omenti

types of stupidity

Recently Pip Coburn circulated a note on the importance of finding the right balance of challenge for your skill level to avoid boredom and frustration.  He noted the linkage to Csikszentmihalyi's work on flow, something that's fascinated me for some time as finding states of flow is important to me.¹  

The note made me think about levels of frustration and feelings of 'stupidity' one can have.  For some time I've identified types of stupidity with one being very useful and perhaps necessary in science.  Here's my response:

 

It’s fascinating to think about these things. Your comment on loving the work - “absolutely loving” it - is so important.

I think it’s important finding a field that no one can possibly master but, at the same time, know parts of it well enough to find where you may have a bit of success in if you apply yourself. As a beginning grad student I had no idea how hard it is to do research. It’s much more difficult than the most demanding courses as it’s an immersion into the unknown. Some people just give up and find easier paths. I find it useful to think about how to be productively stupid.

’Stupid’ is an unfortunate word, but I haven’t found anything better. Productive stupidity isn’t the relative stupidity you may feel in a a class where the other students are doing very well. It’s also not where someone who happens to be very bright is working in an area that doesn’t use their talents. (Stephen Hawking would have been a terrible point guard in the NBA even though the position demands strategic and tactical thinking.) Rather it’s a kind of existential fact .. an absolute stupidity. It puts you in the position of being ignorant by choice (here I use 'ignorant' in the 19^th century sense - realizing there's an area you don't know much about and choosing to pursue it). So it’s fine to have failures here and there as you try to find your way. And sometimes, something wonderful is found. The process can generate serendipity. Of course you end up doing many less challenging things as part of the process. They’re often rewarding, but it’s amazing when you find something totally new. It’s what drives a lot of people.

I’ve had some great conversations with people who push their fields and hear the same thing with different words. Yo-Yo Ma, Sarah Pavan (beach volleyball Olympian), a number of physicists and mathematicians as well as several artists.²  The words were different, but the paths were very familiar.

There’s another piece to this.  Bringing together a very diverse set of people who excel at productive stupidity is something of an amplifier. Often there are interesting hints and even keys that you never would have thought of. That and it’s often a lot of fun.

__________

¹ Years ago I attended a talk by Mihaly Csikszentmihalyi.  Afterwards I had a few questions including "how do you pronounce your name?  He said this is good enough for Americans:

Me-high Cheek-sent-me-high

² There are many others who probably do something like this .. these are just the people I've talked about it with.

 

 

on the visualization of reality

The chunks of plastic held the ghosts of magnetic fields produced by some of our gadgets and appliances.  Shuli's pieces were quite beautiful and had me thinking about visualization on the train ride back home.   

We only see fragments of the real world, but our brain stitches what it can find together into a fleeting proxy for reality.  It's turned out to be good enough of what we do, but I recommend An Immense World by Ed Yong for a look at how evolution has equipped other lifeforms.  (It's a wonderful science for a general audience book - my favorite for 2022)  Science tries to sort out what is real and physics often seems foreign to what we're used to. On the train ride home I was thinking about how we visualize electromagnetic radiation - specifically a question someone had asked a few years back.

What would wifi look like if we could see it?

It seems relevant if you're trying to figure out how walls, refrigerators and other objects in a house or office interact so we know what the signal strength might be.  People who do this either just measure and plot signal strength or have simple models to give them a heat map that hopefully approximates what they would measure.   None of these gets at the real question.. what do wifi signals .. and all the other electromagnetic radiation that surrounds us look like?

 

The equations are straightforward and I suspect many imagine what it might look like when doing calculations - at least the parts we're looking at.  But these representations aren't rich enough.  The real appreciation is in the mathematics.  Feynman commented on it back in the 60s:

I have asked you to imagine these electric and magnetic fields. What do you do? Do you know how? How do I imagine the electric and magnetic field? What do I actually see? What are the demands of scientific imagination? Is it any different from trying to imagine that the room is full of invisible angels? No, it is not like imagining invisible angels. It requires a much higher degree of imagination to understand the electromagnetic field than to understand invisible angels. Why? Because to make invisible angels understandable, all I have to do is to alter their properties a little bit-I make them slightly visible, and then I can see the shapes of their wings, and bodies, and halos. Once I succeed in imagining a visible angel, the abstraction required-which is to take almost invisible angels and imagine them completely invisible- is relatively easy. So you say, "Professor, please give me an approximate description of the electromagnetic waves, even though it may be slightly inaccurate, so that I too can see them as well as I can see almost invisible angels. Then I will modify the picture to the necessary abstraction."

