New perspectives on teaching and learning

As part of a Fulbright grant supporting my sabbatical to Portugal, I taught a "Topics in Astrophysics" course to master's students.  Going outside my usual comfort zone was good for me, and will help me be a better teacher when I go back to UC Davis.

One thing I take back is a renewed appreciation for the fact that each student starts from a unique place.  Having taught at the same university for nine years, it becomes second nature to assume that a student who passed certain courses understands certain concepts thoroughly.  I think I become a bit judgmental when I encounter a student who "should" (according to they courses they have passed) know something but doesn't.  As a fresh arrival in Lisbon, I had no preconceptions about students' prior knowledge.  This helped me create a better learning environment in general, and it also helped me "coach" each student without judging---an attitude I want to maintain when I get back to my regular teaching duties.  Of course, I still have to judge when I assign grades---but not before then.

I also have a renewed appreciation that being a student isn't easy. On my first day in the classroom, I felt like an outsider.  What are the expectations?  Will I crash and burn?  Students deal with these thoughts all the time, and with reason, because so often they are being graded.  When they come to a university they have to learn the system, navigate the courses, work and manage their finances, and learn to function in a new city.  I gained new respect for students by having to do some of these things as well.

Being a visiting professor teaching a topics course was liberating---nothing I did would be considered a precedent in future years, nor did I try to follow any template from previous years.  I tried lots of new things.  To prevent running the course completely outside the comfort zone of the Portuguese students, I just consulted with them rather than wonder what they would think.  I do this back home too, but this experience will help me do more.

Of course, I hope the students got something out of it as well. Having regular homework was apparently an unusual experience for them.  They initially thought it took too much time to do so much homework, but they began to appreciate that doing the homework is the only way to really learn.  But this dynamic worked only because I slowed the pace of the class to make sure they really had time to digest all the lessons they could learn from the homework---and that in turn was made possible by fact that this was a "topics" course without any predefined list of topics to cover.  So I don't know right now what I can change when I teach a more standard courses. But I am changed.

Not one of the ideas here is new, but they tend to fade when I teach over and over in the same setting.  Teaching in an entirely new environment was a great experience that will freshen up my teaching when I go home.  I highly recommend it, and I thank the Portuguese Fulbright Commission for supporting it.

SIRC Solar System

This is a summary of my Science in the River City workshop aimed
at fifth-grade solar system standards.

I started by projecting this opinion piece responding to the release
of a new survey showing that 26% of Americans answered the question
"Does the Earth go around the Sun, or does the Sun go around the
Earth?" incorrectly.  The opinion piece was a bit snarky and
unforgiving; educators know that everyday experience (eg, seeing the
Sun rise and set) is extremely powerful.  People who think the Sun
goes around the Earth are at least processing what they see and
building some kind of model to account for it. The vast majority of
people who "know" that the Earth goes around the Sun are probably just
memorizing something their teachers told them, and probably could not
cite evidence or build an argument to support this statement.  But the
latter is a much more valuable skill in today's society, and the Next
Generation Science Standards (NGSS) call for our kids to practice this skill.
So, I asked the teachers to role-play in pairs, one building an
argument that the Sun goes around the Earth and the other vice
versa. Try it...it's hard!

The #1 argument for Sun-around-Earth is, of course, rising and
setting, but there aren't a lot of obvious strong arguments for the
other way.  Kids will try to invoke authority, such as "NASA launches
rockets that can see what's going on" but I steer them away from that;
people concluded Earth-around-Sun long before rockets were built, so
they should be able to use simpler observations.  One pair of teachers
played out the two models using their bodies so they could see the
consequences of each, and that's exactly what I recommend whenever
possible.  They showed that the Sun's daily motion could be explained
by the Sun-around-Earth model or by a spinning-Earth model (which is
not the same as Earth-around-Sun).  Lesson: don't stop creating models
once you find one that fits, as other models might fit the data too!

If we had no other data, we might not be able to choose between these
models.  In such cases we should try to bring other observations to
bear, and/or deduce further consequences of each model. An example of
the latter: if the Earth is spinning, why don't we feel it?  We feel
dizzy when we spin on a merry-go-round....but if the merry-go-round
took 24 hours to complete one turn the effects might be too small to
feel.  So that was worth thinking about, but inconclusive. Can we
bring other data to bear? Well, just about everything in the night sky
rises and sets and rises again in about 24 hours, so the
Sun-around-Earth model has to become the
entire-universe-turns-around-Earth model.  But it's a lot easier to
believe that one thing (Earth) spins than that everything in the
universe contrives to revolve around Earth in 24 hours. Preferring the
simpler model is called Occam's razor, and we do that all the
time in real life. (Think of situations where someone is caught
red-handed doing something they shouldn't, and they say "This isn't
what it looks like" and tell a complicated story...do you believe the
simple story or the complicated story?)

The real physical proof that the Earth spins is quite subtle. One is the
Coriolis effect: we can measure the effects of being on a slowly spinning
"merry-go-round".  I showed the teachers a great kinesthetic activity
for this (described in a previous post).  Another is the Foucault
pendulum, which is not easy to demonstrate in a school but which kids
may have seen in science museums.  A third is even more modern:
astronomers can directly measure velocities of celestial objects using
the Doppler shift (the same principle is used by radar speed guns),
and we constantly have to correct for the velocity of the observatory.
Because of Earth's rotation, stars that are rising appear to be moving toward
us and stars that are setting appear to be moving away from us.

So the Earth spinning accounts for the daily (apparent) motions of Sun
and stars...so why do we think Earth goes around the Sun? (Do you see
how much reasoning we had to do to even get to this question?) Anyone
who looks at the stars periodically must have noticed that you can't
see the same stars all year round.  And because the presence of the
Sun in the sky defines when we can't see stars, that means that the
Sun moves relative to the stars.  Another way to say this is that,
unlike the Sun, the stars don't take exactly 24 hours to (appear to)
go around; they take 23 hours and 56 minutes.  So the Sun may be
opposite a certain star (ie the star is high up at midnight) at a
certain time of year, but over months will creep around the sky to
prevent us from seeing that star, and after 365 days the Sun will be
back to its original position relative to the stars.  Earth's rotation
period is either 24:00 or 23:56; it can't be both!  (It could be
neither, but considering that possibility may cause cognitive
overload.)

If we again use majority rule or Occam's razor, it's simpler to think
that the stars are fixed and that the Sun moves relative to the Earth
(note how little this conclusion has to do with the basic observation
that the Sun rises and sets each day, which is caused by Earth
spinning).  But there are at least two models which involve the Sun
moving relative to Earth, and again I had teachers play the roles of
Sun and Earth to demonstrate a model where Sun is stationary and Earth
moves, and vice versa.  Both of these models account for the
observations equally well (so far), and the moving-Earth model has the
disadvantage that we don't feel the Earth move.  This is one reason
many ancient Greek thinkers did not endorse the moving-Earth
hypothesis. But Galileo figured out that if you are in a laboratory
moving at constant velocity, you can't feel it move---think of a smooth plane
ride at 500 mph.  Earth doesn't move at constant velocity, but it changes its
velocity so slowly that the effects are really small.

