly. This is where I "me too"-ed the most. This is the peer-to-peer filtering meme, one of the main attractions of the web. Better than any search engine, it leads rapidly and efficiently to the information I need, and to the information I didn't know I needed until I found it. Unformalized, unauthorized, unclassified, ...

(Based on Guiding Attention Produces Inferences in Diagram-Based Problem Solving, by Elizabeth R. Grant and Michael J. Spivey, a paper presented at Diagrams 2002.)

"If we lead one's eyes to fodder, can we make one think?" Experimenters Elizabeth Grant and Michael Spivey at Cornell University investigated this question using a classical reasoning problem, Duncker's radiation problem:

The experimenters supplied participants with a diagram like the one above to look at. By recording eye fixations on the tumor, inside, skin and outside, they learned that those who succeeded in solving the problem looked significantly more at the skin than did those who failed. (Overall, roughly a third of the subjects succeeded.) The experimenters then looked at the effect of visually "pulsing" either the tumor or the skin, and found that, while the former had little effect on the success rate, the latter doubled it.

It's clear that the skin is a critical feature of the diagram; what's not clear is why. The skin is the boundary between the "harm" and "no harm" regions, but overall, the subjects looked more at each of the other regions and so presumably gained relatively little information from the skin. The experimenters conjecture than pulsing the skin caused more "in and out" eye movements in different directions across that boundary, which may have suggested radiation beams pointing from different directions. [Yes, that's the solution: point multiple low intensity beams at the tumor from different directions. Enough sources will make the intensity at the tumor high enough while sparing the surrounding tissue.]

The moral for those of us interested in designing educational diagrams? First be clear which areas of the diagram are relevant for learning (they may not be the obvious ones) and then focus visual attention there.

Diagrams 2002: an international, interdisciplinary conference on using diagrams in all fields of inquiry. I attended hoping to find new directions to pursue, both in geometry (my original academic specialty) and in education (specifically, I was looking for design ideas/principles for onscreen educational materials). I wasn't disappointed; it was one of the most interesting conferences I've attended in a long time and I left with a large number of ideas to follow up.

From the conference proceedings preface: "[Diagrams 2002] attracts researchers from a variety of academic disciplines who are studying the nature of diagrammatic representations, their use in human communication, and cognitive or computational mechanisms for processing diagrams." To my mind, these are precisely the disciplines we need to integrate to design diagrammatic educational content for the web, particularly complex content that must communicate its ideas in the absence of a teacher.

Abstracts for all the talks may be found on the Diagrams 2002 website, and detailed papers for the talks in the conference proceedings. It looks like there'll be a Diagrams 2004 - stay tuned.

Economist David Galenson's new book applying statistical methods to the analysis of avant-garde painting has met with profound indifference in academic cirlces, reports the Chronicle of Higer Education. The article proposes a reason: "The scholarly humanist's eyes tend to glaze over at the sight of such terms as 'labor markets,' 'independent variable,' and 'logarithm' ... Mr. Galenson forthrightly violates an unwritten principle of the humanities: The aesthetic domain must not be quantified."

Is anyone suprised by this reaction? Anyone in academia, at least? Humanists/artists and quantitative/scientific types have been at odds since well before C.P. Snow's The Two Cultures pointed it out. Different worlds, different mindsets; anyone who goes around in academic circles soon absorbs the appropriate perception. We scientific types have always known that those "artsie-fartsies" can't handle any sort of rigorous quantitative thinking, right?

Hmmm. I've been looking at web art recently. Web art combines text, sound, visuals and interactivity in multiple, sometimes complicated forms. My preconceptions are now in tatters: if artistic types can't handle quantitative thinking, how is it they can handle the programming it takes to produce works like those on the LeCielEstBleu site, for example? Flash and Shockwave are by no means simple to use; each requires a pretty daunting, technical scripting language to produce anything non-trivial. I'm impressed: this particular aesthetic domain may not be quantified or even quantifiable, but it does require a quantitatively capable mind to produce it.

Another stereotype bites the dust.