I'm sorry I can't do that for you. I don't know how. I have no picture of this electromagnetic field that is in any sense accurate. I have known about the electromagnetic field a long time I was in the same position 25 years ago that you are now, and I have had 25 years more of experience thinking about these wiggling waves. When I start describing the magnetic field moving through space, I speak of the E and B fields and wave my arms and you may imagine that I can see them. I'lI tell you what I see. I see some kind of vague shadow, wiggling lines-here and there is an E and B written on them somehow, and perhaps some of the lines have arrows on them -an arrow here or there which disappears when I look too closely at it. When I talk about the fields swishing through space, I have a terrible confusion between the symbols I use to describe the objects and the objects themselves. I cannot really make a picture that is even nearly like the true waves. So if you have some difficulty in making such a picture, you should not be worried that your difficulty is unusual.

Our science makes terrific demands on the imagination. The degree of imagination that is required is much more extreme than that required for some of the ancient ideas. The modern ideas are much harder to imagine. We use a lot of tools, though. We use mathematical equations and rules, and make a lot of pictures. What I realize now is that when I talk about the electromagnetic field in space, I see some kind of a superposition of all of the diagrams which I've ever seen drawn about them. I don't see little bundles of field lines running about because it worries me that if I ran at a different speed the bundles would disappear, I don't even always see the electric and magnetic fields because sometimes I think I should have made a picture with the vector potential and the scalar potential, for those were perhaps the more physically significant things that were wiggling.

That said I find myself using my mind to fly through through these abstractions of reality that experience and imagination provides.  I suspect people do this differently.  It isn't something that lends itself to drawing as it's often dynamic and, even though I consider myself extremely visual, not entirely visual. While I find many computer visualizations useful, they don't keep up with the imagination.   I suspect no two people imagine quite the same. 

 

A few nights ago Sukie and I were startled to learn something about her visualization. I was listening to The Case of the Blind Mind's Eye - an episode of an excellent BBC science and math podcast.  I strongly recommend listening to the episode.  They were talking about the mind's eye .. how we conjure up images in our mind.  If I ask you to think about an orange giraffe with purple spots, an image appears in your mind.  Or at least it does for most people.   Aphantasia is a semi-rare condition of people who are unable to generate these images.  It's completely fascinating and you need to listen to the show.  Sukie heard a few comments and had me go back to the beginning.  It turned out she has aphantasia - she's unable to form mental images and is startled that others can.  She's suspected comments about the ability are just a figure of speech.   

 

Aphantasia is more of a difference than a defect.  Sukie is extremely clever and has done impactful work in a few areas.  It turns out there are benefits..  it's another way the brain can imagine.  Ed Catmul (Pixar) has it and managed to create a revolution in how computers create images.  The same for Glen Keane - the Disney character artist who created Rapunzel, Ariel and a score of other important characters.  He describes his process as "thinking with a pencil". The shapes and forms come as he draws in a very fluid and gestural motion. 

 

It may be a spark, perhaps a necessary one, for very out of the box thinking.  In the meantime artists and the process of doing art can unlock the imagination.  It's not a full representation and perhaps that's what makes it so powerful.  It raises so many questions!

 

 

 

six is a surprising number

Ting-Chang Hsien was a dear friend when I was in grad school.  At the time China was beginning to send a few scholars to work with American counterparts and Ting-Chang came to Stony Brook for a few years to work in an area of particle physics the school was noted for.  Near the end of his stay     he decided to see America.  He flew to my parent's home in Montana and, renting a car, spent several weeks traveling through the West.  The day before he flew back to New York my parent's neighbor, Lloyd Erickson, stopped by to chat.  Extroverts by nature, Lloyd and Ting-Chang dove into conversation and homemade ice cream.  Ting-Chang mentioned as a teenager he was a student in Shanghai with an interest in music as well as math.  The school was run by missionaries - his mother was a devout Christian - and his favorite teacher told him he had real talent for math and should give up the piano.  He said "she was from Minnesota .. one of her brothers was technical and worked for a company there."  Lloyd sat up,  "I had an aunt who was a missionary-teacher in China.. she left before the war and and got married."  Ting-Chang mentioned the name of his teacher.  She was in her late 80s and still alive.  Lloyd brought out family photos.  That night old teacher and the physicist talked on the telephone for a long time. 