The other thing the ancient Greeks figured out is that the
moving-Earth hypothesis means that we should see parallax. Think about
sitting in a moving car.  The roadside trees appear to rush by, but
the distant mountains appear to move very slowly.  If the Earth moves,
we ought to be able to see an effect like this by comparing nearby and
distant stars.  I taped some stars around the room and some teachers
played this out.  If Earth is still, we will not see parallax.  The
Greeks looked for parallax but did not see it, so they favored the
stationary-Earth hypothesis; they weren't stupid!  It just turned out
that even the nearest stars are so far away that the parallax effect is
tiny and could not be measured until modern times.

Using parallax we can determine that the nearest star is about 300,000
times more distant than the Earth-Sun distance (that comparison is natural
because it is Earth's motion over that distance that gives rise to the parallax
effect).  It's as if your car moved one mile but you were asked to discern the
difference in your view of mountains 300,000 miles away (eg, on the
Moon).  This can be illustrated dramatically by drawing a one-inch
Earth-Sun model on the board, and then drawing a long line
representing the distance to the next star and asking the kids to stop
me when they thought I had arrived.  Kids (even most adults) have no
idea how much 300,000 times is; they ask me to stop after 5 feet or
so, but I keep going.  When I run out of board, I get a roll of toilet
paper and start unrolling it, as a way to illustrate a very long line.
I keep going even when they tell me to stop. Then, when I run out of
toilet paper, I go to the back room and get a cart full of hundreds of
rolls of toilet paper!  It is really dramatic and fun. It's also fun
to write out the number of miles to the next nearest star on the
board: about 24,000,000,000,000 miles. (Kids and even most adults have
little idea what a "trillion" really means.)

Finally, because a light looks dimmer the greater its distance from
us, we can calculate how much light a typical star emits (as opposed
to the very small amount of its light that we receive).  Correcting
for this distance factor, it turns out that stars emit about as much
light as our Sun!  Some emit more, some less, but the bottom line is
that each star is a sun unto itself, or if you prefer, the Sun is just
another star.  In the 1600's, long before parallax was ever measured,
the astronomer Christiaan Huygens turned this argument on its head:
assuming the star Sirius is as luminous as our Sun, how far away would
it have to be to appear so dim?  His conclusion was 30,000 times as
far as the Sun; this is lower than the true value, but only because Sirius is
actually substantially more luminous than the Sun.  And his number was
certainly big enough to convey some of the vastness of the universe.

The Moon does go around the Earth, so we used that to address the
gravity standard.  I did the donutapult demo, which illustrates that
for anything to move in a circle there must be a force directed toward
its center; therefore the Moon is pulled on by a force directed toward
Earth's center.  We can relate that to gravity (ie, the force we feel
every day here on Earth's surface) by looking at a globe and noting
that, anywhere on Earth, "down" means toward the center of the Earth.
Therefore (Occam's razor again) we don't need to hypothesize about a
mysterious force keeping the Moon in its orbit; it could well be the
same force that makes apples fall. (Proving that it's the same force
goes beyond the fifth grade standards.)

Finally, we discussed scale models, as scale is a crosscutting concept
in the NGSS. There is no better way to make people appreciate how
empty space is than to build a scale model of the solar system! Rather
than write up our discussion, I refer you to my previous blog posts on
the scale model project I led at Peregrine School: intro, poster assignment,
and completion.

A few links you may find useful:

• my previous SIRC workshop on the solar system (8th grade)
• Powers of Ten video to help students appreciate scale
• interactive tool to do the same (Warning: this tool only shows relative sizes, not the huge distances between celestial objects)
• feel free to browse the labels at right

Climate 101

Nice article today in the New York Times: Study Links Temperature to a 
Peruvian Glacier’s Growth and Retreat. It's a good example of how news
about climate change could easily be misread as indicating more doubt
than there really is.  The headline makes it sound as if the link between
glaciers and temperature is so tenuous that this is the first evidence of it,
and that it has been established for only one glacier.  The truth is very
different, even though the headline and article are not wrong once you
understand the context. This post is aimed at helping teachers
and students with the basics, and then use that to parse the news.

Over a century ago, it was known that carbon dioxide impedes the flow
of heat (in the form of infrared light) from the Earth out into space,
while not impeding the flow of heat (mostly in the form of visible
light) from the Sun to the Earth.  If not for this natural greenhouse
effect, Earth would be much colder.  Teachers can demonstrate quite
directly that carbon dioxide impedes the flow of infrared light, but
many teachers may not have the right equipment.  Here's a video
comparing the temperature rise of two bottles, one with elevated
levels of carbon dioxide and the other with standard air.  And here's
a nice video using an infrared camera to show quite directly that
infrared light is largely blocked by carbon dioxide.

Around the same time (1896) Svante Arrhenius recognized that humans
were pumping ever more carbon dioxide into the atmosphere, and that
this would lead to warming.  But "warming" sounded reasonably
beneficial, especially given Arrhenius's prediction that it would take
place slowly over thousands of years.  Arrhenius did not account for
the large increase in population over the ensuing century, nor for the
large increase in per-capita use of fossil fuels (cars, airplanes,
etc). Worldwide, we now emit about 17 times the carbon dioxide emitted
in 1896, so change is coming much faster.  And now we know that an
increase in temperature is not as beneficial as it may sound because
it can radically change weather patterns, which imposes large costs on
humans as well as on many species which cannot move and adapt rapidly
enough.  Apart from that, Arrhenius deserves kudos for his prescience.

Yet if we heard this prediction in 1896 we would be justified in
expressing some skepticism. Earth's atmosphere and oceans (where most
of the excess heat is deposited) form a complicated system, and the
response of a complicated system to a simple input (more heat) may
well not be a simple result (higher temperature).  But healthy
skepticism goes only so far; unless you have a better model, you have
to admit that the best model predicts warming.  Just saying "it's a
complicated system" does not give you the right to reject all models.
In this case, you would have to figure out where the extra heat would
go without causing increased temperatures, and you would have to have
some evidence to motivate belief in that model.

Fast forward to 2014.  Warming is here, and we've learned a lot about
climate models in the meantime. We did find complications (El Nino,
for one), but the simple model was reasonable in its overall
prediction.  More heat does mean a higher temperature.

One way "climate skeptics" (I put the term in quotes because
oil-company funding leads to a kind of "skepticism" different from the
detached sort of skepticism we encourage in science) sow doubt about
this result is to suggest that the warming may be due to natural
cycles.  There certainly are natural climate cycles, but rather than
treat them in detail here I want to make a bigger point about how
science works: When a model makes a prediction and the prediction
comes true, we should gain confidence in the model, and we should lose
confidence in models which made contrary predictions. Yes, it's
conceivable that the greenhouse model's prediction came true through
a fluke of natural cycles rather than accurately modeling how nature works
...but how much confidence would you put on that possibility? 

A prediction is a powerful thing, so let's note the distinction between
a prediction and a retrodiction (or postdiction), which is when you make a
hypothesis after looking at the data.  Using data (rather than laws of
physics or other guiding principles) to generate hypotheses is a fine
thing to do, but because "patterns" can randomly appear in data you
cannot confirm the hypothesis with the same data which generated it;
you must seek out new data. (Admittedly, even scientists sometimes
forget to apply this principle.)  Climate skeptics can suggest
alternative causes for the warming after looking at the data, but we
should have much more confidence in the model which actually predicted
the data.