A visual poem, with words by Kahn Wong and artwork by Torisu Koshiro. I like the way the sound and the visuals give the sense of soaring through very empty space. My favourite line:

From Poetry on Display, on the BBC website.

In support of Derek Powazek's claim that people's dislike of onscreen reading is mythical, Mark Bernstein writes "People love to read on screens. If we didn't like it, why would we all spend so much time reading email?"

Answer: because most of us don't read onscreen, we scan. There's so much stuff coming at us online, we're too busy to do otherwise. Online writers know this, if only instinctively: keep the email or blog post short and focussed or risk having its message overlooked. For anything longer than a few paragraphs, be it email, web pages or PDF files, we scan for personally relevant words or phrases to decide whether to read more closely or not at all. People scan newspapers, too, paying attention to deliberate eyecatchers like headlines, photos and pull quotes to decide whether to read the details.

But the newspaper reader who decides to read more closely will read from the printed page, while the onscreen reader will more likely print the article to peruse later. (As evidence, I'll cite my own four-inch stack of print-outs, including Mark's Patterns of Hypertext.) The choice is typically ascribed to font readability, illumination, etc., but I think it's something more basic: ergonomics. Unlike shorter material like signs, package labels, etc., most of the longer print materials we typically read are placed on a desk, in a lap or in some other less vertical position where our neck muscles have less work to do to maintain our head angle over time. For longer articles, I read or type more comfortably looking at my iBook's tilted screen than I do at my vertical desktop monitor.

A research paper on the Nova Solutions website gives another slant on the subject. The vertical horopter is the vertical locus of points in space that the human binocular vision system fuses easily. Conditioned environmentally from infancy by close objects in the lower visual field and more distant ones in the upper, the human horopter tilts outwards at the top, indicating that computer monitors should do likewise. Not coincidentally, Nova Solutions sells desks which mount computer monitors at the appropriate angle.

"Even if YOU don't know what faith you are, Belief-O-Matic knows. Answer 20 questions about your concept of God, the afterlife, human nature, and more, and Belief-O-Matic will tell you what religion (if any) you practice...or ought to consider practicing.

Warning: Belief-O-Matic assumes no legal liability for the ultimate fate of your soul."

From Beliefnet. I apparently should be practicing Unitarian Universalism (whatever that is).

[Grab it quick - Beliefnet.com has just filed for bankruptcy]

Early applications of new technologies often tend to be "horseless carriage" (HC), i.e. they model themselves on older technologies and fail to use the affordances of the newer ones. The term, derived from early ideas of what an automobile should be like, is heard often nowadays in educational contexts, as in "the classroom metaphor for online education is so HC", for example.

The Froguts website allows students to dissect a frog online. The frog, of course, is virtual, as are the pins, scalpel and scissors used to do the job. The site has won recognition, and deservedly so; the issue here is not what it does or how well it does it as much as why it does it.

The point: why should a dissection be simulated online in the first place? As I understand it, students do (real) frog dissections either to learn dissection skills or to learn frog anatomy. For the former, an onscreen facsimile is at best a pale preliminary: a med-school bound college student will need the real thing, with real tools. For learning anatomy, simulated surgical dismemberment is unnecessary and irrelevant: a squeamish schoolchild could better explore the innards of an intact, "live" virtual frog through a click-driven unlayering of its working biological systems, or, even better, assemble a frog from those systems.

In principle at least, there seems to be little pedagogically advantageous for either student in mimicking a real world dissection online. Other than the HC comfort of the familiar for the teacher, that is.

Tom Lehrer, a mathematician turned entertainer from the '60s, wrote several humorous songs about mathematics. The best known is probably That's Mathematics - here's the text of another with (unfortunately) more than a grain of educational truth. From School Songs.

The Professor's Song

[Tune: "If You Give Me Your Attention" from Princess Ida (Gilbert and Sullivan)]

If you give me your attention, I will tell you what I am.
I'm a brilliant math'matician -- also something of a ham.
I have tried for numerous degrees, in fact I've one of each;
Of course that makes me eminently qualified to teach.
I understand the subject matter thoroughly, it's true,
And I can't see why it isn't all as obvious to you.
Each lecture is a masterpiece, meticulously planned,
Yet everybody tells me that I'm hard to understand,
And I can't think why.

