Liquid nitrogen cannon

This relates to our recent themes of solid/liquid/gas and pressure, but mostly it was for fun.  I brought a steel tube just a bit larger than a tennis ball, a 16-oz plastic soda bottle, a tennis ball, and some liquid nitrogen.  After the usual LN2 demos, we went outside to make the cannon.  I found a large object to hold the tube upright, then I loaded the bottle with about 4 oz of LN2, screwed the cap on tightly, dropped the bottle into the tube, and dropped the tennis ball on top.  Then we waited for the liquid nitrogen to boil into gas, which would make the pressure inside the bottle roughly 100 times atmospheric pressure.  It took about 5 minutes, and then the bottle exploded, propelling the ball hundreds of feet into the air and out into the neighboring field.  Even the ripped-apart bottle shot up in the air, about 50 feet.  It was awesome!

I had a second bottle and ball, but I was too greedy.  I put in about 8 oz of LN2 to get a bigger boom, but after a 7-8 minute wait we heard a hissing.  The bottle had a nonexplosive leak.  Since it was starting to rain as well, we went inside, figuring it would not blow.  But after another 7-8 minutes, we heard it blow.  Bottom line: it's awesome with just 4 oz.

A Sense of Scale

Today we covered three small topics.  It's my last day with the 1-2
graders (I will rotate to the upper graders), and also the last day
before Christmas vacation, so I tried to squeeze in several fun things
and also answer some of the questions which arose last week.  So we
didn't go super-deep in any one topic, but we had a blast. 

We started with a movie: the ten-minute 1977 classic Powers of Ten by
Charles and Ray Eames.  I wanted to show this movie because the kids
had many questions last time, when I mentioned galaxies but didn't
have time to really explain them.  This movie steadily zooms out from
a person on Earth to show how big and how far apart astronomical
objects are.  The movie then zooms progressively in to show the sizes
of microscopic things.  (Note: if you want more snazzy modern special
effects you might try the more recent Cosmic Voyage, but Powers of Ten
feels more intense.)  I stopped the movie many times to answer
questions as they arose, but eventually there were too many questions.
We had four 3-4 graders in the room, and I will be doing astronomy
with them in the spring.  It looks like showing this movie would be a
good way to start my three months with them.  They could generate
questions, and we could take our time answering them.  The best
question today was, "How do we know all this?" and I hated not having
time to give a real answer.  My three months of astronomy with the 3-4
graders will be the answer to "How do we know all this?"

I then showed a website for visualizing the sizes of things, where you
control the zoom. This is a great site for showing how much
bigger the Sun is compared to Earth, how much bigger some stars are
compared to the Sun, etc.  But be warned: they do NOT show the space between these objects, so don't be fooled.  The space in between stars
is VERY, VERY BIG compared to the stars themselves. 
Apart from that, it's a great tool.  (One caveat: I did not zoom out all the way to the "estimated size of the universe"...there is no estimated size of the universe.)  Exploring this site also generated many questions, so that may also be a good icebreaking activity.  Because you can zoom in as well, it could even be a good icebreaker for life sciences as well.

A few more links for interested parents:

• http://www.nikon.com/about/feelnikon/universcale/  is a similar idea as the previous link but with a different feel.  It's worth checking out, but it mostly focuses on microscopic things rather than astronomical things (it has a few mistakes too.

• http://www.wordwizz.com/pwrsof10.htm is another site along these lines; it's not as pretty as the previous two, but it does properly show the vast space between things.

• http://www.powersof10.com/ is a site (currently in beta) by the Eames Office. I just discovered while gathering links for this post, so I can't say much about it other than it looks promising.

Activity number two was understanding orbits with a donutapult demo and play with the coin funnel.  Since I've blogged about this before, I won't describe it in detail here.  If you'd like to read about it, search for these terms (donutapult and coin funnel) in this blog's search box.

Activity number three will be the next blog post.

Origins Part II

(This is a continuation of a previous post.)

After a short break, we tackled the Big Bang.  I asked if we needed to
extend the timeline even further back than the origin of the Earth.
They were clear on the need to do so, since the Earth and the solar
system formed from a pre-existing cloud of gas.  I showed them some
pictures of galaxies and then some of a fly-through movie of galaxies from
the Sloan Digital Sky Survey.  I just wanted to roughly establish that
galaxies are like neighborhoods: we have ours, and we can see where
some others are too.  It was clear later that they have no real idea
that galaxies are much bigger than the solar system (even though I
said it); they kept mixing up planets and galaxies.  But I didn't
dwell on that; I figure there's only so much I can do in one morning,
and it was more important to establish that "things are moving apart"
than to work on a sense of scale.

Next, I took a long loose spring (almost like the helical telephone
cord that used to be on all landline phones) to which I had attached
galaxies (each galaxy had the name of a kid).  Starting with the
spring scrunched up, I extended the spring 12 feet or so and got the
galaxies far apart.  I did this a few times so they could see how all
galaxies moved away from each other.  This was actually the first time
I had done this demo, and now I'm sure I will do it with my college
kids.  I always use a balloon, and I still will, but there's something
nice about also doing the one-dimensional case.  It just makes
everything a lot more visible, especially in a big room.  With the
spring stretched I asked how we could figure out how long it has been
since everything was together.  One kid figured it out right away.  He
explained that knowing how fast they are moving and how far apart they
are, we can calculate the time it took.  He used the example of a
speed of one inch per year.  In that case, the number of inches apart
is the number of years it took to get that far apart.  This was an
amazingly good answer; it was almost as if I had rehearsed it with him
beforehand and planted him in the audience (I swear I didn't).

Next we went outside and practiced doing the same thing with our
bodies.  With Teacher Pa as the Milky Way, we all ran away from her.
Teacher Ethan ran the fastest and became the most distant galaxy by
the time I said stop.  I wanted to make clear that the most distant
galaxy is further away not because it has been traveling for a longer
time, and that there was a time when all galaxies were together.  We
went back inside and I drew a diagram of us as someone above the field
would have seen us.  If that person came upon that scene and saw us
moving very slowly, would he guess that we had started a long time
ago, or a short time ago?  (Long.)  If that person came upon that
scene and saw us moving very quickly, would he guess that we had
started a long time ago, or a short time ago? (Short.)  Using the same
logic, astronomers have found that all galaxies were together and
started moving apart (the Big Bang) 13.7 billion years ago.  I made a
big show of extending the timeline into the hallway and out of the
school to emphasize that that is a long time.

Now, the kids may have gotten the wrong impression that we are at the
center because everything is expanding away from us.  To combat that,
I had prepared two transparencies.  One has a smattering of galaxies, each
with a different shape so that it's recognizable.  I had prepared this by putting
graph paper behind the transparency and drawing galaxies at random coordinates.
I prepared the second one by drawing the same galaxies (now in red instead
of black) at the same coordinates multiplied by 1.5.  You can pick one galaxy
in the middle to represent the Milky Way and show the initial (black) transparency
to show where the other galaxies are around it.  Then overlay the red one, matching
up the Milky Way's position, to see how everything moved away from us.  Here's the
cool part: now you pretend you are an alien in another galaxy, match up that galaxy
across the two transparencies, and you ALSO see that everything is expanding away
from the alien!  This blew everyone's mind, including the teachers.  We spent a fair
amount of time with the kids picking a galaxy, and me matching that galaxy and
showing that everything moved away from it.  Bottom line: just because we
observe everything moving away from us doesn't mean we're at the center.  When
there's more space  everywhere, EVERY galaxy can observe this.  We're not special.
People often ask, where did the Big Bang happen?  It happened everywhere,
including here!  All the places that all the aliens in the universe could call "here"
all overlapped , in the distant past, with what WE call "here"!  I reinforced this with
the traditional balloon demo of the expanding universe: draw some dots on a partially inflated balloon, then fully inflate it and show how each dot is further from each other dot, but none is in a special or central position.  If the balloon could really be completely collapsed, then all the dots would be in the same starting position without ever really moving away from its position on the balloon.

But, should we believe that we can extrapolate that far back in time?
We should look for evidence! I made a lame show of demonstrating how
things are hot when compressed (I brandished a bike pump and asked
them to notice how the valve gets hot next time they pump up a tire),
so that the whole universe would have been red hot at some point in
the past when it was highly compressed.  We actually see that light:
it's called the cosmic microwave background.  This is fossil evidence
of a hot early universe.

We were running out of time but the kids voted to do the marshmallow
activity rather than just stop early.  I had brought white and yellow
mini marshmallows and toothpicks.  These represent the building blocks
of atoms (technically, protons and neutrons, but I didn't use those
words).  These building blocks can be stuck together only at very,
very high temperatures which the universe experienced only in the
first three minutes, when it was even hotter than red hot.  (Kind of
like marshmallows will stick together if heated.)  We had talked about
solids, liquids, and gases in previous weeks, so I sketched out
hydrogen (just one proton) and helium (two protons and two neutrons),
which they knew were gases.  I had set up cups half full of a mix of
protons and neutrons, in a 7:1 ratio.  I gave them the cups and told
them they had only three minutes to build as much helium (ie stick two
white and two yellows on a toothpick) as they could.  Because of the
paucity of neutrons, they were typically able to build only 3 helium
atoms, with about 36 hydrogen atoms left over.  This is actually the
ratio we observe!  So the atoms themselves are additional fossil
evidence of a very hot early universe.  [Parents: if you're curious
where the 7:1 ratio came from, that came from the even hotter
conditions in the first fraction of a second, and the observed ratio
agrees with the Big Bang model, thus providing even further fossil
evidence. If you want to read more, search "Big Bang nucleosynthesis."]