Now, the news: a reconstruction of the timeline of growth and
shrinkage of a Peruvian glacier shows that shrinkage is most highly
correlated with temperature and not with other factors such as
precipitation.  You have to get halfway through the article to get the
background:

land ice is melting virtually everywhere on the planet...the pace seems to have accelerated substantially in recent decades as human emissions have begun to overwhelm the natural cycles. In the middle and high latitudes, from Switzerland to Alaska, a half-century of careful glaciology has established that temperature is the main factor controlling the growth and recession of glaciers. But the picture has been murkier in the tropics. There, too, glaciers are retreating, but scientists have had more trouble sorting out exactly why. 

So, you may have started reading the article thinking that scientists
understood very little about glaciers if they were just now finding a
"link" between glacier shrinkage and temperature, but you now see
that a lot of important knowledge has already been established.
Newspaper articles are designed to tell you what's new first, so it's not
the writer's fault that this background was buried deep in the article.
Nevertheless, in practice many readers will just read the headline and
skim the first part of the article, thus missing this crucial background.
Teachers and students should be aware of this when reading science news.

But wait, there's more! The article goes on to explain how the details
of tropical glaciers are different from most glaciers (intense
sunlight can vaporize the ice directly, and the sunlight lasts
year-round) but that one group of scientists has studied the matter
and still concluded that temperature is the driving factor in
shrinking tropical glaciers. "But a second group believes that in some
circumstances, at least, a tropical glacier’s long-term fate may
reflect other factors. In particular, these scientists believe big
changes in precipitation can sometimes have more of a role than
temperature."  In other words, this is a legitimate scientific dispute, but
it is about the details of a very specific type of glacier and has little or
nothing to do with overall concerns about glaciers (or sea ice)
melting worldwide, much less about the reality of climate change.  Yet
someone who wants to sow doubt about climate change can point to this
and say "scientists don't really understand why glaciers melt" and people
who don't read the article carefully may well be snookered by that.
Please make sure you (and, if you are a teacher, your students) don't
get snookered.

My next post discusses two more aspects of the nature of science---uncertainty
and model-independent statements---in the context of climate.

One Percenters

We've been bombarded all winter with stories of cold and snowy weather in the eastern US, but the news was just released that January 2014 was the fourth-warmest January on record.  How can this be? The eastern US covers less than 1% of the Earth's area, so (as this essay nicely puts it) "if the whole country somehow froze solid one January, that would not move the needle on global temperatures much at all."  That essay is worth reading because it goes on to explain how subjectively people do perceive global warming: something as unrelated to global warming as being in a cold room does have an influence on the opinions voiced in a survey.   Educators should be aware of this, and actively work on making students think objectively and use data.

Mostly Harmless

In the Hitchhiker's Guide to the Galaxy, "mostly harmless" is the
Encyclopedia Galactica's assessment of Earth (which is not important
enough to merit a longer entry).  This made me think that looking at
the solar system through alien's eyes might help students learn about
it.  I conducted Science in the River City workshop for earth science
teachers based on this idea, and this is a list of resources for such
teachers.

First, I highlighted a graphing activity I had done with elementary
kids; that experienced is described in great detail here. (Feel free
to download and copy the graph.)  I extended the activity to graphing the
surface temperatures of the planets as a function of distance from the
Sun, which led to the greenhouse effect discussion below, but now it
occurs to me that a great way to extend this activity would be to jigsaw
it: assign one group of students to graph size vs distance from the Sun,
another to graph temperature vs distance from the Sun, another to graph
density vs distance from the Sun, etc, and then the groups come together
to think about what it all implies for the formation of the solar system.

Second, when discussing the formation of the solar system and

describing how small grains of dust started to stick together, I

wanted to show a video clip but had some technical difficulties.  Here

is the link; start at 3 minutes into the video and go for 2.5 minutes.

(If you have time, the whole episode is worth watching.  It's from the

How the Earth Was Made series, which has some really nice

visualizations and is constructed around evidence, which is a key

feature missing from many science documentaries.  It tells science

like the detective story it is.  That's generally a good thing, but in

this case the implication that this particular astronaut doing this

particular demonstration singlehandedly saved the theory is a bit of

an exaggeration.)

Pocket solar system: https://nightsky.jpl.nasa.gov/download-view.cfm?Doc_ID=392

Peppercorn Earth: http://www.noao.edu/education/peppercorn/pcmain.html

Extrasolar planets: http://exoplanets.org/ has the most up-to-date

info. Even better, they have built-in graphing tools so you and

your students can easily explore the data.

Earth's surface temperature: I got my plot from the most authoritative

source for modern temperatures, NASA's Goddard Institute for Space Studies.

This link only scratches the surface of climate change data because it deals

with modern temperature measurements (as opposed to long-ago temperatures

inferred from ice cores etc) but as the greenhouse effect was not the focus of

the workshop I won't try to compile a list of links here.  (For those

not attending the workshop: we graphed planets' surface temperatures

vs distance from the Sun, and we saw the general pattern that farther

from Sun equals colder, but we also saw that Venus is a real outlier

from this pattern.  That's because Venus has had a runaway greenhouse

effect.  Earth also has a natural greenhouse effect which keeps us

from being frozen, but which is now being augmented by a manmade

greenhouse effect.  I did tell the teachers that Earth has a "carbon

cycle" which will absorb the extra carbon dioxide through the oceans

into rocks, but I forgot to mention that it will take hundreds of

thousands of years; I didn't mean to imply that humans can carry on

regardless. Venus' greenhouse effect is "runaway" because its

carbon cycle shut down when its oceans boiled.)

Finally, a few links I didn't get time to show but which will help you

appreciate the size of the universe (and the sizes of things in it):

the classic Powers of Ten video and an interactive tool.

Planet Posters

Two weeks ago each student chose a planet (or other solar system object) to research and make a poster about. Today they brought in their posters, and each student told the class what they learned in their research.  The kids were very engaged and asked so many good questions that we spent all morning doing this.  So next week we will put up the posters at the appropriate distances from the Sun poster (which I made and put up near the school entrance today) to make a scale model of the solar system.   The discussions today were so full, frank, and wide-ranging that I can't hope to capture them in a blog post.  I will simply leave you with a short video with amazing images of Jupiter's moon Europa.

I think the posters were quite successful as a learning experience. The kids learned by researching and making them, but they also learned by listening to other kids talk about their posters, and they all learned when I answered numerous questions in more depth as they arose.  I think a key to real learning is that the posters should not be just a laundry list of facts, but should really be based on the students' questions.  When I issued the assignment, I offered some questions they might be interested in answering:

• What would it be like to visit?  What is the temperature?  Is there a solid surface? Would the Sun look bright from that distance?  If the temperature is extreme, think about ways to convey how extreme it is.
• Does the planet have moons or rings? If you chose a moon to begin with, briefly describe the host planet.
• What are seasons like on that planet? This depends on how tilted the planet is with respect to its orbit.
• How long is the year on that planet?  How long is a day?
• Are there volcanoes? Rocks? Rivers/lakes/oceans?  (If so, are they made of water or some other substance? Moons of Jupiter and Saturn are especially interesting in this respect.) Clouds? Earthquakes? Storms? Lightning?
• Could you possibly find life there? 

I think these questions helped prevent a "laundry list" result.  One thing I would like to do better next time is have some kind of first draft with feedback and then a final draft. I don't know how to do this with posters, but I would like to give kids feedback before it's too late to change the final result.