 

We all know it's a small world.  We find common connections with total strangers even if we don't consider ourselves well connected.  I tend to be shy and don't have that many friends, but continue to be surprised.  Of course there's the Six Degrees of Kevin Bacon.

 

Steven Strogatz does applied math at Cornell.  In the mid 1990s he and his student Duncan Watts were interested in the problem of synchronization.  How is it that some insects like beetles manage to synchronize their sounds at night?  Why is it that types of glowworms blink in unison?  Why is it .. there are many of examples of this in nature ...  They weren't trying to solve the underlying science, but rather were looking to see if there was a deeper mathematical underpinning.  Cornell has lots of beetles in the Summer.  They could make measurements. 

 

They considered two extreme cases.  One is very regular.  Think of a checker board.  Each square on a checker board filled has an immediate group of neighbors - small neighborhood cluster.  To "talk" to square on the other side there are about eight degrees of separation.  Each square is only directly connected to the nearest neighbors.   This kind of model is well-studied in physics where a force may only have a very short range. None of us have connections that limited.    Next consider complete random connections. If you know 150 people, each of them would know a similar number so the total number of people connected by two degrees of separation is about 150² - 22,500.¹  Most of us know that's wrong (except for some early models of Internet connectivity:-) 

 

Their gut feeling was the real answer would le somewhere in between.  Technically they were looking for the phase transition .. the point at which most of the beetles were suddenly in synchronization, or everyone seems to have these seemingly random connections. The small world problem.  What they found was a fairly simple mathematical description that said you only need a small bit of randomness.  In your circle of friends an outlier or two.. or knowing someone with such connections.   

 

Their short paper showed examples ranging from power plant distributions to noise making bugs and is one of the one hundred most cited papers in all of science and math.   They also noted that given connections afforded by travel these days pandemics could spread rapidly.  It's very difficult preventing that phase transition.

 

The small world problem is very beautiful.  It's one of those topics that should be taught in a course that doesn't seem to exist.  In college you get music and art appreciation.  I'll never do good art or music, but 101 level non-major courses gave me a better appreciation for those very human endeavors.  

 

Math is important enough that students in high school get either working on mindless problems they'll never see again or pre-STEM work that only a small percentage will use.  People are properly recommending and even building 'data science' courses.²  These seem like a great idea - familiarity with tools, presentation and interpretation - but I'd like to see a math appreciation path.

 

Math appreciation would have history from a number of civilizations.  Was math the first example of external memory?  Why did people use different number systems?  Why was double-entry accounting so important to Italy?  What about zero?  Imaginary numbers?  The Fibonacci sequence in nature.  Why does a slice of pizza fold the way it does?  What is infinity, are there different kinds?  How are coffee mugs and doughnuts similar? What is the fairest way to vote in a democracy? It's easy to think of dozens of lecture topics that might even inspire a few students and create links to what they finally go into.

 

Just an idle dream I guess..

__________

¹ Apologies to Robin Dunbar - he'd say the number is almost always used out of context anyway).  

² I hate the term.  Generally anything that adds 'science' to a label isn't science.

 

unexpected arcs

Three weeks ago breathless reports on the 'breakeven' fusion event at the National Ignition Facility began to appear.  While a remarkable achievement, it's more a marker of progress along a long and difficult path rather than an event promising a hopeful turn in decarbonizing the energy supply. 

 

Nuclear fusion has been hyped since the late 1950s.  Fission based reactors turned out to be straightforward.  All you had to do was refine uranium ore and then control a reaction to heat water and run an otherwise conventional power plant.  The cold war led to any number of atoms for peace projects suggesting a nuclear everything future.  The realization that nuclear fusion was much more efficient and "cleaner" than fission led to optimism and work on controlled fusion.  After all, it happens all the time in stars doesn't it?

