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.