Overall, it was a VERY successful day.  The kids had many additional
questions about planets and orbits which I didn't take time to answer,
and this could form the basis of an activity for my next visit, and I
do think they gained an appreciation of the basic idea that we can
tell the age of the universe from how fast it's expanding.  How well
they'll remember that or be able to answer questions on an assessment,
I'm not confident.  But the basic idea is not beyond the grasp of 1st
and second graders.  The movie we saw before the break was great, the
spring demo was great, and the marshmallow activity was good.  (We
didn't have enough time to do it really well, but it's a very
promising idea and I will develop it further.  Given more time, I
would have the kids make mini posters with "raw ingredients" "elements
after cooking in the heat of the Big Bang.")  The radioisotope dating with
dominoes (see previous post) came up a bit lame, but I think it's a good idea
that just needs more refinement.

As the kids went to lunch, I worked on the science section of Teacher
Pa's poster comparing different religions' creation stories.  Because
I wanted to emphasize the LACK of parallelism as discussed in my
previous post, I ripped off the science column and posted it on the
wall next to the window where the religion part of the poster was.  I
also did not carry over the formatting of the rows in the religion
column.  I hope to post a picture here rather than describe it in
words, but I think I achieved the right balance in emphasizing how
science is different from religion while respecting both.

Update: if you look at this picture full size, you will be able to read the poster.

Origins Part I

Teacher Pa's class as been studying various religions, including their
creation stories, this week, so she asked me to review the scientific
"creation story" with the kids.  She had made a big poster with
Hinduism, Judaism, Christianity, and Islam as column headings, each
with entries in rows titled [Name of] God, [Sacred] Book, Creation
Story, Golden Rule, What Happens After Death, and Holidays, and she
wanted me to fill in a Science column for Creation Story, Golden Rule,
and What Happens After Death.

I wanted to make very, VERY clear to the kids that science is not
another religion, so I refused to tell a "creation story" and instead
made a detective story about our origins.  (It turns out I was
justified: even after spending the whole morning with the kids and
emphasizing how science works, as the kids went to lunch I began
ripping the Science column off the religion poster and my own son
Linus said, "Dad, what holidays does science celebrate?")

I started the morning by discussing what kinds of questions science
can answer and what kinds of questions it can't.  If you're about to
bite into your last cookie and someone asks you to share it, can
science help you figure out if you should share it?  No.  If your best
friend moves away and you're lonely, can science help you figure out
what to do?  No.  Religion might help you with those questions.  But
if you have a question about nature, such as "When did the Earth
begin?", then science can help.  I think it's super-important to help
kids draw these distinctions.  Because religion tries to say something
about our origins, and so does science, it's tempting to make
parallels between them.  But the differences are more important then
the superficial parallels, and we need to help kids see that.  Science
and religion are simply about different things.  If we had a poster
comparing different sports, we wouldn't put Sudoku on it!

The kids had done a timeline of the history of Davis, so I started
with a blank timeline with "Now" on the right and "?" on the left.  I
put a few recent events (the years they were born) close to "Now" and
asked how we could know about the distant past using evidence (clues).
Because they had recently been to Yosemite and seen a slice of a tree
with about 1,000 rings, I started with that: we know that trees grow
one ring each year, so this tree tells us that Earth is at least 1,000
years old.  In fact, the oldest trees in the world live in California
and they are over 4,000 years old, so I marked that too.  (Aside: by
matching long-dead trees with just-felled trees [using ring thickness
as an indicator of how good for growth each year was], scientists have
been able to put together tree-ring histories going back about 10,000
years!)

Next, we moved on to rocks. They had studied some geology in
preparation for Yosemite, so we reviewed how long it takes millions of
years for a river to carve a canyon, based on how fast we observe it
carving today.  So Earth is at least millions of years old.  One kid
knew that some rocks are at least 1,000,000,000 (one billion...I wrote
out the number to impress them) years old.  But how, I asked.
"Dating."  OK, but how do we do that?  I did a very simplified version
of radioisotope dating.  I took some dominoes and stood them up on a
desk.  Standing up, they have some potential energy, because they have
the potential to fall.  Once fallen, they don't have potential energy.
(We had talked a bit about this concept previously.)  Now some atoms
in your bones (or in rocks) have this extra potential energy, but as
time goes on more and more of them lose this.  I knocked down a few to
illustrate the passage of some time, then a few more to illustrate the
passage of more time, etc.  They quickly got the idea of "more down
equals older" (I gave them many scenarios and they got the relative
ages right) but I'm not sure what they were really visualizing when we
said "more energy" or "fall down" because I got questions about
whether the atoms are dead or had changed into something else.  A nice
thing about these dominoes was that they came in different colors, so
it was easy to point out that this domino is still a red domino with 5
and 2 dots, it's just that it doesn't have extra energy now.  So I
think the got the idea that we were using small particles in the rocks
as a clock, but not much else.  Which is probably ok; you can't do
everything.  (If I had planned this whole semester better I probably
would have brought in a microscope very early on, and established the
concept of atoms so that I could safely refer to it throughout the
semester...last year all the kids in the school studied atoms but only
one of those kids is in this room this year.)

So I extended the timeline all the way across the other (very long)
whiteboard and wrote 4,500,000,000 as the age of the oldest rocks on
Earth.  I then mentioned meteors, which they had heard of, and how
their slamming together would generate heat.  (I slammed clay lumps
together for visual effect.)  We think Earth was formed by meteors
slamming together and creating so much heat that they melted together.
The rock-dating clock starts when the rock solidifies, so the age of
the Earth is 4,500,000,000 years.  I then wanted to show them a movie
rendering of this process, and I showed the first few minutes of the
Birth of the Earth episode of How the Earth Was Made; in the first
several minutes they have some really nice visualizations of this.
But they like it so much that we kept watching, well into break time,
and almost finished.  But with about 10 minutes left in the 43-minute
episode, I really wanted them to stretch their legs so we encourage
them to go outside but left the option of continuing to watch. Half
the kids watched to the end.  I highly recommend this episode, and in
fact this whole series.  It emphasizes the use of evidence to test
ideas.

The kids had MANY questions in response to the video.  It was great to click Pause as soon as a question arose so I could deal with it right away.  I felt like the movie was an awesome way to keep their attention (which is sometimes a struggle), but I could still provide an interactive teaching environment.  It was the best of both worlds.




I have a lot more to say about what we did after break, but I'll make
that another post.  To be continued....

Written in Fire

We split Friday morning into two unrelated activities: sound, and a
review of states of matter.

For sound, I brought a lot of toys: tuning forks, a xylophone, etc.
The standard I wanted to cover was "sound is made by vibrating objects
and can be described by its pitch and volume" so I started with the
tuning forks and steered a discussion of pitch and volume (they
noticed right away that the tuning fork vibrated).  It was pretty
funny, as the kids focused entirely on pitch, and I could not get them
to guess, despite numerous hints, that VOLUME or LOUDNESS was a
difference between the two sounds I was playing, even when they were
the same pitch.  (It didn't help that when I really whacked the
xylophone hard, it did change its pitch somewhat as the whole thing
shook.)

Then I turned the kids loose to play with the tuning forks, the
xylophone, and a few other toys:

• bathtub flutes, which you can fill with water and then blow on while they drain.  The pitch corresponds (inversely) to the length of the wave which just fits in the air-filled part of the tube, so the pitch starts out high and then drops as the water level drops.

• plastic hoses flared on one end.  You whirl them around quickly, and they make an eerie whistling noise.  Same principle as the flute, only this time the length is fixed, and we make the air flow by whirling the tube rather than blowing on it.

• a "thunder stick" which is a long spring connected to a drum membrane stretched over one end of a tube (the other end is open).  Holding the tube and shaking it results in surprisingly loud boingy sounds.

Then we got back together as a group and talked about how our
observations are explained by sound being a wave.  To visualize this,
Teacher Pa and I stretched a very long spring (like the coiled wire
that ran from a telephone to its handset, in the old days before
wireless phones) across the room, and I bunched up my end and released
the bunch (still holding my end).  It was very clear that a pulse
traveled the length of the spring to Teacher Pa, and it bounced off
her and came back to me.  I related this to the behavior of the
thunder stick (what do you think happens if you hold the spring rather
than the tube?)  We also had two plastic cups linked by a string, and
Teacher Pa gave a chance for each kid to hear her voice carried along
by the string.

For the piece de resistance I brought a Rubens tube.  This is a long
tube with many small holes drilled in a line, connected to a propane
tank.  You turn the propane on and light the holes so it looks like a
row of 100 candles.  Now comes the cool part: there is a speaker
attached to one one.  Hooking it up so that music plays on the speaker
makes waves in the propane in the tube and pushes it out more in some
places than in others.  The fire dances to the music!  Music has lots
of tones mixed together though, so it's best to start with some pure
tones.  I brought a function generator to generate tones of any
desired frequency and amplitude, which is a really great
visualization.  Into and beyond break/snack time, kids and teachers
from all the rooms in the school were cycling through our room and
watching this.  The upper graders were transfixed.  They wanted to try
all their favorite songs.  We found that (to the dismay of some)
Gangnam Style was a really good one for making the flames dance.  We
eventually had to shut it down at close to 11:30, about an hour after
I first fired it up.  To see a Rubens tube in action yourself, check out this educational video.