Picnic Day

UC Davis Picnic Day is a giant open house attended by 50,000 or more
people. The Physics Club runs a "magic show" and a demo room where
visitors can do some hands-on experiments, but there hasn't been a
kid-friendly physics room in the past.  This year, I decided to make
one with the help of kids and parents at Peregrine School.  The day
before Picnic Day, I brought all my toys (Coriolis effect demo,
balloon in a bottle, infrared camera, mixing colors of light,
airzooka, etc) to the school and spent the morning training students
and parents so they would be able to explain the ideas behind them to
visitors on Picnic Day.  On Friday night my wonderful wife Vera and I
set the demos up in a room on campus, and on Saturday we had a ton of
visitors.

I think we did a really good thing here.  We didn't have our kids
explaining physics to other kids as much as I had imagined, for
various reasons: our kids were having fun playing too; they wanted to
visit other exhibits on Picnic Day; and most of the visitors to our
room were actually adults.  So the kids got less practice in
explaining physics than I had imagined, but we did a great public
service.  As an educator, I'm always thinking about ways to tweak
things, so if there is a next time (or as advice to others thinking
about doing this kind of thing), one way to get kids really deeply
invested might be to have them develop their own unique demos.

Planets and their orbits

Today I brought the coin funnel to Primaria and we learned how orbits work.  One four-year-old came up with Kepler's 3rd Law all by himself!  With 6-8 kids in each group I organized it by having each child raise a quiet hand, tell me something they know about planets or the Sun, and then I gave that child a marble to roll in the well.  That gave me a chance to comment on what we could learn from each marble.  Depending on how it was thrown, the lesson could have been about escape velocity, about planets not falling straight into the Sun, about all the planets going around in the same direction, etc. Most of the time we pretended the Sun was at the center but if you want to make it more exciting you can pretend it's a black hole. 

Then  we shifted gears and built a scale model of the solar system: I showed a basketball representing the Sun, and I laid out various objects (a tomato, a grape, a small candy, etc) for them to choose which one they thought was the right size to represent Jupiter, the Earth, etc.   The correct answers are truly amazing: learn more at this well-written site. We didn't have time to go outside and put the correct distances between the planets, but I did describe the highlights of that aspect: Earth would be roughly in the office if we were with the Sun in the P1 room.

Aside on teaching and learning: one aspect of orbits is that all objects in a given orbit go at the same speed, regardless of their mass (technically, as long as the mass is much smaller than the mass of the thing being orbited).  To reinforce that, I dropped two marbles of very different mass and asked which one would hit the ground first.  Of course, many had the misconception that the heavy one would hit first.  I remark on it now because I had done the same demo with the same kids eight weeks ago in a slightly different context, so it's clear that they forgot.  These misconceptions are persistent!

Science in the River City

K12 teachers in the Sacramento area should check out the Science in the River City professional development workshop.  It happens roughly monthly throughout the school year and the last one this year is coming up on April 24.  Teacher feedback indicate that it's pretty useful.

All Charged Up

Continuing with the theme of different forms of energy, yesterday at
the elementary we did some static electricity experiments.  I knew
some students in the upper grades had studied electricity before, but
I decided to start at a pretty basic level here, to make sure everyone
really understood what they thought they understood.  Along the way I
hoped to add some physics context which would be lacking in most
elementary experiences of this topic.  You can do all of these at home
too.

We started with the classic: rubbing a balloon on someone's hair.  Of
course this makes the hair stand on end, but I expect few people will
have thought about it this way: the mass of the entire Earth is
pulling down on that hair with gravity, yet it only takes a few rubs
with a balloon to get the balloon to exert a stronger upward force on
the hair.  That demonstrates how remarkably strong electric forces can
be, compared to gravity.  (By the end of the activity we'll see why
they aren't always stronger.)

Why is the hair attracted to the balloon?  The older kids will shout
out some version of "the balloon has negative charge" but I make it
clear that giving it a name doesn't explain anything by itself.  We
could just as well call it magic if all we want is a name for it.  So
let's proceed through some other experiments to see if we can learn
more about it.  I wrote "Observations" on the board and jotted down
the result of each experiment as we did it.

One experiment we can do is see if other things besides hair are
attracted to the balloon.  I brought some ground pepper and shook some
out on the table, and a well-rubbed balloon will make that pepper just
jump up and if the kids are quiet they will hear a nice kind of
raining sound as the pepper hits the balloon.  (Warning: the balloon
loses its "magic" over time, so you need to give some good rubs before
each experiment.  If you get tired of rubbing people's hair, come
equipped with some tools for it.  Rabbit fur is usually recommended,
but if you find that difficult to come by you can google for
alternatives.  It's a good idea to bring a few pieces for the kids to
share when they do their own experiments.  Otherwise, it's unfair to
the girls with long hair!) Apparently sawdust is another good material
to try, but I have not tried it.

Next, we did a ping-pong ball.  The balloon doesn't lift it up, but
the kids knew right away that's because the ball is heavier.  But the
balloon can pull the ball sideways across the table.  You can wave the
balloon back and forth and make the ball dance, or keep pulling the
ball in one direction clear off the table.

For fun, I did soap bubbles too.  Blow some bubbles and put the
charged balloon above one.  The bubble goes up rapidly and dashes
itself on the balloon.  With some practice, you can pull the bubble up
without breaking it instantly.

So is the rubbed balloon always attractive?  If we rub two balloons
then maybe they attract each other even more strongly?  I had a second
balloon ready, tied to a string so I could hold the string with the
balloon hanging straight down.  Any attraction would then be visible
to the whole class by seeing the string depart from perfectly
vertical.  I had two students rub the two balloons, then brought them
near each other.  They repel!  They do not attract.  So how do we
explain that?

If two similar things repel, then we might think that opposites
attract. Kids can come up with this idea just as well as the
18-century geniuses who provided the foundation for electromagnetism.
We can call these opposites positive and negative, or up and down, or
blue and red, or whatever.  The basic picture is that in normal matter
these two kinds coexisting closely, so that from the outside they
appear to cancel out and have no net charge.  But the balloon, when
rubbed on hair or fur, tends to tear off and acquire one kind (which
we happen to call negative) more easily than the other.  This makes
the balloon negative and the hair net positive, and they attract.  But
two rubbed balloons, each being negative will repel.  (By this logic
the hair of two rubbed people will repel, which is an interesting
experiment I did not think of at the time.)  I drew all this out on
the board, with a bunch of mixed + and - signs initially, moving some
- signs away to show how the hair is net positive, etc.

(By the way, this +/- picture explains why gravity appears to be more relevant than electric forces in, say, holding you down to the earth.  Electric charges can be much stronger when one kind is isolated, but usually the two kinds are mixed and deliver no net effect.)

So far, so good.  Now I wanted to challenge the kids.  I rubbed the
balloon vigorously and stuck it on the wall, where it stayed.  How do
you explain that, when I didn't rub the wall?  (Astute observers will
note that I didn't rub the pepper, the ping-pong ball, or the soap
bubbles either, but I didn't remind them of that, just to minimize
confusion.)  Explanations were offered, but none that really worked.