 

Power from nuclear fusion has been twenty years in the future ever since.   Science and technology have made serious progress, but the challenges are great.  There's also the issue of cost per kilowatt-hour. If you build a plant that produces more power than it takes to run it, will it be economically competitive with other low or no-carbon solutions? 

 

There's a simple invention → development → application → change arc of technology mindset that doesn't describe the real world. It turns out you can travel a good distance along the arc only to see failure.  The garage-worthy helicopter, supersonic airliners and dirigibles have all be built and commercialized, but have failed for a number of reasons that are likely to keep new versions failing for the foreseeable future.  Somehow the dream factor persists in many of these and new failures continue to rhyme.

 

There are hugely successful inventions that have solved a problem, but turn out to have a deadly flaw that caused great harm.  DDT,  chlorofluorocarbons and leaded gasoline are obvious examples.  Less obvious are inefficient transportation.  Standardizing on 3,500+ pound vehicles to move people a handful of miles on average trips has led to millions of traffic deaths, a large pollution toll and a structuring of infrastructure and where and how people live that is proving difficult to change even though much better choices exist.  Sometimes tend to fire, aim and then look for damage around the target.  Sometimes it takes decades and change is expensive. 

 

There are a wishlist class of inventions where some progress is made, but effective deployment is a long way off (fusion power plants)  or unlikely as the underlying approaches are flawed at this point (self-driving cars, hyperloops, neural implants, space elevators, nuclear powered cars and airplanes, etc.)

 

It's very easy for an imaginative artist or writer to sketch an idea that is technically impossible.  We have a tendency to sort through the chaff and find amazing predictions from a hundred years ago.  It's useful to realize how much noise there is and that these things didn't happen in the author's or their children's lifetimes.  Still - every now and again you run across something that stops you.  This is one of my favorites - not only because it suggests wireless video communications, but it also nails the social element of two people ignoring each other so they can look at a screen.

 

Then there's a class of necessary arcs that we need to get past application and into wide-spread use.  Flexible power distribution to move power from generation to where it can be stored or used.  An advanced worldwide pandemic monitoring system.  Water management and treatment to deal with changing local climates as well as use needs.  More efficient agriculture that can deal with changing climates.  Changes to transportation and housing infrastructures that are less damaging.  Some of these aren't terribly sexy, but require social and political will - areas where progress is often very difficult.  And in the background there needs to be investment in fundamental science.  While progress can be difficult to predict, historically it's easy to predict that there will be unexpected discovery that can lead to something useful down the road.

 

It's fun to enjoy the stories writers and artists weave.  Creating your own is a great way to work with the imagination of a nine year old on a rainy day. Time and interstellar travel won't come along with scores of other ideas that help move stories along.  

 

 

fusion!

The short version is it’s a remarkable technical and scientific achievement. Controlled human-made power from fission is easy - last week marked the 80th anniversary of the first reactor. Fusion - what powers the stars - is much more difficult on Earth. The necessary temperatures are far greater than those found in most stars. A star like the Sun gets away with a very slow and low power density reaction that would be impractical for generating power on Earth. It turns out the power density of the core of the Sun is about a quarter of your metabolic density and close to that of a lizard. To get around this different fuels are needed along with much higher temperatures. Still, if you can only figure it out, it sounds like the perfect goal - fuel from water and no radioactive waste.

 

The first serious attempt at magnetic confinement look place about 60 years ago .. it, along with every other effect until this last week failed. Those that achieved fusion required more input power than the power they produced. The figure of merit is Q: the ratio of the power of the fusion reaction divided by the power required to ignite and sustain it.

 

The laser confinement experiment announced focused a short pulse of light - about a billionth of a second - from 192 lasers onto a small gold container that vaporized to generate x-rays that collapsed a diamond coated fuel pellet of deuterium and tritium. The power of the laser light was 2.05 MJ (megajoules — a gallon of gasoline is 121 MJ, a hot dog in a bun is about 1.5 MJ) and the fusion explosion yielded 3.15 MJ. A Q of about 1.5. What isn’t mentioned is the efficiency of the lasers or the facility isn’t included - or the efficiency of extraction power from the miniblast. The lasers are less than one percent efficient so you can see there’s a long way to go.