From 11:30 to 12:15, Teacher Pa led us through some activities
reviewing the states of matter.  For example, she passed out images of
many different things and the kids had to paste them onto a poster in
the Solid column, the Liquid Column, or the Gas column.  This was
really useful for the kids, and for the teachers to assess how much
the kids got it.  In discussing this with the kids, we found a great
idea for next week: clarifying what we mean by "amount" or "size" of
something.  The difference between "size" (which most people would
take to mean a diameter, a distance across, or a height) and volume
came up in the context of gas expanding to fill its container, and it
was clear that we didn't have time to address it that day.  So we'll
do that next time.  It will play into one of the Investigation and
Experimentation standards about measuring length and volume, but I
want to keep it conceptually rich as I did last year.

It's a Gas

Last Friday I discussed solids, liquids and gases with the 1-2
graders.  I brought in samples of each to provide a basis for
discussion.  In addition to the obvious (a wood block, a glass of
water, a balloon filled with air), I brought some things designed to
stimulate their thinking: a rubber band, a cloth, a balloon filled
with water, and sand.  We took about 35 minutes to discuss how we
could define solid, liquid, and gas.  It's not as obvious as you might
think at first; for example, if liquids and gases flow unlike solids,
why can you pour sand?  Does that mean sand is a liquid?  I wish I had
time to document our discussion here!  I'll just document that at the
end it is important to note that substances can change from one state
to another and back depending on the temperature.  Water is the best
example: we talked about glaciers, lakes and rivers, and rain, which
they have already studied this year.  But it's worth mentioning other
examples lest they think this is peculiar to water.  The metal parts
of their desks were once liquid, which was poured into a mold.

In the hour after snack break, we did a more extensive experiment.  I
handed out cups with (small amounts of) vinegar, and they wrote down
observations: it smells funny, it's liquid, etc.  They did the same
with cups of (small amounts of) baking soda.  They also weighed both
cups on the scale together.  I found it easier to use a kitchen scale
which read grams rather than a scientific scale which reads to a
hundredth of a gram, so that I didn't have to explain decimal points.
Then they mixed the two and observed the reaction.  They drew it and
wrote down their observations.  Then we observed what was left: it
smelled different, and it weighed less (typically by a few grams out
of about 100 to start with).  We figured out together where the
missing grams went: when the bubbles popped, the gas escaped (in the
earlier session we had talked about this with respect to balloons).

They did all of the above in small groups (individuals, actually,
because we had three adults and four kids!), but we spent a few
minutes at the end summarizing what we learned.  One child wrote on
his worksheet "V [vinegar] +B [baking soda] = air" so that was a great
place to start discussing.  Did we know that gas was air?  What else
was produced?  They seemed to not recognize that what was left in the
cup was also a result of the reaction and should go on the right hand
side of the equation.  Is the stuff left in the cup just leftover
vinegar and baking soda?  No, because it smelled different.  The
equation written by the child was a great insight, but by the end we
produced a more accurate equation with more words.

Finally, it is important to note that this is a chemical reaction: we produced
some new kinds of substances! This is very much unlike water going
from liquid to gas, which they might have in mind as a model
transformation of a substance.  Because some of them like explosions,
I related it to the chemical reactions in explosions.  Explosions (ok,
most explosions) are chemical reactions too; they just happen faster
and give off more heat.

Then the kids spent 5-10 minutes drawing a scene with as many
different solids, liquids, and gases as they could think of, labeling
each.  At the very end, I rewarded them with a show of Diet Coke and
Mentos fountains.  It was raining, so I bought 1-liter bottles whose
fountains could be contained in the sink.  This turned out to be a
great capstone for the morning, as carbonation seemed to be a new idea
to many of the kids.  We tasted the Diet Coke before and after
defizzing to see the effect of gas on our taste buds.  We also
discussed how the gas was in the liquid and normally comes out slowly
(you can see small bubbles coming out when the cap is off) but comes
out quickly with the help of Mentos.

For those who want more: Here is an entertaining video of Diet Coke and
Mentos reactions.  If you want to go beyond entertainment and learn more
about why it happens, you should watch the Mythbusters episode on Diet
Coke and Mentos.  They do experiments with different ingredients to figure
out what is most responsible for the reaction.

Balance, and floating vs sinking

Today in the 1-2 grade room we had a blast with some of the ideas we
need to use in making the water feature.

First, balance.  I brought in two-meter-long sticks on pivots, along
with sets of weights of various sizes, and had the kids hang weights
in different places and then see where they had to place other weights
to balance it out.  They quickly discovered that a small weight can
balance a large one, IF it is placed at the end of a long arm.  This
was a really good exercise because, in contrast to some of our
previous ones, I had enough equipment for each child to explore
completely on his/her own. 

The pre-snack period culminated with two capstone events:
(1) I gave the kids a worksheet in which I drew balance beams
with a weight on one side (varying the size and position of the
weight), and they had to draw the weight (size and position) they
would put on the other side.  Mostly they got it right, and in the few
cases where there was confusion we had the equipment right there to
check if their drawing represented reality.  (2) I demonstrated how
balance facilitates rotation.  You can see a video I made about this
demo at the end of this blog post from last year.  As kids went to break,
some of them commented how this demo is like the Moon going around the
Earth, and asking whether the Earth wobbles a little as it does so.
The answer is yes, and so does the Sun as the planets (Jupiter has the
biggest effect) go around it.  Therefore, if you saw a star which was
wobbling, what could you conclude about it?  Right, it has planets!
This is really how astronomers do it; the vast majority of planets are
too faint to see directly given the glare of their host stars.

Post-snack, we switched to fluid mechanics.  We started by reviewing
what we learned about pressure last time, focusing on why water
doesn't fall from a straw when you cover the top with your finger.  I
then showed the same idea in slightly different form: with two 2-liter
soda bottles screwed together, water does NOT fall from the top one to
the bottom one (it may drip, but it doesn't make the waterfall you
might expect in an open-bottle situation).  The water doesn't fall
because for the water to go down, the air in the bottom bottle has to
move up, and the two get in each other's way.  We then figured out how
to make them not get in each other's way: swirl it to make a "tornado
in a bottle."  The air goes up through the middle while the water
swirls down around the outside.

We then took some time for each kid to make his/her own tornado in a
bottle, with the option of coloring and/or glittering the water.  This
was great fun; the kids were really into it and came up with some
pretty (and/or Halloweeny) combinations. 

Next, we studied floating and sinking, following more or less the
script from one of my Primaria sessions last year (adding a bit of
sophistication such as introducing the word density).  But we had time
only to get to the egg in the salt water.  We'll do the rest next time.

At the last minute, we stumbled into a nice connection between the egg
and geology.  Teacher Pa said that the way to tell if an egg has gone
bad is to see if it floats (in non-salted water).  Linus had said just
5-10 minutes before that pumice is a rock that floats because it has
lots of gas bubbles in it.  So the connection is that an egg which
floats (without the help of salt) probably has gas bubbles in it,
which clearly is a sign that it's going bad.

Hydrodynamics 101

Today I worked with the 1-2 graders to extend their concepts of force
and motion to include work and energy, and then, after the break,
fluid dynamics.

While waiting for the kids to come back from chorus to start science,
I sat with one child who hates chorus, and we interleaved the pages of
two phone books.  When science started, we talked about friction and I
used the phone books as a demo.  The friction of 200 pages trying to
slide past 200 other pages is so much that two strong adults cannot
bull the books apart.  Mythbusters had a great episode on this, in
which they used bigger (800 page?) phone books and couldn't pull them
apart even with cars.  They finally resorted to military tanks, and
found that it took a force of 8,000 pounds to separate the books!

We then talked about work, which is applying a force over some
distance.  Sitting in your chair, you are applying a force (your
weight) to the seat of the chair, but you are not doing work.
Exerting a large force (eg lifting a heavy weight) over a large
distance makes for a lot of work.  We related this to irrigation
because the kids are studying the community, and are about to learn
that farming really took off around here when large pumps became
available to move the water.

Energy is the ability to do work, and we spent a looong time talking
about different forms of (mostly stored) energy: food, chemicals,
light, heat, electricity, etc.  We spent a loooong time figuring out
what makes the electricity that comes to our houses!

Then came break.  After break we finished up a few more forms of
storing energy: magnets, rubber bands, springs, etc.  But mostly we
moved on to discussing how water moves (fluid dynamics).  I did the
"three-hole can" demo (see paragraphs 3-4 of this post) to introduce
pressure and the relationship between pressure and water height.  Then
I did the finger-on-the-straw demo (paragraph 6 of that post) to show
that the air also exerts pressure.  Next was a siphon tank demo, to
show that air pressure can sometimes help quite a bit in moving water.
This demo did not work well, possibly because of a leak, so see this
video.  Finally, I did the balloon in a bottle demo (paragraphs 6-8 of
the post linked to above) which is very analogous to the
finger-on-the-straw demo but far more dramatic....I could see Teacher
Ethan do a double-take when he first saw it.