Is it something about the wall?  I went to the sink, ran a thin stream
of water, and brought the balloon close (but not enough to get it
wet).  The balloon attracts the water!  This is pretty cool and you
should do it at home if you've never seen it.  So now we have many
different types of material (including the pepper, the ping-pong ball,
and the soap bubbles) which, even in the absence of rubbing (i.e.
presumably uncharged; we would never get a spark from them, for
example), are attracted to a rubbed (i.e. charged) balloon.

This stumped the kids, but it turns out we don't need a radically new
model of how charge works; we just need to think in more detail about
the implications of our existing model.  Our model is that water (or
the wall, the pepper, etc) contains a mixture of + and -, mixed so
thoroughly that from the outside we experience it as uncharged.  But
as the water nears the balloon, maybe the balloon can push the -
particles toward the far side of the water stream and attract the +
particles to the near side of the stream:

With the + part of the water nearer the balloon, the water has a net
attraction to the balloon.  Yes, the - part of the water is repelled
from the balloon, but more weakly than the + part is attracted.
Therefore, the stream of water moves toward the balloon.  Physicists
say that the water is polarized by the charge in the balloon.  Not all
materials are polarizable, but apparently many are.

Grades 1-3 asked me some very good questions about this.  They asked
if I could get the balloon to repel the water by rearranging the
charge in the water.  I said I guess so, but I don't know how you'd
prepare the water with the negatives on the balloon side.  As I said
that, I did think of a way, but thought it was too complicated to explain.  Then
a girl raised her hand and suggested the same thing I had thought of:
prepare a stream with negatives on, say, the left-hand side by
passing it near a balloon on the right-hand side (as shown above).  Then pass
 that stream by a balloon on the left-hand side and see if it repels.  I
was floored.  This was pretty good thinking for a second-grader!  It
illustrates one of the main ways science progresses, by using the
results of one experiment to set up a more elaborate experiment.  And
in retrospect, it demonstrates that she really understood the model of
how charges behave.  She had not just memorized the buzzwords.

With about 15 minutes left, I set the kids free to work on an
experiment of their choice.  The one that I recommended was building
an electroscope.  I demonstrated a sturdy one: a vertical piece of
metal branching into two vertical pieces of aluminum foil, in a
protective glass container.  This is just a more sensitive version of
the repelling balloon demo.  When any charge is brought near the piece
of metal, the aluminum foil lifts up.  I showed them how to build a
cheaper version out of a clay base, a flexible straw support, and
pieces of aluminum foil attached with string and tape.  They could
take those home.  Other students chose to try to find unpolarizable
materials, experiment with charged balloons and running water, examine
the motion of pepper across a charged balloon, etc.  They seemed to be
very engaged in these activities.

All in all, this activity was a hit.

Epistemology 101

A little side note on yesterday's class: I knew some students in the upper
grades had studied electricity before, but I decided to start at a
pretty basic level with static electricity, to make sure everyone
really understood what they thought they understood.  At some point a
student volunteered an answer to one of my questions, and I asked,
"How do you know that?" with the intention of highlighting how the
student's conclusion followed from the things we had just observed.
But the student said she knew it because a teacher (in a previous
year) had told her, so I asked how the teacher knew that.  The
response: that teacher had a teacher at some point.  So where did THAT
teacher learn it?  The whole class was eagerly getting in on the act
and shouting out different answers by this point, but one answer was
"From scientists."  So how did scientists learn it? Eventually we came
back to the idea of doing experiments and learning from them.

I think this was really useful because too many people are stuck at
the first stage of epistemology: knowledge comes from an authority,
and that's that.  Of course, it's normal at this age (grades 4-6), but
I'd like to do whatever I can do to move the kids on through the next
stages.  It goes to the very nature of science: is it just a set of
results, or is it the process?  It's both, of course, but the process
too often gets short shrift in education.  It's difficult to
teach---it can't be a unit by itself, rather it has to be built in to
every science unit, which makes the logistics very difficult---and
it's difficult to write a test question about it. But it has to be
done.

If you're interested in what thoughtful people have discovered about
the stages of epistemology, you might start with this quick summary of William G. Perry's research.

The second group, grades 1-3, would have missed out on this except
that at some point I said, "Here's what I think is going on," and one
student said. "You're the teacher, you should KNOW what's going on" or
something like that.  So that was a good chance to have a similar
discussion with that group.

Rainbows keep falling on my head

Yesterday Vera guided the 4-6 graders in exploring how different
colors of light mix.  As you may recall, in her previous outing she
showed how light from a white bulb can be split into various colors
like a rainbow.  This shows that we perceive as white is actually a
mixture of all colors, so a natural question to explore next is: What
happens if we mix different combinations of colors, like red+green,
green+blue, etc?  This is a lot like mixing colors of paint, but not
exactly because many colors of paint mixed together approaches the
appearance of black, but many colors of light mixed together
approaches the appearance of white. 

She directed the students to play with this computer simulation which you are free to do at home with your child.  After about 5 minutes of just playing
with it, the students were guided through a set of thought-provoking
questions which made them go back and test the hypotheses they had
formed regarding light.

The concepts are similar to many I wrote about in my last visit to Primaria, so I won't belabor them here.  I will remark instead on the
importance of testing students' mental models.  It's one thing to say
that "white light contains all colors" but it's quite another to probe
what students really think that means in practice.  If you put white
light into red glass, did the glass turn all the other colors into
red, or did it just block all the non-red colors from passing through?
The answer is the latter, but the most interesting thing is the
thought process which leads to that.  How would we devise an
experiment to distinguish between these two hypotheses?  By passing
the resulting red light through blue glass: if the conversion
hypothesis is true, a lot of blue light will come out, but if the
blocking hypothesis is true, basically no light will come out (because
the blue glass will block the red light).  It's not enough just to
tell kids that "white light contains all colors."  If you tell that to
7 kids, they will have 7 slightly different ideas about what that means
in practice. So a teacher has to find ways for students to test and
modify their mental models.  Not only does this lead to a better
mental model, but it make students practice that most important skill:
using evidence to improve their model of how the world works.

Big Study Links Good Teachers to Lasting Gain

This New York Times article discusses recent research which shows that good elementary and middle school teachers (as measured by the jump in their students' standardized test scores) have a lasting effect on their students (as measured by students' lifetime earnings, avoidance of teen pregnancy, and college enrollment).  Although the difference between having an average teacher and having an excellent teacher for one year yielded only $4600 in additional lifetime earnings for each student, the cumulative effect is large: A 30-student class would yield $138,000 extra in lifetime earnings as a result of excellent teaching.  (The difference between a poor and an average teacher was quoted as $266,000 in one year of teaching.)  And avoidance of teen pregnancy?  Priceless!  Although admittedly the difference is small (8.1% vs 7.7% chance of teen pregnancy), when it happens to your family it is not a small thing.

I realize that not everything good in life can be quantified, but I think this kind of research is important because good teachers are not paid enough, and this kind of research helps us advocate for them.  (And, one might argue, gives us greater motivation to get rid of poor teachers more quickly.)  Many of us might think from personal experience that the effect of a good teacher is permanent, but previous studies have not supported that conclusion.  Those previous studies looked only at test scores in future years, not these positive long-range life outcomes.  Therefore, this more sophisticated research is an important step forward.

The study addressed fourth through eighth grade, and found that the grade in which a student encountered a good teacher did not matter.  About a year and a half ago, a roughly similar study was conducted for kindergarten teachers, which concluded that good ones brought a $320,000 economic benefit to their students.
You may wish to read the New York Times column addressing the earlier research, and if you wish to dig deeper into the data you can see some slides presented by those researchers.