 

This facility has been running for about a dozen years. It has been solving enormous engineering and applied science problems along the way as well as contributing to the pure science of understanding plasmas at this scale and temperature. Even the fuel pellets are very difficult to make and test. They have to be extraordinary symmetrical and most are rejected in testing - usable pellets are very expensive.  What has been achieved is not a prototype, but rather a triumph of instrumentation technology along with the control of some difficult to manage parameters. 

 

Moving this line of fusion forward seems like a long shot. A factor of five improvement may be possible with a redesign, but they need more than a thousand. Not impossible, but there’s a long path ahead.

 

There are other design types that confine microwave heated plasmas. Getting it right and working will take many years - probably decades and then there’s the issue of what does a facility cost compared to other types of power stations.

 

I hope work goes forward, but not at the cost of workable technologies that address global warming *now*. There’s so much that can be done, but very little political will.

 

2 december 1942 - a half watt

4:00 PM 2-Dec-1942

Compton: The Italian navigator has landed in the New World.

Conant: How were the natives?

Compton: Very friendly

 

Almost exactly 80 years ago the first human-made self sustaining nuclear reaction took place under the stands of a vacant football field at the University of Chicago.  It's a remarkable story and something of a race with Nazi Germany.  Fortunately so many competent physicists had fled Germany that the Nazi bomb project was a failure.  It turns out the wikipedia article on the subject is an accurate and readable summary of the dawn of the pre-manhattan project effort to get a handle on nuclear fission.  The piece  gets a bit technical here and there, but you can skip over those parts if you like.  I'll try and give a high level view of what a sustained reaction is without going into the quantum mechanics of the process.

__________

 

Nuclear fission is the process where the nucleus of an atom breaks apart into smaller pieces releasing energy (often quite a bit) in the process.  The nucleus of an atom is a tight cluster made up of positively charged protons and electrically neutral neutrons.

All of those positive changes packed together - you might ask why don't they fly apart? After all, electromagnetism causes like charges repel.  It turns out a very strong force, cleverly called the strong force, acts on neutrons and protons binding them together.  A difference between the two forces becomes important.  Electromagnetism is a long range force while the strong force only acts over short subatomic distances. To first order you can think of the strong force acting only between immediate neighbors while the electromagnetic force acts on all of the protons.

If the nucleus gets big enough the total repulsive electromagnetic force of all of the protons overcomes the closest-neighbor strong force and breaks the nucleus up in to smaller pieces releasing energy in the process (nuclear fission)

Most of the elements we're used to have nuclei that are too small to break up on their own.  A few are on the borderline. In the case of uranium the addition of a single neutron is enough to undergo fission and in the process it releases energy and, on average somewhat more than two extra neutrons along with a couple of nuclei that are smaller than uranium.¹

In nature uranium usually isn't concentrated and the neutrons from fission are absorbed by other materials.² But if you concentrate uranium you get to a point where each fission has a high probability of striking another uranium nucleus and so on - creating a sustained reaction. Unchecked, with enough purity, and you get an atomic bomb. At lower concentrations and with the ability control the amount of neutron absorption you get a controlled nuclear reactor that liberates a useful amount of energy.

 

A number of interesting characters were involved in these early experiments.  Enrico Fermi was the key player in the first sustained reaction.  Physics was getting specialized and he was probably the last physicist who was both a great theorist and experimentalist. 

__________

¹ I've left out a lot.  I haven't mentioned which isotope of uranium for example.  Also - you may have noticed that adding an extra neutron shouldn't increase the electromagnetic force as the neutron is neutral.  It turns out the uranium nuclei is so close to breaking up that the kinetic energy of the addition neutron adds just enough to break things up.  A nice example is adding a neutron to U-235.  One possible fission chain  n + U-235 → Rb-92 + Cs-140 + 3n + 200 MeV of energy. 

Also details of the strong force weren't known at the time..  A clear hypothesis didn't arrive until about 30 years later and deeper details of the mechanism took almost thirty more..

² A natural formations of uranium t pure enough to initiate sustained nuclear reactions was discovered in Gabon.  It operated for a few hundred thousand years producing less than a megawatt of power.  