My diagrams are models of true art, you must agree,
And my handwriting is famous for its legibility.
Take a word like "minimum" (to choose a random word),

[ This was performed at a blackboard, and the professor wrote: /\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/ ]

For anyone to say he cannot read that, is absurd.
The anecdotes I tell get more amusing every year,
Though frankly, what they go to prove is sometimes less than clear,
And all my explanations are quite lucid, I am sure,
Yet everybody tells me that my lectures are obscure,
And I can't think why.

Consider, for example, just the force of gravity:
It's inversely proportional to something -- let me see --
It's r cubed-- no, r squared -- no, it's just r, I'll bet --
The sign in front is plus -- or is it minus, I forget --
Well, anyway, there is a force, of that there is no doubt.
All these formulas are trivial if you only think them out.
Yet students tell me, "I have memorized the whole year through
Ev'rything you've told us, but the problems I can't do."
And I can't think why!

I've been playing recently with a cute Mac OS X toy called Tape Measure X. Its function is simple: to measure distances on a computer screen. What interests me most is its visual transparency: not only can the backgrounds of the individual arms be adjusted to any degree of transparency (their markings always remain visible) but the whole object itself can be made arbitrarily transparent. The effect is not unique to Tape Measure X; it's part of the Mac Aqua GUI, and appears in other OS X applications.

Transparency has interesting possibilities for onscreen educational content design. Imagine a human anatomy lesson in which the student could make the ribs transparent to reveal the lungs. Visuals with transparency preserve contexts (e.g. just where are the lungs positioned relative to the ribcage?), something that mere layer manipulation (hide the rib layer entirely) doesn't do as well. Student-controlled transparency would also allow more natural "discovery" explorations, beyond the "click here to see the lungs" genre.

Are there examples (online or CD-ROM) of successful transparency use in educational contexts? More important, are there interface design principles yet for transparency use? I couldn't find anything in Apple's Aqua Human Interface Guidelines beyond the tehcnical stuff. Tufte's ideas on the visual display of content vs context could apply here, for example, modified for dynamic media instead of static. Ideas or information, anyone?

Most of us (my generation, anyway) learned in geometry class how to use the standard tools (straight edge and compass) to bisect angles, draw equilateral triangles, etc.. Some of us went on to learn why certain constructions (trisecting angles, duplicating cubes) are impossible with the standard tools.

But suppose we change the tools, to something as deceptively simple as paper folding (origami). An axiom system for origami geometry was developed by the Italian-Japanese mathematician Humiaki Huzita in 1992. By judicious use of these axioms, it is indeed possible to trisect angles and duplicate cubes. This page shows how.

Walter is Director of Global Marketing, Research and Education at SGI.

"The Personal Identity Game

Objective: The aim of the game is to stay alive!

How to play: There are three rounds. In each round, you will be presented with a scenario and then offered two choices. The decisions that you make determine whether you stay alive or perish. You should always base your decisions on nothing more than the desire to keep yourself in existence. ...

At the end of the game you will discover if you have stayed alive or not, although, being a philosophical game, the answer won't be that straightforward... "

From the Philosopher's Magazine Online. I survived!

"Communication is king, not course content."

A statement originally about online publishing re-jigged by Stephen Downes (OLDaily, March 29) to apply to online education. This one irritates me every time I hear it (and that's often nowadays): it seems that, as long as an online course is set up with conferences, email, chat and other forms of interpersonal communication, the content and how it is communicated is unimportant, or just "happens" somehow.