Then I led the kids through designing different water systems on the
whiteboard.  I supplied basic ideas such as water flowing into a
shovel on a pivot, and asked them to predict what would happen (when
the shovel fills with water, that end pivots down, dumping the water
out).  We went through a bunch of these ideas, and I made sure to lead
them to realize the need for a pump to cycle the water back from the
bottom to the top.  By this point they were very eager to start
drawing their own ideas, which played right into my plan.  We had a
great time making posters of our ideas.  In the last five minutes, I
unveiled the hydrodynamics kit which they will use in free-choice time
(or whenever Teacher Pa deems fit) to actually implement their ideas.

Overall, I think it went really well.  We discussed a lot of ideas,
without overwhelming the kids, and the poster-drawing session was both
fun and educational.

Understanding the gravity of the situation

Last Friday with the 1-2 graders we reviewed and extended our
observations of force and motion which we began two Fridays ago,
before the Yosemite trip.  Because it had been two weeks, we started
with quite a bit of review, which I did by asking the kids questions
rather than lecturing to them.  We observed the motion of a rolling
ball in order to change the context from last time (when we used a
hoverpuck or a marble shot out of a blowgun).

I had them observe and draw some motions.  This addressed California
Grade Two science standards 1a and 1b, as well as built the case for
the following argument.

By observing a ball thrown up in the air, we concluded that there is a
force on it even when I am not touching it, and that that force is
simply gravity.  I then repeated the donutapult demo to refresh their
thinking on how something goes in a circle only when there is a force
on it; if there is no force on it, it will go off in a straight line.
Then we talked about the Moon and how there must be a force on it
because it goes in a circle around the Earth.  That force is also
gravity!

(I think the following was too advanced, but we did discuss it.
Gravity always points to the center of the Earth.  One student is
going back to Korea soon, so I drew Davis and Korea on a globe and
showed how this must be the case.  Then I noted how the force on the
donut also points to the center of its "orbit" because that is the
only direction the string can pull.  So there is very strong reason to
think that the force on the Moon is Earth's gravity, the same force we
know and love, that makes things fall when we drop them! [Standard 1e])

After the break we discussed how to send forces in different
directions and in different amounts by using simple machines such as
levers, pulleys, and gears.  I had brought in the Gears!Gears!Gears!
toys earlier in the week, so they easily got the basic idea of this
standard (1d).  But I lost them    when I got into the details of
levers...they weren't able to predict where to place a lever and a
fulcrum to perform a given task, nor were they able to draw arrows
indicating the sizes of the forces at the different ends of the lever.
And I didn't really have the equipment handy to do real hands-on work
with that, so I may do this again this Friday with better equipment.

Newton's laws for 1-2 graders

Friday I spent an intensive morning with the 1-2 graders working on Newton's
laws.  The format was quite a contrast from last year when I had 20 minutes with
each of three mixed-age groups!  That was insanely rushed.  Still, I followed more
or less the same format as last year, with the hoverpucks, the donutapults, and the carts, so I won't rewrite all that here.  I added a few things, which I will describe here, but mostly we used the time (45 minutes before morning break and an hour after) by going a lot more slowly and thoroughly.

I knew it was going to be a long morning for them, with a lot of different things to
pay attention to, so I started with an overview.  I started by writing three goals on the board and we talked about what goals are.  I told them that if we accomplished all the goals they would get a present at the end.  The goals were:

• observe how things move
• make a model to explain these observations.  I phrased it this way because the previous time I had worked with them one-on-one, we made models of how the mystery tubes work.  I wanted to draw an explicit parallel: a few simple connections will explain and unify a whole lot of observations.
• figure out how to measure pushes and pulls (forces)

We did a lot of observations of the hoverpucks and the donutapult before break. One thing I could do better next time with this age group is let them play with the hoverpucks first, and then ask for their observations; that might be easier than holding their attention through some demos and then letting them play to build on that.  In any case, by break time we had figured out that objects don't change their speed or direction unless acted on by a force (note that friction is a nearly ubiquitous force, which always acts to slow things down), and the donutapult reinforced that.

After break, I brought out a new toy which I had made earlier in the week by softening a PVC pipe in boiling water and bending it into a circle (curving it around a bit more than one full loop so that it clearly has two ends).  A marble fits inside the pipe and I blow on it like a dart gun.  When the marble comes out, does it continue curving around that circle, go in a straight line, or something in between?  This was another really fun demo.  [Note that I spent two hours making the darn thing, because this was my first attempt at softening PVC, and I ruined two pots.  Dedicated teachers spend much, much more prep time than most people imagine!]

Then we turned to the carts as per last year's agenda.  A new ingredient I added here is the leafblower on a skateboard.  We can trust the leafblower to always push against the air with a constant force, so stacking the skateboard with different weights nicely demonstrates Newton's second law.  We also heard an interesting misconception from one child: that the leafblower/skateboard had to be near a wall to push off the wall.  So we discussed how to design an experiment to test that, and how the experiment showed that the leafblower pushes against the air, not the wall.

This completed the "make a model to explain these observations" goal: objects don't change their speed or direction unless acted on by a force (Newton's first law); a bigger force produces a bigger effect on a given object, and a given force produces a bigger effect on a lighter object than on a heavier object (Newton's first law).  These kids aren't really ready for a deep understanding of Newton's third law, so I summarized it as "things push back when you push on them."  That way of summarizing it may do more to prevent injuries than to improve their understanding of physics, but I felt that I was starting to lose them and that we should move on to our third goal.

The leafblower was indeed a nice segue to "figure out how to measure pushes and pulls" because when students pushed a heavy cart and then a light cart with the same force to observe the same pattern (a given force accelerates a light object more than a heavy object), they had some trouble really pushing with the same force on each cart.  Their muscles weren't very well calibrated.  So I asked how we could measure the size of a force.  I pointed to the scales we had used earlier, but this didn't generate any ideas other than "use a scale."  So I got a popsicle stick and showed that if I press on both ends lightly, it bends a little; if I press more it bends more; and if I press very hard, it breaks.  This is a rough way to measure force.

We can make it more precise by using a spring.  I hung a spring from the whiteboard tray and asked them what would happen if I hung a small weight on it, two small weights, etc.  (A weight is another thing we can trust to always pull [down] with the same force.)  I had taped a piece of blank paper hanging down from the whiteboard tray, and I used that to start to build up a scale with tickmarks and numbers.  Then we broke into two groups (I had only two springs) to construct two scales. Unfortunately, my group overloaded their spring and broke it rather quickly.  Note to self: bring more, and stronger, springs next time.  In any case, we did construct reasonable scales so they achieved their third goal and earned their reward: each child got a brand new, professionally manufactured spring scale.  Before they could play with them, Teacher Pa made them record some of what they had learned in their science journals.

I'd never done the "measuring force" activity before, and I think it went well.  The kids did play with the scales after recording in their journals, even a bit into recess time, so that was a good sign of engagement.  Linus and Malacha experimented with multiple springs set up in parallel and in series.  They observed, for example, that when two springs hold up a weight, each is extended only half as much as it is when it has to hold the same weight alone.  This is because each has to hold only half as much weight.

Some kids expressed interest in having a hoverpuck at home.  They are only $20 and are sold under the name Kick Dis.

Glaciers, Plate Tectonics, Rock Cycle and Fossils: The Geology and Yosemite

Friday was jam-packed with science this week as Teacher Carol and I
helped the upper graders demonstrate the geology of Yosemite to the
younger children, in preparation for our field trip there.  I stayed
in the 1-2 grade classroom, so I will mostly report from there.

Carol set up four half-hour activities:

• glaciers
• structure of the Earth (crust, mantle, core) and plate tectonics
• [snack/recess]
• the rock cycle
• making fossils

In each activity, the upper graders kicked it off by explaining the
topic with the aid of posters they had made (you can read more about
Carol's work preparing the upper graders on her blog).  The upper
graders knew their stuff but had not been trained in pedagogy, so
Teacher Marcia and I facilitated by asking questions and repeating
explanations with simpler words and examples when necessary.  (Teacher
Marcia was really excellent in this regard!  At some point after
discussing erosion, the movement of rocks came up again and instead of
assuming the students instantly made the connection to erosion, she
asked "Do rocks have legs?"  This was funny but also made the children
stop and make connections to what they had learned earlier.)  Then
each topic turned to a related hands-on activity or demonstration:

Glaciers: we went outside as the upper grades made a block of ice
slide down a "mountain" of sand in the sandbox.  The kids sketched it,
then returned in the afternoon to sketch it after the glacier melted.
The point was to observe the pile of soil and rock left at the point
of the glacier's farthest advance.  We will see moraines like this in
Yosemite.  Often, they serve as dams for rivers which form in the
channel left by the glacier, and thus have lakes right behind them.
This phenomenon of course wasn't visible in the sandbox demo but I
wonder if we could tweak the demo next time so that it is.
  
Structure of the Earth and plate tectonics: we used a hard-boiled egg
to demonstrate a really thin crust (the shell) over a mantle (the
white) and a core (the yolk).  The Earth's crust really is that thin
relative to its bulk!  Slicing the egg in half also fractured the
shell into "tectonic plates."  We further demonstrated different ways
in which plates interact at their edges (convergent, divergent, and
transform boundaries) with pieces of cardboard, paper, and our hands.