 

 

a delightful word

Mr Gross was my high school German teacher.   He was big on learning language through song and would regularly point out a German word or phrase that might be more appropriate than American English. I wasn't a good student of the language, but he made an impact on me.  I still remember and use some of his offerings.  Today I saw one of my favorites used in the New York Times - the first time I've seen it used in America.  

freudenfreude

Mr Gross described it as the opposite of schadenfreude. It's the joy and delight we feel when someone else finds success - even when, particularly when,  their success doesn't involve us.  He used to describe the joy he felt when a blind student from our school won a state piano competition.   It's an important social glue - a gift to be aware of and take delight in.  Mr Gross has been gone for a long time, but thinking about this word turned out to be one of the best things I learned in high school.

 

28 years!

This is the original holiday card I sketched back in 1994 along with a bit of midi. I borrowed a Wacom tablet and made a few strokes with the pen - a minute's worth of work.  The music took longer.  The web was new so I posted it on a server under my desk at work.  I didn’t have enough time or imagination to make some new the next year so it returned unchanged.  The third year rolled about and I would have forgotten about it, but there were a few dozen requests.  From then on it went up every year on or soon after Thanksgiving in honor of the Winter Solstice holidays.  It’s a survivor having been hosted on five different servers over the years.  For the silver anniversary I though about drawing something new and held a vote.  The old card stays - it was unanimous.

Now it’s 28 years old.

the original holiday card

Autoplay didn’t exist then so you’ll have to start the music yourself.

Happy Holidays! - peace, friendship, and health to all.

Steve

 

 

finding what works

 

It’s a clear winter night as I write these last words. I’ve stepped out to look at the sky. With the stars up above and the blackness of space, I can’t avoid feeling awe.

How could we, Homo sapiens, an insignificant species on an insignificant planet adrift in a middleweight galaxy, have managed to predict how space and time would tremble after two black holes collided in the vastness of the universe a billion light-years away? We knew what that wave should sound like before it got here. And, courtesy of calculus, computers, and Einstein, we were right.

That gravitational wave was the faintest whisper ever heard. That soft little wave had been headed our way from before we were primates, before we were mammals, from a time in our microbial past. When it arrived that day in 2015, because we were listening—and because we knew calculus—we understood what the soft whisper meant.

                 Steve Strogatz 

 

Chatting with a friend the subject of gravity came up.  It isn't a conventional force, but we perceive it as one.  Rather it's connected with the curvature of a four-dimensional object called space-time that we happen to live in.  The fundamental concepts aren't intuitive and the math that describes it is beyond what you might get in an engineering degree.  So how to describe it?  There are very high level books and videos that speak in terms of balls on rubber sheets and then jump to oddities like clocks running faster on mountain tops and blackholes.  I usually find them a bit fluffy and disappointing.  They're bound by assumptions about their audience, so it's not a major criticism. It's just that I was talking with one person and something a bit deeper seemed appropriate.  Just how do you find the right level and where do you go from there?

 

A few years ago Wired produced the 5 Levels series.  An expert would try to explain something about a complex subject to five people:  a child, a teenager, an undergrad majoring in the same subject, a grad student and, finally, a colleague.   My favorite is Donna Strickland on lasers - give it a watch, she's excellent:

I went through something like this just before my Ph.D. thesis defense.  I was to explain my thesis work to a group of high school students. At first I thought it would be easy, but that notion quickly evaporated.  It turned out to be one of the most difficult and embarrassing things I've done.  My advisor stepped in near the end of the talk and summed everything up beautifully.  Afterwards he told me if you understand something deeply, you can explain the gist of it to a high school student.  That's high on the list of the most important things I've learned. 

 

My friend has a BA in biochemistry and knows about differential equations so I figured I should aim for something between the vague videos and what a senior level general relativity course offers.  The more I thought about it I realized it would be best to talk about the history and use a bit of calculus and geometry.  I had the advantage of knowing her well and she can stop and ask questions or give a blank look along the way.  And afterwards she was going to talk about something where she has serious expertise. 