I first encountered this attitude during a pre-conference workshop some years back (run by Stephen, coincidentally). New to the game of online education and prattling on enthusiastically to a fellow participant about new forms of communicating mathematical content online, I was told in no uncertain terms that that was not "real" communication; real communication was only between people, and I should read so-and-so's definitive book on the topic. (She didn't quite say "so there", but it was in her tone.) She wasn't alone: of about 40 of us there, only one other appeared to have any interest in using technology to communicate content, a geologist who explained plate techtonics to her students via an applet. The others, primarily from professional or "discussion-based" subject areas, saw only the potential for enhanced interpersonal communication.

My reaction at the time was "but that's nothing new; they teach by discussion in the classroom, all they're doing is moving it online. Better and faster in some respects, but the pedagogy's the same". In the intervening years, I've come to appreciate more of the issues involved in discussion-based pedagogy, and have seen the education community develop theoretical and practical foundations for its use online. But I've yet to see the education community exhibit much of an appreciation for online content design beyond porting paper content to web pages. Either it's assumed that the medium makes little difference to how content is communicated, or the issue is avoided altogether in the expectation that someone else (outside the education community) will figure out how to do it right.

And in case you're wondering, yes, that's my main interest in online/onscreen education: how do we use the medium to design effective, engaging educational content? There are issues here, folks.

Need to relax after a hard day? Need to relax before starting a hard day? Try these all too brief moments from Abbey National Bank. "Because life's complicated enough."

I watched the Oscars a week or two back, or rather, they were on in the background while I was doing other stuff; I wasn't really paying attention. One thing that did catch my eye, though, was the performance by the Cirque du Soleil. It struck me as one of the few examples I've seen of just what multimedia should be - integrated media - rather than what is usually called multimedia, but is often merely simultaneous multiple media, each engaging in its own right but unconnected to the others.

If you missed it, there were the usual Cirque-type acts - clowns clowning, acrobats leaping, etc. - all performed with the elegance one expects from this troupe. In the background, on huge screens, were fast-paced cuts from various films, most brief enough to be unidentifiable, but - and this is the important point - visually integrated very closely with what was going on onstage. So an acrobat wheeling around on one of those double wheeled things (do they have a name?) was mirrored on screen by the rotating space station from 2001 (I think it was), and another spinning large hoops was accompanied by what looked like rings of electric energy from one of those generator things popular in old science fiction movies. And so on: film clips onscreen visually reflecting (and thereby enhancing) the action onstage. Someone put a great deal of thought into that performance.

Very little multimedia I come across on the web or elsewhere is so well integrated. It's understandable, I suppose; most of what's happening with the new technologies is still very experimental, whether for the arts, for teaching or for other forms of communication. There are few clear criteria yet for how onscreen visuals should be used to enhance a poem (few do), or how interactive simulations should be used to enhance mathematics learning (most don't). Perhaps there never will be clear criteria - perhaps media integration is destined to be more of an art than as science, or a craft, discovered by trial and error and passed along informally to apprentices. Whichever it is, I find the whole subject fascinating, and one I'd like to learn much more about.

Welcome to the inaugural post of Afterhours. I'd like to tell you what this weblog is going to be all about, but I'm not sure I can at this point. I have an eclectic collection of interests in education, mathematics, design, art and other subjects, and I'm sure I'll write about most of them eventually, especially about how they play themselves out online. But to a large extent, I expect the content to evolve over time into something other than I presently conceive it, or even other than I can presently conceive it.

A major purpose of this blog is education: mine. Other than a Ph.D. in mathematics and a quarter century teaching it, I have no formal training or qualifications in these subjects. I've learned a great deal over the past several years simply by reading what I find on the web, going to conferences and working in academia. From maintaining this blog, I expect to learn more, both from the writing itself (one of the best ways I know to understand anything) and from interactions with you, the readers.

Postings will probably be sporadic until I get my numerous notes transcribed. To organize them, I'm using Tinderbox, an innovative personal content management system from Eastgate that fits my working style perfectly: lots of "scraps" of ideas organized associatively rather than hierarchically, which Tinderbox's attributes and agents pull into coherence. An amazing tool for working with "piles, not files"!

I encourage you to email me about any of the posts I make here (click the "Comment?" link) or about any related topic you think I might be interested in.