Update: Nicholas Kristof has written an eye-opening column about this.

Let there be light!

Back to Primaria (pre-K/K) this week.  The teachers asked me to
explain how lenses work, because the kids had been making toy
eyeglasses out of pipecleaners and were curious about it.  I had long
wanted to do some demos with light anyway.  It takes a lot of trouble
to make a room really dark (so that the light relevant to the
demonstration is more visible) during school hours, so I figured I
would go to that trouble and combine topics.  Linus (my son in
Primaria) had asked just a week or so before about the Moon.  He
thought the phases of the Moon were due to Earth's shadow falling on
the Moon.  I pointed out that the crescent Moon appears not too far
from the Sun, so the Earth's shadow cannot be falling on it.  He came
up with some crazy stuff about light bouncing back and forth, back and
forth between Earth, Sun, and Moon "like an air hockey puck."  So I
had a motivation to do phases of the Moon with the kids, but first I
had to build on basic concepts of light, like the difference between
emission and reflection (the Sun emits light and is the source of
light in our solar system; the Moon reflects some fraction of the
light it receives, but not enough to illuminate the other bodies in
the solar system).

So I set up in the kids' bathroom, which is the only room with no
windows.  I still had to spend a lot of time taping up the open
doorway with black plastic to prevent a lot of light coming in.  In
groups of 5-6, the kids came in and we started by talking about how we
couldn't see anything without a source of light.  I then turned on an
unexpected source of light: a laser pointer.  We discussed how they
couldn't see the source of light directly, but they could see the
light when it reflected off the ceiling. Next, a flashlight.  I
pointed it directly at them, then pointed it at the ceiling.  So a
given light source can be seen directly or indirectly (reflected)
depending on your relationship to it.

Now I turned on the "Sun": a naked light bulb.  Unlike a flashlight or
laser pointer, it emits in all directions.  But can we always see the
Sun?  We discussed various reasons for not seeing the Sun, such as
clouds.  But when is it really dark?  At night.  And what is night?
"Clouds" were again offered as a reason, so we discussed what happens
just before night: "the Sun goes down behind the mountains."  So then
we each pretended we were the Earth, and slowly turned around so that
the Sun came into and out of our field of view.  [The next day, my
wife Vera offered a really good suggestion: have them extend their
arms to make a "horizon" which turns with them.]  To be honest, a lot
of kids spun way too rapidly and weren't really getting it.  I
repeated the whole thing with a globe.  We agreed on the location of
California and looked at how California varied between bright and dark
as the Earth turned.  A problem with this is that light reflecting off
the walls provides a non-negligible amount of illumination for the
back side of the Earth, and the effect is not nearly as dramatic as
you would thing.  Vera suggests decoupling the day/night concept from
the light demo, just pasting up a picture of the Sun in a regular
classroom and doing the horizon thing.  I think she's right about
that! Another possibility is to build a little model.  If the Sun were
a Christmas-tree bulb and the Earth a nearby marble, relatively little
light would bounce off the walls of the room!

Next, we tackled phases of the Moon.  I had one child volunteer to be
Earth while I took a volleyball Moon and moved it around Earth,
showing how the Earth-person sees a fully-illuminated Moon when it is
opposite the Sun, and sees (rather, does not see) an un-illuminated
Moon when it is more or less between Earth and Sun.  However, this did
not work well for several reasons.  Each group had a bunch of other
kids who were not the Earth and saw the whole thing from a variety of
vantage points.  It was very difficult to steer the kids into seeing
what they were "supposed" to see.  One girl said "now I'm the Earth"
when the Moon happened to come close to her.  In one group, the
Earth-volunteer gave the wrong answer when I asked him whether the
side of the Moon he was seeing was bright or dark; I think he just
didn't know what to compare to, so I need to be more careful about
exactly how I word my questions.

Finally, the lens.  The key to a good visualization is to avoid using
all three dimensions. I put a flashlight on a table so they can see
how the light spreads out by looking at the light and dark patterns on
the table.  I put a comb in front of the light to give a visual
impression of light rays spreading out on the table.  Then I set a
special lens on the table, which is like a slice of a lens so that it
can sit flat on the table.  This shows that the light rays which go
through the lens are bent so that they converge back together rather
than continue diverging.  It's quite striking if set up right.  I had
a card which I pretended was a movie screen, and projected the focused
image there. We talked about movie theaters and where they would sit,
did they ever look behind them and see the bright light coming out of
the lens, and what would happen if there was no screen.  For the
groups which had a bit of time left at the end, I moved the lens
around to show that if it's too close to the light, it's not powerful
enough to converge the light.  It might be powerful enough to stop
further spreading of the light, though, and I showed how a second lens
could then converge that light.  The idea was to show that there are
many combinations and possibilities.

I felt that the kids were more disengaged than usual, and I felt that
it was directly attributable to the "demo" rather than "hands-on"
nature of the activity.  I made the "demo" decision because I felt it
would be chaos to have 4- and 5-year-olds handling flashlights and
lenses in teams of two or three.  That may have been correct, but I
should have found some way to prevent the whole 20 minutes from being
all demo.  One possible structure is the sandwich: an initial demo
followed by hands-on activities with a more complicated demo or
summary discussion at the end.  But this didn't fit with the list of
topics I wanted to cover.  I realize now that I was too much in
"professor" mode: practicing inquiry is more important than covering a
list of topics!  The kids brought up (indirectly) one thing I had
thought about last year but forgotten: setting up a light so that they
can make shadows themselves.  They love doing this, and if I set it up
right they can actually explore different aspects of light.  For
example, I could set up two lights of different colors and they could
see how to control the color of the shadow.  Or they could explore how
a given object can have differently shaped shadows depending on its
orientation to the light.  I can even imagine setting up a "light
studio" which they could play with during the week before or after my
visit.

Playground Design 101

The motivation for the next few visits to the elementary school is
that the kids are going to help design the playground for their new
school site, so I'm going to show them a bit about how things work, ie
classical mechanics.  One thing I love about this school is that the
teachers frame things this way.  Instead of just hearing that "today
we're going to be learning Newton's laws of motion" students have this
wonderful backdrop to keep them motivated and (perhaps more important)
foster creativity.  The laws of mechanics will be a springboard to
creating something wonderful, not a straitjacket of rules we have to
memorize. 

We set the foundation last time with
Newton's laws of motion exemplified in the simplest possible situations,
to make them as clear as possible.  This time, we added complications to
show how interesting it can be when forces interact. I concocted three
different examples of interacting forces and set up three
stations. Each group of 6-8 kids split into 3 groups of 2-3 and spent
5 minutes at each station, with 5 minutes left at the end for group
discussion.

Station 1: I repeated the pulley activity from last week at Primaria.
I rigged up different pulley arrangements to lift identical 20-pound
weights.  One arrangement was just a single pulley at the top as you
might expect, reversing the direction of the rope so that the kids
could stand on the ground and pull down on the rope to make the block
go up.  The second arrangement had the end of the rope tied at the
top, running down to an "upside down" pulley attached to the block,
and then back up to a pulley at the top which acted much like the
single pulley, just reversing the direction of the rope.  The kids
tried both setups and compared the difficulty of lifting the block.
The second arrangement is much easier, but why?  I challenged the kids
to go beyond simple explanations like "two pulleys are better than
one" and "there are two ropes pulling up the weight so it's twice as
strong."  The latter statement starts to get to the answer, but is by
no means a complete answer.  If I have to drag something with a rope,
tying two ropes to it doesn't make it any easier.