 

Steve Strogatz is an applied math professor at Cornell. He's written several books including the one the opening quote is taken from: Infinite Powers: How Calculus Reveals the Secrets of the Universe.   It's an excellent read that makes no assumption on your mathematical background.¹  Thinking about it gave me some hints of how I might proceed.

 

First why do I have to use math?  Why won't words work?  The laws of nature obey logic that we can make predictions from.  Math, calculus in particular, is a logical calculating tool.  In a way it's a prothesis. Math lets us take a bit of logic, write down and perform logical manipulations that far exceed what we can do in our head, and then interpret the results. The logic can be crafted to represent something about the science.   Every once and awhile the results can make predictions that can lead to new discoveries, but more often they're used to to solve an enormous range of problems.   And why calculus?  Much of the  underlying structure of nature has been successfully expressed with the corner of calculus known as differential equations.  "Why?' is a deeper question.  If one encountered an intelligent alien who understood some aspects of how the Universe worked, I suspect they'd use math.  I suspect, but it's only my suspicion, that it's deeper than an artifact of how we think about things. 

 

I spent a couple of hours writing.  How general relativity came about, a bit on the structure of space-time, what goes into the main equation and how a simple prediction could be calculated.  Then about an hour of chatting that left me with two delightful philosophical questions that will probably lead to more discussions as well as her turn to teach me something. 

 

Whatever your expertise, it can be fun to try and explain the gist and maybe even the beauty of it (those can be he same thing) to someone with a very different background.   If you're like me you'll fail at first, but eventually you'll get to a point where you can find the right grounding, the right words and perhaps the right drawings or even music (some of you are artists and musicians).

__________

¹ By all accounts he's a wonderful teacher and has become a popularizer of math.  Among other things he's created a college course at Cornell for people who think they're afraid of math and have generally put it off until their senior year. 

 

 

 

 

teaching and technological change

On Saturday my niece sent a few photos from her son's robotics competition.  While standardized components and a simple scripting language are used, there's a lot of room for learning and creativity.  Winning teams spend much of their time modifying the basic design along with moving to more sophisticated programming languages.  The basic hardware is quite expensive so many of the groups are sponsored by their schools and sometimes local businesses.  It seems like a great way for kids who like to build things and play computer games to learn a bit of engineering and programming.  My niece's son isn't interested in math or science at school, but loves to build things.  It's a path that has him learning.  Given the right advisor there could be a lot of learning.  It's also a nice example of using technology as a teaching vehicle.

 

Sometimes new technologies are seen as a way to cheat.  I've read that some teachers worry that language models like GPT-3 will give students easy access to ghost-written essays.  Of course there's still the problem of students buying essays from human, but this may be cheaper and easier to get.  One can imagine a number of ways to deal with this.  A teacher might ask something like "based on the debate Adam and Sally were having last Tuesday, how would you.."   Indexicality - pointing to something in the context in which it occurs - is something these language models can't address.  Better still may be to focus on critical thinking in the class and test on something that isn't a simple regurgitation. Essays seem like a very uncreative way to teach writing and critical thinking skills.   And there will probably be teachers who figure out how to involve language models as a tool to teach creative thinking.  That would be a leap similar to using robotics competitions to teach shop, basic engineering and programming skills.

 

I'd like to see changes in how science and math are taught in high school.  Currently there's a push to eliminate areas most students will rarely use.  I'd counter that by teaching the subjects in a way to improve critical thinking.  Pure math may be a way forward, not the "new math" that crashed and burned in the 60s, but big ideas.  It doesn't have to be about formal proofs and calculations (although tools like Mathematica and Maple can be useful experimentation and visualization tools).  I've probably belabored it before, but simple topology, infinity, the continuum, maps, abstraction, inference and model building. None of this has to be taught with rigor, but playing with them can open pathways students have never thought about.   Sure teach some of the regular math, but it doesn't have to be so repetitive. I'd add probability and risk analysis as well as personal finance. The same with science - do away with memorization and focus on the concepts and how we arrived at them.  Of course extra points for making it playful like the robotics competition.

 

But I'm not a k-12 educator so all of this may be silly.

 

And for college here's something by the wonderful Woodie Flowers of MIT on changes he'd made to university engineering curriculum and beyond.  (Woodie was one of the most innovative teachers at MIT)