The rock cycle: we grated crayons to represent erosion, then we
deposited the grains into a riverbed of aluminum foil.  We did this
for a few different colors to make distinctive layers of sedimentary
rock, then we wrapped up the foil and added pressure (with kids'
hands) and heat (with a torch).  When we opened the foil we found
metamorphic rock!  The torch was my idea because kids love flame, but
it melted the outside without melting the inside, so I would recommend
Carol's original suggestion of a hot-water bath to supply the heat.

Making fossils: we transitioned from the rock cycle to this by
discussing how older layers of rock are deposited first and buried
further down, so we can relate the rock layers to the ages of fossils.
The 1-2 graders are really into dinosaurs, so this was a great
transition: training for dinosaur hunters.  Beforehand, Carol and I
half-filled small paper cups with clay and coated the flat top of the
clay with a bit of Vaseline.  The kids chose from a selection of
animal figurines and pressed their animal into the clay.  They removed
the animal to simulate the decay of the flesh, but the imprint
remained.  Then a mudslide came along (me pouring wet plaster from a
large cup) and buried the imprint.  They took the cups home and
excavated their fossils the next day.

It seemed like a great experience for the kids, but it would also have
been great if it had been a little more spread out, say over two
Friday mornings.  We were asking the 1-2 graders to absorb a lot of
information in one morning!  Teacher Marcia found a good way of
spreading it out after the fact: Carol provided worksheets for the
kids to fill out, but we didn't have time for that because we had to
go slower for the 1-2 graders, so Marcia decided she will use them to
reinforce and review over the next week.  Apparently the 3-4 graders
were able to complete their worksheets in the morning.

The upper graders certainly learned a lot in the week leading up to
this Friday, first learning from Carol (with the worksheets asking
them to articulate their knowledge), and then making posters and
rehearsing demonstrations to prepare for teaching the lower graders.
(If you want to read more about Carol's work with the upper graders,
see her blog.)  However, because the upper graders had no training in
instructive strategies (asking questions, asking students to come up
with additional examples, etc), the teachers in the room had to
intervene a lot (Carol confirmed that this happened in the 3-4 grade
room too) and by the end the upper graders had become somewhat
passive.  I wonder if we could improve this next time by asking the
upper graders to fill a more specific role rather than a general one,
for example each doing a certain experiment or demo which was
self-contained enough for them to feel expert in.  They were certainly
good in helping the kids one-on-one, for example in making the fossils
and, in the 3-4 grade room, in responding to questions asked by the
worksheets.

Desperately Seeking Distances

One of the most shocking things about astronomy is that when we take a
picture of celestial objects in the night sky, we have very little
idea how far away they are.  This is utterly different from everyday
life, where our brain automatically processes distance-related clues
and instantly supplies us with correct judgements.  The brain knows
the true sizes of everyday objects, so it can use the apparent size
of, say, a car to infer its distance: the smaller the car appears, the
further away it must be.  The same can be done with the apparent
brightness of lights: if we see headlights but they appear faint, we
know the car is still far away.

But in astronomy, we can only figure out the true sizes and distances
of things with a lot of effort.  One difficulty is simply that
everything is so far away: apart from the Sun, no star is close enough
to ever appear larger than a point, so we can't judge their distances
by their apparent sizes.  Another difficulty is that the universe is
far less standardized than our man-made world: most cars have more or
less the same true size, but stars and galaxies come in a vast range
of intrinsic sizes, preventing us from forming a rule of thumb about
"if it appears this big, it must be about that far away."  Imagine if
some trickster built a 50-foot iPad, faithful in every detail.  If you
mistook it for a real iPad, you would guess that it's much closer to
you than it really is.  The universe is full of the equivalents of
50-foot iPads---stars 100 to 1000 times bigger in diameter and
millions of times bigger in volume than our Sun---as well a
50-millimeter iPads---dwarf galaxies containing thousands of times
fewer stars than does our own Milky Way galaxy.

Astronomers have painstakingly built up a vast store of knowledge
regarding the sizes and distances of things, which I won't attempt to
describe here (but at the end of this post I provide a few links to
sites which help you visualize these things).  The point is that when
a new technique to estimate distances comes along, it's a potentially
powerful tool for astronomers.  Today's episode describing a recent
paper of mine shows how I explored a new idea for determining
distances and showed that it was interesting, but ultimately less
powerful than other ideas that have already been developed.

The new idea is actually an old problem turned on its head, which is
often a useful way to make progress in science.  Imagine that you're
the assistant to a seventeenth-century scientist, put in charge of
monitoring his inventory of chemicals.  You get really frustrated
because you can't tell how much alcohol is in the narrow-necked
bottle---it keeps expanding during the day and contracting at night.
You could continue to view this as a problem, or you could turn the
problem on its head and invent the thermometer.  In science, we often
approach relationships between two or more variables (in this case,
temperature and volume) with a predetermined notion of which variable
is important or worth measuring.  But when measuring that variable
gets frustrating, brainstorming a new goal often results in a valuable
new tool.  That's easy to point out in retrospect but difficult to
apply in practice because on an everyday basis we are often too caught
up in reaching our immediate goals.

In this case, the original "problem" arises from using an effect
called gravitational lensing in which light from background
galaxies is bent by the gravity of an intervening mass concentration
such as a cluster of galaxies.  We can use this effect to determine
the mass of the cluster, if we know the distance to the background
galaxies.  In certain contexts, it's very difficult to know the
distance to the background galaxies accurately enough, and overcoming
this difficulty is an ongoing area of research for major gravitational
lensing projects now in the planning phase. 

At some point my colleague Tony Tyson suggested to my graduate student
Will Dawson that he look into how well the distances to background
galaxies could be pinned down by studying the lensing effect around a
few well-studied mass concentrations.  At the least, it might be
possible to distinguish between sets of galaxies which are more or
less in the foreground (and thus are not lensed) and sets of galaxies
which are more or less in the background (and thus are lensed).  With
different lenses at different distances, it might be possible to infer
something more specific about how galaxies are distributed in terms of
distance from us.

We tried different ways of pulling this information out of the data,
but none of them worked very well.  So I suggested something nearly as
good, at least as a first step: assuming that some solution exists,
let us compute how precise the solution could be in a best-case
scenario.  This would tell us whether continued searching for the
solution would even be worth it.  Now, the ability to compute the
precision of an experiment which has not even been performed yet seems
like magic, but in my previous post I explained how it works.
For me, the best thing about this whole project was that I did a
calculation like this for the first time (they don't teach you this
stuff in school) and therefore really understood it for the first
time.  It's really a pleasure to come to understand something which
previously seemed like a bit of a black box.

The result: lensing can be used to infer how galaxies are distributed
in terms of distance from us, but only roughly.  The precision gets
better and better as you add more data, but to do as well as other
methods which have already been developed requires a very large amount
of data indeed.  For a given amount of telescope time, the other
methods are more precise.  That doesn't mean this method will never be
used: because it piggybacks on a lot of data which will be taken
anyway for other purposes, it may someday be used to double-check that
the other methods are not way off due to faulty assumptions or other
"systematic errors."  It's always good to have multiple different ways
to check something as important as the distances of galaxies.  It may
be somewhat disappointing that this method won't be the primary method
people use, but we can take some satisfaction in definitively
answering the question "how good will this method ever be?" rather
than getting bogged down searching for marginal improvements. 

A few resources about the sizes of things in the universe:

• Scale of the Universe is a neat visualization which lets you zoom smoothly from very small things like atoms all the way to the size of the observable universe, and has nice accompanying music.  But it doesn't show you the distances between celestial objects.  Most tools don't, because the distances are so large that 99% of your screen would be empty space!  Scale of the Universe 2 is by the same people and honestly I can't see much difference between the two. 
• Nikon's Universcale is a similar approach, but with more accompanying text information so you can learn more.  The presentation is a little weak on the astronomical end of the scale, but strong on the micro end of the scale.
• Powers of 10 is a classic documentary which does the same zoom trick and does show you the distances between things.  A much more slick attempt at the same thing called Cosmic Voyage was made decades later, but I still prefer the classic.

This work was supported by the University of California (and therefore to some extent by the State of California) through my salary.  I thank California for investing in research.  It ultimately pays off because research apprenticeships are how we train the next generation to become independent thinkers.

The Phisher Matrix

This is the post I've been dreading.

As regular readers know, I'm writing a blog post for each paper I
publish, in an effort to help the public understand the scientific
research that they pay for.  That research is often communicated only
to other scientists in papers which are impossible to decipher unless
the reader is already an expert on the subject, so a gentle intro to
the topic is the least I can do to give something back to the citizens
who help fund my research.

It's nearly a year since I decided to do this, but at that time I was
working on a paper based on the Fisher matrix, and I was very
reluctant to try explaining this to novices.  At one point, I was
reading the Dark Energy Task Force report to review how they used the
Fisher matrix, and I came across this sentence:

My daughter looked over my shoulder and said, "Really, Dad? The Fisher
matrix is simply ....?"  So I've been procrastinating this one. 