The trick is to observe closely what happens when you pull.  The
moving pulley makes it so that if I pull my end of the rope one foot,
the weight moves up half a foot.  This means that you only need half
the muscle that you need with the fixed pulley.  (This is called
"mechanical advantage" but I did not use that term.)  This was not too
easy for the elementary kids; in fact I think last week the pre-K/K
kids did better, possibly because the three-station setup this week
was very distracting.  They were able to extrapolate how to make it
even easier to lift (add more pulleys) but we didn't have time to
discuss how we would connect those extra pulleys, which would really
probe understanding.  This could be a good home activity for
interested parents and kids: set up a 4-pulley system so that it's 4
times easier to lift a given weight.  How do you set it up, and how
much rope will you have to pull to lift the weight 1 foot?  (Advice:
don't try to connect 4 separate pulleys, because the ropes will easily
get twisted and tangled.  Buy two "double parallel pulleys" so that
everything stays more or less aligned.)

Also note that in each case, one pulley exists only to reverse the
direction of the pull.  You could simplify the comparison by thinking
about standing on a deck and pulling a weight straight up (no pulleys)
vs. tying one end of the rope to the deck, running it around a pulley
attached to the weight, and then pulling up on the other end.  Here it
is clear that to get the weight up to the deck, you will need to pull
a length of rope which is twice the height of the deck.  But the
benefit is that you need only half the strength to pull the rope.

Station 2: an overhead pulley with an adjustable amount of weight
attached to the rope on each side.  This can be used to emphasize a
few different concepts.  First, balance: when the weight on each side
is the same, neither side moves.  This might seem boring, but it is
actually an easy way to move weight up and down.  In balance, it takes
only a tiny amount of strength to move one side up or down, because
you are not moving any net weight up or down.  This is how elevators
work: there is a counterweight so the motor doesn't have to work so
hard.  This also provides safety in case the motor breaks: the
counterweight is always there and needs no power to function.
Wouldn't it be fun to have some kind of human-powered elevator on the
playground?

Second, this station can serve to reinforce ideas about force and
acceleration (Newton's laws of motion).  When there is only slightly
more weight on one side, the net force due to gravity is small, and
that side accelerates downward quite slowly.  But with a relatively
small counterweight, the the net force due to gravity is large, and
the heavy side accelerates downward quite rapidly.  It's kind of like
a seesaw with rope, which makes it relevant to the playground.
See the Wikipedia article on the Atwood machine for a nice diagram,
and this video demonstrating the small acceleration when the weight is nearly the same on each side.


Third station: this was very much like a small seesaw, with a meter
stick balanced on a pivot at the center.  The kids could hang weights
of various sizes at various distances from the center.  They were
supposed to figure out that a small weight placed far from the center
could balance a much heavier weight placed close to the center.
However, five minutes was not enough time to absorb this.  In many
cases it took them just a few minutes to figure out that if one side
of the balance beam is down, piling more weight on that side doesn't
help balance it!  And others were not cognizant that the weights came
in different amounts, from 5 to 50 grams, and just counted the number
of weights rather than the total amount of weight.  (OK, I know the
gram is not technically a unit of weight, but we have to keep things
simple!)  So in the future I would structure the balance beam as a
complete activity in itself, and define a series of goals starting
from a very basic level.  This time, I can forgive myself because I
only had one setup, which wouldn't have worked for 6-8 kids.  Anyway,
with the balance beam station I also brought the discussion back to
the playground.  What fun things could they design which might involve
balancing big things on one side and small things on the other?  Maybe
a balance beam for kids to hang from and balance each other?

After the stations, we had a 5-minute wrapup for each group of 6-8
kids, discussing some of the nuances I wrote about above, which were
missed in the quick 5-minute rotations.  This is the first time I
tried having small groups work on different things, and I have to say
it was hectic.  Thank goodness the school is well staffed!  I had at
least one teacher or or aide or intern rotate in with each group,
which saved the whole thing from being a complete organizational
disaster.

After all the groups rotated through, the kids reassembled in one big
group for circle time, and I asked for 5-10 minutes to do a few demos.  I
did these in the big group because (1) there was no time to do it
during the rotations; and (2) the kids would have fought over these
things if it had been a hands-on activity.  First, I showed a rod with
a heavy ball on one end and a light ball on the other end, and I asked
how I should place the rod so that it balances on my finger.  Not
everyone answered near the heavy ball!  So it was worth demonstrating.
But the really cool part is that if something is well balanced, it
will rotate nicely.  So I showed how it spins about its balance
point very smoothly and for a long time, whereas it clearly would not
spin nicely about the center of the rod.  Here's a video: (apologies for
the appalling quality of the video.  I figured it was more important
to help parents see what their kids saw than to worry about looking
good.)

Second, I demonstrated Newton's cradle.  This again relates to forces,
and a large version would make a really cool addition to a playground. 
(Note: if your kids have studied pendulums, Newton's cradle may be best
understood as a kind of pendulum.)

To wrap up, I asked for their ideas on the playground.  After taking a
few, we ran out of time, and we agreed that kids would draw their
concepts during free-choice time.  In two weeks, I'll return to the
elementary for some activities involving rotation, and the next day
we'll take an optional family field trip to the Berkeley Adventure
Playground which has "many unusual kid designed and built forts,
boats, and towers."  Then we'll get to work more seriously on
designing our own playground!

Length, Area, Volume, Dinosaurs and Giant Insects

One of the strengths of Peregrine School is that learning is
integrated.  Instead of offering isolated concepts and facts, we try
to design activities which build on and reinforce each other.  The
past week or two, the elementary students have been reviewing how to
measure.  They have measured their bedrooms and mapped their rooms on
pieces of graph paper, for example.  So I thought I would follow that
up by specifically relating length with area and volume and leading
into one of the most underappreciated aspects of science: scaling
relations. 

(By the way, I didn't get time to talk about a lot of these ideas with
each group, so this post may be a good resource for parents who would
like to follow up the activity and investigate more.)

Beforehand, I printed out some graph paper with 3/4" squares to match
the unifix cubes we had on hand.  This way, when we shifted from
measuring area (counting squares of graph paper) to measuring volume
(counting cubes), the kids would not be distracted by a change in the
size of the measuring unit.  To people who understand area and volume,
a change in grid size might be an irrelevant detail, but to those who
don't, it could be the start of a long and unnecessary detour.

First, I asked them to draw a map of a room which was 1x1, and measure
the area by counting the squares inside (1, obviously).  This should
be really boring and easy, but it took a surprisingly long time to get
everybody clear on what I was asking.  Maybe next time, to speed
things up, I would start by drawing the 1x1 square so they would
instantly see my intention rather than having to parse my words.
Next, I asked them to draw a 2x2 square, predict the area, and then
measure it.  Over the course of 3 groups, I learned to be stricter
about making the prediction.  Once kids understand what the task is,
they are quite happy to do bigger and bigger squares and get answers
without making any predictions.  But they don't learn anything this
way; only if they see that their prediction was wrong will they take
the trouble to figure out what's really going on.  (In principle, the
discomfort of realizing that reality conflicts with his/her beliefs
("cognitive dissonance") will spur the student to make the effort to
really understand.  In practice, it doesn't always work that
way...perhaps the subject of another post.  But the fact that some
students won't care when their predictions are wrong should not stop
us from trying; they are even less likely to care if they never commit
to a prediction.  And if we get even a bare majority of the class
modifying their beliefs based on evidence, we are doing a great
thing!)