Instead of focusing on the mathematical manipulations, let's focus on
what purpose they serve.  Imagine you work in a mail room, and your
boss gives you two boxes to weigh, and two chances to use the scale.
Naturally you will weigh each box once.  But suppose that your boss
intends to glue the boxes together and ship them as one item, and
furthermore that you need to know the total weight as precisely as
possible and the scale has a random uncertainty of +/- 0.5 pounds.
Should you weigh the boxes separately and then add the numbers, or
weigh them together, or does it not matter?  Assume the glue will add
no weight, and remember that you have two chances to use the scale to
attain the best accuracy.

If you weigh the boxes separately, you have 0.5 pound uncertainty on
the weight of the first box and 0.5 pound uncertainty on the weight of
the second box.  The uncertainty on the sum of the weights is not 1.0
pound as you might expect at first.  It's less, because if errors are
random they will not be the same every time.  For example, the scale
could read high on one box and low on the other box, so that the error
on the sum is very small.  However, we can't assume that errors will
nicely cancel every time either.  A real mathematical treatment shows
that the uncertainty on the sum is about 0.7 pounds.  (Note that we
are not considering the possibility that the scale reads high every
time.  That's a systematic error, not a random error, and we can deal
with it simply by regularly putting a known weight on the scale and
calibrating it. Scientists have to calibrate their experiments all the
time, but for this paper I am mainly thinking of random errors.)

If you weigh the boxes together, you have a 0.5 pound uncertainty on
the sum, and furthermore you can use your second chance on the scale
to weigh them together again and take the average of the two
measurements, yielding a final uncertainty of about 0.35 pounds (0.7
divided by 2, because you divide by two when you take the average of
the two measurements).  So you are twice as precise if you weigh them
together!  This may not seem like a big deal, but it can be if
procedures like this save the mail room money by not having to buy a
new high-precision scale.  Similarly, scientists think through every
detail of their experiments to squeeze out every last drop of
precision so that they can get the most bang for the buck.

Now bear with me as we examine one more twist on this scenario, to
illustrate this point in more detail.  Suppose your boss changes her mind
and decides to ship the boxes separately after all.  If you were smart
enough to follow the procedure which yielded the most precise total
weight, you would now be at a complete loss, because you have no
information on the weights of the individual packages.  If you know your
boss is indecisive, you might want to devise a procedure which is nearly
optimal for the total weight, but still gives some information about the
individual weights.  For example, you could use your first chance on the
scale to weigh the boxes together, which would yield a 0.5-pound uncertainty
on the total (better than the 0.7 pounds provided by the naive procedure of
weighing the boxes separately and then summing), and use your second
chance on the scale to weigh one box alone (yielding an uncertainty of
0.5 pound on that box, the same as if you had performed the naive
procedure).  You can always obtain the weight of the second box if
necessary by subtracting the weight of the first box from the total!
We had to give up something though: the weight of the second box is
now more uncertain (0.7 pounds) because it comes from combining two
measurements which were each uncertain by 0.5 pounds.

You probably hadn't suspected that an experiment as simple as weighing
a few boxes could become so complicated! But it's a useful exercise
because it forces us to think about what we really want to get out of
the experiment: the total weight, the weight of each box, or something
else?  Similarly, a press release about an experiment might express
its goals generically ("learn more about dark energy"), but you can
bet that the scientists behind it have thought very carefully about
defining the goals very, very specifically ("minimize the uncertainty
in dark energy equation of state parameter times the uncertainty in
its derivative").  This is particularly true of experiments which
require expensive new equipment to be built, because (1) we want to
squeeze as much precision as we can out of our experiment given its
budget, and to start doing that we must first define the goal very
specifically; and (2) if we want to have any chance of getting funded
in a competitive grant review process, we have to back up our claims
that our experiment will do such-and-such once built.

If you made it this far, congratulations!  It gets easier.  There's only one
more commonsense point to make before defining the Fisher matrix,
and that is that we don't always measure directly the things we
are most interested in.  Let's say we are most interested in the total
weight of the packages, but together they exceed the capacity of the
scale.  In that case, we must weigh them separately and infer the
total weight from the individual measurements.  We call the individual
weights the "observables" and we call the total weight a "model
parameter." This is a really important distinction in science, because
usually the observables (such as the orbits of stars in a galaxy) are
several steps removed from the model parameters (such as the density
of dark matter in that galaxy) in a logical chain of reasoning.  So to
say that we "measure" some aspect of a model (such as the density of
dark matter) is imprecise.  We measure the observables, and we infer
some parameters of the model.

Now we can finally approach the main point head-on.  The Fisher matrix is a way of predicting how precisely we can infer the parameters of the model, given that we can only observe our observables with a certain precision.  It helps us estimate the precision of an experiment before we even build it, often before we even design it in any detail!  For example, to estimate the precision of the total weight of a bunch of packages which would overload the scale if weighed together, we just need to know (1) that the precision of each weighing is +-0.5 pounds, and (2) the number of weighings we need to do.  We don't actually have to weigh anything to find out if we need to build a more precise scale!

The Fisher matrix also forecasts the relationships between different things you could infer from the experiment.  Take the experiment in which you first weigh the two boxes together, then weigh one individually and infer the weight of the second box by subtracting the weight of the first box from the weight of both boxes together.  If the scale randomly read a bit high on the first box alone, then you not only overestimate the weight of the first box, but you will underestimate the weight of the second box because of the subtraction procedure used to infer its weight. The uncertainties in the two weights are coupled together.  Those of you who did physics labs in college may recognize all this as "propagation of errors."  The Fisher matrix is a nice mathematical device for summarizing all these uncertainties and relationships when you have many observables (such as the motions of many different stars in different parts of the galaxy) and many model parameters (such as the density of dark matter in different parts of the galaxy), such that manual "propagation of errors" would be extremely unwieldy.

The great thing about the Fisher matrix approach is that it gives you a best-case estimate of how precise an experiment will be, before you ever build the experiment ("best-case" being a necessary qualifier here because you can always screw up the experiment after designing it, or screw up the data analysis after recording the data). Thus, it can tell you whether an experiment is worth doing and potentially save you a lot of money and trouble. You can imagine many different experiments and do a quick Fisher matrix test on each one to see which one will yield the most precise results. Or you can imagine an experiment of a type no one thought of before, and quickly show whether it is competitive with current experimental approaches in constraining whatever model parameters you want to constrain. It's a way of "phishing" for those experiments which will surrender the most information.

That's the Fisher matrix, but what did I do with it in my paper? Well, this has been a pretty long post already, so I'll deal with that in my next post.  Meanwhile, if you want to follow up some of the ideas here, try these links:

• The report of the Dark Energy Task Force contains a solid review of
  the Fisher matrix for professional physicists
• The Wikipedia article on Design of experiments goes through an
  example of weighing things in different combinations, as well as
  clarification of statistical vs systematic errors and lots of other
  terms.
• A very informal guide I wrote to introduce the Fisher matrix to, say,
  undergraduate physics majors.

Mystery tubes 2012

This year I have a new title (scientist in residence) at Peregrine School, and a new format: every Friday morning with grades 1-2 for three months, then with grades 5-7 for three months, then grades 3-4 for three months.  This should allow me to go much further in depth with each group, and to facilitate really substantive projects on their part.  Today was my first day with the five first and second graders, and to break the ice I brought some "mystery tubes" which are basically like the one shown on this short video.

The students got their hands on the tubes, did any experiment they wanted to (short of looking inside the tubes), and drew what they thought was inside.  Most students went through a couple of iterations as they realized that their first model wouldn't reproduce their observations.  When a student was satisfied with his/her drawing, I brought out toilet paper tubes, strings, beads, etc so they could build a model and show that it behaved like the real thing.  The point: science is about building models (usually mental models rather than physical models), and this activity allows us to practice many aspects of this in one session, including thinking of experiments to test the model, performing those experiments, generating predictions from the model (hypothetico-deductive reasoning), and comparing the results of the experiments to predictions generated from the model.  Furthermore, since I never allowed them to look inside the tube we had ample opportunity to discuss how science is less about knowing the right answer than about the process of finding answers.  After all, nature never tells us the right answer directly.  Kids at this age are very much in the mode of gaining knowledge from books, but it is worth making them stop and think about how every bit of the knowledge in books was, at some point, figured out by someone who had to figure out by reasoning and then convince other people that it was correct. 

You can also read about the way I did this activity with mixed ages (grades 1-6) last year.   A note for teachers using this activity: it took much more time this year, 45 minutes, because the 1-2 graders did not have the fine motor skills to easily build their little toilet-paper-tube model with strings and beads.  With mixed ages last year, it seemed as if the young ones contributed equally intellectually, but the older ones probably did the actual tying of strings and beads.  And the 45 minutes was with two adults helping four kids!  If you try it with a larger group of 1-2 graders, you'll have to bring full-size materials. I do this activity with college students (who find it interesting and beneficial) so this activity is remarkable for the range of ages who find it suitable!

I learned something from Teacher Marcia too.  With five minutes remaining in the period, I wanted to have a wrap-up discussion with the kids.  She showed me a way to make kids pay full attention to the wrap-up discussion rather than surreptitiously keep working on their model: move them from the material-strewn desks over to the rug where they listen to stories etc.  This was brilliant.  Now if I can figure out how to do this with college students, I'll be set!