So, in the spirit of being strict about making predictions, I ask you
to make a prediction.  If you were to double the length and the width
of the 1x1 square, what would happen to the area?  What if you were to
triple the length and width?  Quadruple? Make your predictions before
reading on!

With the kids telling me the results of their experiments, I compiled
the following chart:

length   area
---------------
1        1
2        4
3        9
4        16
5        25

Note that very few of them made correct predictions of these results!
I also made a graph of this, to show them the pattern in different
ways.  The graph shows a very rapidly rising curve.  The area
increases much faster than the length!

Why should we care? None of them knew.  And I think this is a weakness
with traditional math education.  These concepts are taught in a
vacuum and never made relevant. (A recent op-ed in the New York Times
agrees with me on this.)  So I gave two examples:

(1) if you wanted a room twice as long and twice as wide, how many
times more carpet would you need to cover the floor?  Amazingly, no
one got this right even after I pointed to the chart, with numerous
hints!  The most popular answer was double.  They were not able to
transfer the idea from the chart to a new situation, so they had not
really learned.  Good thing I asked them instead of just moving on!
So we discussed this a bit before moving on.

(2) Suppose you have a small room full of people, like at a party.
There's a door in each wall so if there's a fire, they can easily
escape.  But now imagine a room 3x longer and 3x wider.  It can fit 9x
the people, so now there will be long lines at the doors when the fire
alarm goes off.  People could die if architects didn't think about
these things!  We'll need a lot more doors for rooms which, if we
thought only about length, might not seem so much bigger.  Or you
could think about evacuating a city.  A city 10x the size in each
direction has 100x the people, but nowhere near 100x the number of
escape routes, so it may be simply impossible to evacuate in a short
amount of time.

(3) I didn't have time to show that surface area behaves the same way
as area, but it does.  That is, if we wanted to carpet ALL the
surfaces of a room, a room twice as big in each dimension would still
take only 4x the carpet of the smaller room.  (In either case, the
total surface area is 6x the area of the floor.)  Furthermore, the
same is still true of an irregular surface such as the surface of your
body.  A dog twice as large in each dimension will have quadruple the
surface area you need to brush!

Ok, now we moved on to volume.  I asked them to model a 1x1x1 room
with cubes and tell me the volume, then predict the volume of a 2x2x2
room.  Again, no one got it right!  We quickly built up a new column
in our chart:

length   area    volume
---------------------------
1        1               1
2        4              8
3        9            27
4        16          64
5        25        125

(actually, I filled in the 64 and 125 due to lack of time).  Again,
why do we care?  Well, the volume increases much faster than even
the area does, so if you wanted a swimming pool twice as long,
twice as wide, and twice as deep, it would have quadruple the surface area but
you should plan to pay for eight times the water to fill it up!

And here's where I finally got to the applications I thought would
interest the kids.  If you have a dog twice as long, twice as tall,
and twice as wide as another dog, it will have 8x the volume and thus
8x the weight.  You will have to feed that dog 8x the amount of energy
if it's going to have the same activity level as the smaller dog (thus
explaining why larger dogs generally do NOT have the same activity
level as larger dogs).  If you have a dinosaur four times the size of
a rhino in each dimension, it will weigh as much as 64 rhinos!

This actually explains why there is a limit to the size of animals.
If you double the size of an animal in each dimension, it has eight
times the weight to support.  But the strength of its bones grows only
as their cross-sectional area (4x in this example).  So the bones will
not be able to support the weight.  To make that bigger animal, we
will need to change the plan and not just scale everything up.  This
explains why we don't see mice the size of elephants.  To go from
mouse-size to elephant size requires a drastic redesign and
proportionally much thicker bones.  And there's a limit to how thick
you can make the bones, so there's a limit to the redesign strategy.

So why are the largest whales so much bigger than elephants?  Because
whales don't really support their weight; the ocean does.

And why were the largest dinosaurs so much bigger than elephants?  I'm
guessing that it's due to a more subtle limit on the elephant.  If you
double each dimension of an elephant so it has 8x the weight, then it
also produces 8x the heat because it is warm-blooded.  But this heat
can only escape through the surface area, which is 4x the original surface area.  So the 2x-scale elephant would overheat!  Dinosaurs (at least the biggest
ones, as far as we know) were cold-blooded, so they didn't have this
issue.  (It was also thought at one time that the largest dinosaurs
were swamp-dwellers so that water supported some of their weight, but
this seems to be unsupported by evidence.)

(Why don't blue whales overheat?  I'm guessing it's because
water conducts heat away much better, and because they are very long
and skinny, thus maximizing their surface area; they are NOT 2x-scale
copies of smaller whales.)

This heat consideration explains a couple of other things.  Larger
mammals have slower metabolisms than smaller mammals, to avoid
overheating.  (Slower metabolisms, by the way, enable longer life
spans.  Ever notice how the life span of a mammal correlates roughly
with its size?)  And elephants have enormous ears to help them get rid
of heat.

Finally, let's think about insects.  Insects get oxygen by absorbing
it through their "skin", so if we make a 2x-scale insect, we can get
oxygen in at 4x the rate, but we have 8x the amount of flesh which
needs oxygen.  That's not gonna work.  So we have a limit on the size
of insects.  But why were there 2.5-foot dragonflies hundreds of
millions of years ago?  Turns out that there was much more oxygen in
the atmosphere at that time, so each bit of surface area could take in
oxygen more effectively, and the area/volume tradeoff could be pushed
further towards more volume.

Now, I didn't get time to talk about many of these ideas, at least in
any detail.  We will expand my science slot to 30 minutes with each
group.  But here are some resources for parents to follow up:


Wikipedia article on the square-cube law: this repeats much of what I said about volume vs area, but has links to related articles in more depth.  In particular, if you like to think about animals, you might want to check out....

Wikipedia article on allometry: allometry is the fact that most larger animals are NOT scaled-up smaller animals, but have changes in body plan.

Wikipedia article on Meganeura: this is the 2.5-foot dragonfly.  It has a link to the 1999 Nature article linking insect gigantism to oxygen supply.

The World in a Grain of Sand: Science Through Beginners' Eyes

To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.

                         --William Blake


This blog will be about my adventure in bringing science to preschool and elementary-school children over the coming year.  Although Blake is known as a mystic rather than a scientist, this quote captures how I feel about bringing science to kids: too many people have never seen that curiosity and a sense of wonder are a big part of science.  Everyday objects like grains of sand and flowers, if looked at with a genuine sense of curiosity, lead us to think about bigger and bigger ideas, until we practically hold infinity in the palms of our minds.

The main purpose of this blog is to record something about my activity each week: what I did with the kids, how well it worked, how it could be made to work better, etc.  This is for my own benefit so that I have a journal to refer back to if I do this again next year.  If others happen to look at this blog and use it to help bring science to kids, that would be fantastic.