Cosmic Magnification

In the previous post I noted how wide-area surveys of the sky like the
Deep Lens Survey serve the dual purposes of finding rare objects and
surveying a representative sample of the universe (to determine its
average density, for example), and I described one of our successes in
the former category while promising to post an example in the second
category.  Here it is!

First, we have to understand that the path of light is bent by gravity
and therefore, if we can observe some consequence of this bending, we
can learn about how much mass is between us and the source of light.
I'm not going to explain this in any detail here, but if you wish you
can watch my YouTube video on the subject, or just skip to the part
where I do a demo showing that this bending can lead to magnification.
In that demo I don't specifically point out the magnification, but at
one point you can clearly see that the blue ring on the whiteboard has
been magnified.

If we observe this magnification while looking in one very specific
direction as in the video, we can find how much mass is lurking in the
object which provides the magnification (usually a specific galaxy or
cluster of galaxies).  A few galaxies happen to have background
sources of light lined up just right so that we can see the
magnification easily, so we can learn about those specific galaxies.
But are they representative of galaxies in general?  Probably not,
because the most massive galaxies provide the most magnification and
are more likely to get noticed this way.  Also, having the mass more
concentrated toward the center of the galaxy helps, so if we just
study these galaxies, we will be looking only at the more massive,
concentrated galaxies.

In our wide-field survey, a team led by graduate student Chris
Morrison measured the very small amount of magnification around the
locations of hundreds of thousands of typical galaxies.  Their
statistical analysis doesn't measure the magnification caused by each
galaxy (which would be too small to measure), but it measures the
typical magnification caused by the galaxies in aggregate.  For this
reason, this type of analysis is called "cosmic magnification" which
sounds mysterious but can be thought of as "magnification caused by
the general distribution of mass in the cosmos rather than by a
specific identifiable lump of mass."

The amount of cosmic magnification tells us not only about the
distribution of mass in the universe, but also about the distances
between us, the magnifying masses, and the sources of light.  (Imagine
watching the wineglass demo in my video, but having me move the
wineglass much closer to the whiteboard...you can probably predict
that the magnification will be less.)  These are two very fundamental
things about the universe which astronomers are trying to measure,
because they are both affected by the expansion rate of the universe,
and the expansion rate is unexpectedly accelerating.  Three
astronomers won the 2011 Nobel Prize in Physics for their role in
discovering this acceleration, and ever since they discovered it
(1998), many astronomers and physicists have focused on figuring out
why.  Some attack this question from a theoretical point of view (a
theorist coined the term "dark energy" which has become the popular
term, but be warned that it may not be caused by a new form of energy
at all), and others attack it from an observational point of view: if
we can get better and better measurements of how the expansion is
actually behaving, we can rule out some of the theories which have
been proposed to explain it.  Cosmic magnification has a real role to
play in that process, and Morrison's paper is the first one to
measure, even in a crude way, how cosmic magnification increases as we
increase the distance between us and the masses causing the
magnification.

Colliding clusters of galaxies

One of the questions generated by my previous post describing the
Deep Lens Survey is: Why do such a large survey of the sky?  What
do you hope to accomplish that the Hubble Space Telescope can't?

HST is great at some things but not others.  Expecting HST to be great
at everything in astronomy is like expecting a great criminal-defense
attorney to also be great for cases involving bankruptcy law, probate
law, torts, and tax law.  Novices would put all of these things under
the single category of "law," but people closer to the legal system
recognize that these are very different specialties.  Similarly, if
you look closer at "astronomy" or "telescopes" you realize that
there's such a wide variety that no one telescope can do it all.  And
whether it's attorneys or astronomy, the few performers which become
known outside the field are those with some combination of high
performance in the field and a good public relations machine.

So what is HST great at?  It was launched into space primarily because
turbulence in Earth's atmosphere makes images blurry.  Above the
atmosphere, HST can take really sharp images.  The flip side of
capturing these really fine details is that it can't capture a very
wide panorama.  So we need very big, wide surveys from other
telescopes to find things which are interesting enough to follow up with
HST and other specialized telescopes such as X-ray telescopes (which,
like HST, need to be above the atmosphere and are therefore similarly
expensive and rare).

But wide surveys are more than just rare-object finders for HST and
other specialized telescopes.  Equally important, they give us a
representative sample of the universe.  Just as an anthropologist
could not fully understand how humans live by studying only the "most
interesting" countries (the ones with revolutions underway, for
example), astronomers could not understand the universe in general
just by studying the most interesting objects.  I'll give an example
of the rare-object-finding capability of the Deep Lens Survey in this
post, and an example of the understanding-the-universe-in-general
capability in my next post.

Rare objects are scientifically interesting for many reasons. Some of
them tell us about extremes: knowing the mass of the most massive star
or the luminosity of the most luminous star tells us something about
how stars work.  In other cases, what is rare and interesting is not
so much the object itself as the stage it happens to be in right now.
Because the lifetimes of celestial objects are millions or billions of
years, we can't follow a single star, say, over its lifetime to
determine its life stages.  Instead we have to piece together their
life cycles from different stars seen in different stages.  Imagine an
alien anthropologist who pieces together the human life cycle from one
day's visit to Earth: because a small fraction of humans are babies
right now, that must mean that people spend a small fraction of their
lives as babies.  In the same way, a certain star or galaxy may not be
intrinsically special, but if we happen to be seeing it at a special point in
its life cycle, that helps us understand all objects of its type.
Finally, in some cases objects are particularly interesting because we
have a particularly clear view of them.  Just as an overhead camera's picture
of the top of a person's head is less informative than a picture of their face,
Earth's view of many celestial objects is not fully informative. Objects
which happen to expose their "faces" to us give us more insight, which
can then be applied even to those objects which do not face us.

Today I want to highlight a collision between two galaxy clusters
which was discovered in the Deep Lens Survey.  Imagine observing a
head-on collision between two large trucks.  You will observe a lot
more of the details if you are standing by the side of the road (a
"transverse" view) than if you are driving behind one of the trucks.
My student Will Dawson was the first to realize that we have a
transverse view of this collision. This immediately makes it
interesting because if the component parts of the clusters (galaxies,
hot gas, dark matter) become separated, a tranverse view gives us the
best chance of seeing that separation and therefore learning more
about those components.

In particular, separation between the dark matter (which carries most
of the mass) and the hot gas (which is the second-most-massive
component) is important because dark matter has never been observed
very directly.  Astronomers infer the existence of dark matter when
orbits (of galaxies in a cluster, for example) are too fast to be
explained by the gravity of all the visible mass (stars and gas).
Therefore, the conclusion goes, there must be some invisible component
with substantial mass: dark matter.  Observing a clear separation
between dark matter (ie, the bulk of the mass) and normal matter would
boost our confidence in this conclusion, and help refute competing
hypotheses (for example, that what we understand about gravity and
orbits from studying the solar system may not fully apply to these
other systems).  This "direct empirical proof of the existence of dark
matter" was first done for a transversely colliding galaxy cluster
called the Bullet Cluster, which you should definitely read about if
you are interested in this topic.  A good place to start for beginners might
be this Nova Science Now video.

Finally, we get to the research paper I wanted to highlight.  It's an
examination of the evidence we collected regarding the aforementioned
collision in the Deep Lens Survey, including not only the original DLS
images but also data from HST, the Chandra X-ray telescope, the
10-meter Keck telescopes and other telescopes.  The conclusion:
it is indeed a transversely viewed collision of galaxy clusters
with a substantial separation between the dark matter and the hot gas.
My student Will Dawson is the principal author who assembled all these
pieces, with substantial help from many co-authors.  This is something
of a teaser paper: it's not an exhaustive analysis, but it's enough of
an analysis to establish that it's an important system worthy of
further study.  All further studies of the system (including proposals for more
telescope time) will cite this paper because it lays out the basic facts.

Closeup of our colliding clusters, with the location of the mass (mostly dark matter) painted on in blue and the location of the hot gas painted on in red. If you look closely you can see that there are many galaxies colocated with the mass, but not with the hot gas.  This (temporary) ejection of the hot gas allows us to study dark matter more clearly.

I've tried to keep this post short and relatively free of technical details, so
some readers may want more.  A good place to start is Will's research page.
And  feel free to ask questions in the comments below!  I may give a quick
response in the comments, or I may use them to motivate a future post.

Finally, to bring this back to the question which initially stimulated
this post: this is a special system which we did study with HST, but
we never could have found it without a wide survey like the Deep Lens
Survey.

What does figure skating have to do with astronomy?

With this post I add a new ingredient to this blog: explaining my
astrophysical research.  I do feel that scientists have an obligation
to help the public understand their research, most of which is funded
by the public, and I'm afraid that science journalism is generally not
up to this important task.  Science journalism does well when a story
easily fits into the "gee whiz, isn't this a cool result" format, but
tends to be too focused on "breakthroughs" rather than the
investigative process which really makes science what it is (Radiolab
excepted).  Most "breakthroughs" are overstated because scientists
have an incentive to overstate the importance of their specific
contribution to the field, and the media generally give science
journalists too little space to explore the process of science more
deeply.  Maybe scientists blogging about their work can help fill a
gap.

That's easy to say, but harder to deliver.  I don't pretend that I
will have a lot of spare time to write thorough, clear explanations of
many aspects of my research.  But I do hope to convey, for each paper
I author or co-author, why that paper is important or interesting, how
that fits into the bigger picture of astrophysics research, and maybe
something interesting about how the research was done.  I'll try to do
a paper every week or two over the summer, eventually describing all
the papers dating back to last fall when I first wanted to do this.

The first paper I'd like to describe is (like all my papers) freely
available on the arXiv. It solves a problem which
appears not only in astrophysics, but also in figure skating and many
other contexts.  (I'm not claiming that I came up with the original
solution to such a far-reaching problem, just that this paper
describes how to apply it in a certain astrophysical niche.)  So let's
start with the figure-skating version of the problem.  Say you have a
figure-skating competition with hundreds of skaters.  Evaluating each
skater with the same panel of judges would be too time-consuming; the
competition would have to be stretched over a period of weeks and the
judges would have judging fatigue by the end.  You need to have
several panels of judges working simultaneously in several rinks, at
least to narrow it down enough so that the top few skaters can have a
head-to-head competition for the championship at the end.

Given that, how do we make it as fair as possible?  One panel of
judges may be stricter than another, so that a skater's score depends
as much on a random factor (which panel judged her) as on her
performance.  Remixing the judging panels from time to time, by
itself, doesn't help.  That does prevent situations such as "the most
lenient panel sits all week in Rink 1," but it generates situations
such as "a lenient panel sat in Rink 1 Monday morning, a slightly
different lenient panel sat in Rink 2 Monday afternoon, yet another
lenient panel sat in Rink 3 Tuesday afternoon," etc.  But remixing the
panels does enable the solution: cross-calibration of judges.

Let's say judges Alice, Bob, and Carol sit on Panel 1 Monday morning
and judge ten skaters.  We can quantify how lenient Alice is relative
to Bob and Carol, just by comparing the scores they give for the same
ten skaters.  The mathematical description of leniency could be simple
("Alice gives 0.5 points more than Bob and 0.25 points more than
Carol, on average") or complicated ("Alice scores the top skaters just
as harshly as Bob and Carol but is progressively less harsh on
lower-scoring skaters as captured by this mathematical formula")
without changing the basic nature of the cross-calibration process
described here.  At the same time, judges David, Ethan, and Frank sit
on Panel 2 Monday morning and judge ten other skaters.  We can
quantify how lenient David is relative to Ethan and Frank by comparing
the average scores they give for the their ten skaters.

But we still don't know how lenient Alice, Bob, and Carol are as a
group, compared to David, Ethan, and Frank; if Panel 1's scores were
higher than Panel 2's on average, we can't tell if that's because
Panel 1 is more lenient or because the skaters assigned to Panel 1
happened to be better on average than the skaters assigned to Panel 2.
So in the afternoon session we switch Alice and David.  Now that we
can measure how lenient Alice is relative to Ethan and Frank, and how
lenient David is relative to Bob and Carol, we know the relative
leniency of all six judges, and we can go back and adjust each
skater's score for the leniency of her judges.

This system isn't perfect.  However we choose to describe leniency,
that description is probably too simple to capture all the real-life
effects.  For example, perhaps Alice isn't that much more lenient than
Bob and Carol overall, but she is very lenient on the triple axel,
which most or all of Monday morning skaters happened to do.  We would
then be making a mistake by looking at Alice's Monday morning scores
and concluding that she is overall more lenient that Bob and Carol.
But this system is surely better than no system at all.  Scientists
make models of reality, and we know those models never capture all the
complexity, but we are satisfied if they capture the most important
features.  The purpose of a model is to simplify reality enough to
understand its most important features.  Over time, if we are not
satisfied with a model, we can increase its complexity to capture more
and more features of reality.  As Einstein said, "make things as
simple as possible, but not simpler."  In this example, we could
improve the model, at the cost of additional complexity, if the
judges wrote down their scores for each performance element.  We could
then compute the leniency of each judge for each element and apply these
more finely tailored correction factors.  But in practice, a simple correction
for an overall leniency factor probably gets rid of most of the unfairness
in the system.

In astronomy, assessing the brightnesses of stars and galaxies is like
assessing the performance of figure skaters in this giant
competition.  We have too many stars and galaxies to assess to do it
one at a time, so we build a camera with a wide field of view, which
can look at many thousands of stars and galaxies simultaneously.  But
different parts of the camera's field of view may not be equally
sensitive.  By taking one picture and then repointing the camera a
little bit so that each star or galaxy is assessed first by one area
of the camera and then by another, we can compute the relative
leniencies of the different areas of the camera.  Then we can infer
the true brightness of any star or galaxy by correcting for the
"leniency" of the area(s) upon which its light fell.

The social sciences are actually years ahead of the physical sciences
in applying these kinds of models.  The reason is that social sciences
very often face the problem of calibrating the relative difficulty of
things for which there is no absolute standard.  For example, on an
exam or evaluation how much more difficult is Question 10 compared to
Question 1?  There is no absolute standard for difficulty, so social
scientists have developed methods, as described above for the figure
skaters, for calibrating relative difficulty from the test responses
("the data") themselves.  This is quite different from a typical
physics experiment in which we directly compare the thing to be
measured with some kind of objective reference; for example, in
astronomy we could more or less directly compare the measured
brightness of a star or galaxy ("the science data") with a calibrated
light source ("the calibration data").  So astronomers typically had
no reason to try to use the star/galaxy data alone for the kind of
self-calibration that social scientists do.

But this is changing for several reasons.  First, sky surveys are
getting more massively parallel.  We now have cameras which take
images of millions of galaxies in a single shot, spread over a billion
pixels.  We can no longer do the most direct calibration---shining a
calibrated lamp on the very same pixel---for each star or
galaxy.  Second, we never really did such a direct comparison all the
time.  We often fooled ourselves into thinking that an occasional
direct comparison was good enough to keep things on track, but we are
now demanding more precision out of our sky surveys.  Third, precisely
because the science data have so much information, we should use that
information for self-calibration as much as possible, rather than rely
on a different and smaller set of data deemed to be "calibration data."
This point was established by Padmanabhan et al (2008) who christened
this approach (using the main data set to do as much internal calibration as
possible) "ubercal" and applied it to the Sloan Digital Sky Survey (SDSS).

My paper adopts this general approach for calibrating a sky survey
called the Deep Lens Survey (DLS), but because DLS is implemented very
differently from SDSS, the choices made in implementing ubercal for
DLS were very different.  One goal of the paper was of course to
document what was done for DLS, but this was a goal which could have
been accomplished in other ways (a section of a larger paper on DLS,
for example).  The reasons for making it a standalone paper were (1)
because most surveys are more like the DLS than like the SDSS, provide
a detailed worked example of applying the ubercal idea to this type of
survey; and (2) raise awareness that ubercal MUST be done to get the
most out of our surveys, because the errors left by the previously
standard calibration techniques are surprisingly large.  Only by
applying ubercal were we able to quantify just how large they were.

If you want to learn more about how social scientists calibrate tests
and evaluations, look up "psychometrics."  This is a pretty broad
area, so you may find it easier to focus in on one specific technique
called Rasch modeling.  I learned from a book on Rasch modeling that
to become a medical examiner in Illinois, doctors were randomly
assigned a case study from a pool of cases, and randomly assigned an
evaluator from a pool of evaluators.  But it turned out that the main
factors influencing whether a candidate passed were (1) which case
study was assigned, as some were easier than others; and (2) which
evaluator was assigned, as some were easier than others. This was
discovered by doing Rasch modeling to determine the relative leniency
of each evaluator and the relative difficulty of each case.  After
correcting a candidate's score for these factors to obtain a "true"
score indicating candidate quality, it was apparent that candidate
quality was not a very important factor in determining who passed and
who failed!  (Aficionados should be aware that candidate quality is
really a model parameter rather than an afterthought as this
description may imply, but novices need not care about this
distinction.)

Rasch modeling can be used not just to compute the overall leniency of
each judge, but also to help flag scores which are unexpected given
that judge's leniency.  For example, perhaps a Russian figure skating
judge at the Olympics is consistently a bit tougher than the other
judges, but then is no tougher than the other judges when a Russian
skates.  Without statistical analysis such as Rasch modeling, we
wouldn't know exactly how much tougher the Russian judge is normally,
and therefore we wouldn't know how much of a break the Russian judge
gave the Russian skater by giving a score in line with the other
judges.  The Russian judge could argue that she showed no favoritism
because her score for the Russian skater was no higher than the other
judges' scores.  But a statistical analysis such as Rasch modeling, by
quantifying the Russian judge's strictness for most skaters, could
provide strong evidence that she did do a favor for the Russian
skater, and quantify the size of that favor.  There are analogies for
this too in astronomy, where ubercal helps flag bad data.  (Maven
alert: ubercal is not Rasch modeling because the underlying
mathematical model is linear in ubercal, but the concept is the same.)
If you want to read about the application of Rasch modeling to some
figure skating controversies, start here.

Work on the Deep Lens Survey has been funded by Bell Laboratories/Lucent Technologies, the National Science Foundation, NASA, and the State of California (indirectly, but importantly, through University of California faculty salaries and startup funds).  Thank you!  Our data were obtained at Kitt Peak National Observatory in Arizona and Cerro Tololo Inter-American Observatory in Chile, which are wonderful scientific resources funded by the National Science Foundation.