One of the strengths of Peregrine School is that learning is
integrated. Instead of offering isolated concepts and facts, we try
to design activities which build on and reinforce each other. The
past week or two, the elementary students have been reviewing how to
measure. They have measured their bedrooms and mapped their rooms on
pieces of graph paper, for example. So I thought I would follow that
up by specifically relating length with area and volume and leading
into one of the most underappreciated aspects of science: scaling
relations.
(By the way, I didn't get time to talk about a lot of these ideas with
each group, so this post may be a good resource for parents who would
like to follow up the activity and investigate more.)
Beforehand, I printed out some graph paper with 3/4" squares to match
the unifix cubes we had on hand. This way, when we shifted from
measuring area (counting squares of graph paper) to measuring volume
(counting cubes), the kids would not be distracted by a change in the
size of the measuring unit. To people who understand area and volume,
a change in grid size might be an irrelevant detail, but to those who
don't, it could be the start of a long and unnecessary detour.
First, I asked them to draw a map of a room which was 1x1, and measure
the area by counting the squares inside (1, obviously). This should
be really boring and easy, but it took a surprisingly long time to get
everybody clear on what I was asking. Maybe next time, to speed
things up, I would start by drawing the 1x1 square so they would
instantly see my intention rather than having to parse my words.
Next, I asked them to draw a 2x2 square, predict the area, and then
measure it. Over the course of 3 groups, I learned to be stricter
about making the prediction. Once kids understand what the task is,
they are quite happy to do bigger and bigger squares and get answers
without making any predictions. But they don't learn anything this
way; only if they see that their prediction was wrong will they take
the trouble to figure out what's really going on. (In principle, the
discomfort of realizing that reality conflicts with his/her beliefs
("cognitive dissonance") will spur the student to make the effort to
really understand. In practice, it doesn't always work that
way...perhaps the subject of another post. But the fact that some
students won't care when their predictions are wrong should not stop
us from trying; they are even less likely to care if they never commit
to a prediction. And if we get even a bare majority of the class
modifying their beliefs based on evidence, we are doing a great
thing!)
So, in the spirit of being strict about making predictions, I ask you
to make a prediction. If you were to double the length and the width
of the 1x1 square, what would happen to the area? What if you were to
triple the length and width? Quadruple? Make your predictions before
reading on!
With the kids telling me the results of their experiments, I compiled
the following chart:
length area
---------------
1 1
2 4
3 9
4 16
5 25
Note that very few of them made correct predictions of these results!
I also made a graph of this, to show them the pattern in different
ways. The graph shows a very rapidly rising curve. The area
increases much faster than the length!
Why should we care? None of them knew. And I think this is a weakness
with traditional math education. These concepts are taught in a
vacuum and never made relevant. (A recent op-ed in the New York Times
agrees with me on this.) So I gave two examples:
(1) if you wanted a room twice as long and twice as wide, how many
times more carpet would you need to cover the floor? Amazingly, no
one got this right even after I pointed to the chart, with numerous
hints! The most popular answer was double. They were not able to
transfer the idea from the chart to a new situation, so they had not
really learned. Good thing I asked them instead of just moving on!
So we discussed this a bit before moving on.
(2) Suppose you have a small room full of people, like at a party.
There's a door in each wall so if there's a fire, they can easily
escape. But now imagine a room 3x longer and 3x wider. It can fit 9x
the people, so now there will be long lines at the doors when the fire
alarm goes off. People could die if architects didn't think about
these things! We'll need a lot more doors for rooms which, if we
thought only about length, might not seem so much bigger. Or you
could think about evacuating a city. A city 10x the size in each
direction has 100x the people, but nowhere near 100x the number of
escape routes, so it may be simply impossible to evacuate in a short
amount of time.
(3) I didn't have time to show that surface area behaves the same way
as area, but it does. That is, if we wanted to carpet ALL the
surfaces of a room, a room twice as big in each dimension would still
take only 4x the carpet of the smaller room. (In either case, the
total surface area is 6x the area of the floor.) Furthermore, the
same is still true of an irregular surface such as the surface of your
body. A dog twice as large in each dimension will have quadruple the
surface area you need to brush!
Ok, now we moved on to volume. I asked them to model a 1x1x1 room
with cubes and tell me the volume, then predict the volume of a 2x2x2
room. Again, no one got it right! We quickly built up a new column
in our chart:
length area volume
---------------------------
1 1 1
2 4 8
3 9 27
4 16 64
5 25 125
(actually, I filled in the 64 and 125 due to lack of time). Again,
why do we care? Well, the volume increases much faster than even
the area does, so if you wanted a swimming pool twice as long,
twice as wide, and twice as deep, it would have quadruple the surface area but
you should plan to pay for eight times the water to fill it up!
And here's where I finally got to the applications I thought would
interest the kids. If you have a dog twice as long, twice as tall,
and twice as wide as another dog, it will have 8x the volume and thus
8x the weight. You will have to feed that dog 8x the amount of energy
if it's going to have the same activity level as the smaller dog (thus
explaining why larger dogs generally do NOT have the same activity
level as larger dogs). If you have a dinosaur four times the size of
a rhino in each dimension, it will weigh as much as 64 rhinos!
This actually explains why there is a limit to the size of animals.
If you double the size of an animal in each dimension, it has eight
times the weight to support. But the strength of its bones grows only
as their cross-sectional area (4x in this example). So the bones will
not be able to support the weight. To make that bigger animal, we
will need to change the plan and not just scale everything up. This
explains why we don't see mice the size of elephants. To go from
mouse-size to elephant size requires a drastic redesign and
proportionally much thicker bones. And there's a limit to how thick
you can make the bones, so there's a limit to the redesign strategy.
So why are the largest whales so much bigger than elephants? Because
whales don't really support their weight; the ocean does.
And why were the largest dinosaurs so much bigger than elephants? I'm
guessing that it's due to a more subtle limit on the elephant. If you
double each dimension of an elephant so it has 8x the weight, then it
also produces 8x the heat because it is warm-blooded. But this heat
can only escape through the surface area, which is 4x the original surface area. So the 2x-scale elephant would overheat! Dinosaurs (at least the biggest
ones, as far as we know) were cold-blooded, so they didn't have this
issue. (It was also thought at one time that the largest dinosaurs
were swamp-dwellers so that water supported some of their weight, but
this seems to be unsupported by evidence.)
(Why don't blue whales overheat? I'm guessing it's because
water conducts heat away much better, and because they are very long
and skinny, thus maximizing their surface area; they are NOT 2x-scale
copies of smaller whales.)
This heat consideration explains a couple of other things. Larger
mammals have slower metabolisms than smaller mammals, to avoid
overheating. (Slower metabolisms, by the way, enable longer life
spans. Ever notice how the life span of a mammal correlates roughly
with its size?) And elephants have enormous ears to help them get rid
of heat.
Finally, let's think about insects. Insects get oxygen by absorbing
it through their "skin", so if we make a 2x-scale insect, we can get
oxygen in at 4x the rate, but we have 8x the amount of flesh which
needs oxygen. That's not gonna work. So we have a limit on the size
of insects. But why were there 2.5-foot dragonflies hundreds of
millions of years ago? Turns out that there was much more oxygen in
the atmosphere at that time, so each bit of surface area could take in
oxygen more effectively, and the area/volume tradeoff could be pushed
further towards more volume.
Now, I didn't get time to talk about many of these ideas, at least in
any detail. We will expand my science slot to 30 minutes with each
group. But here are some resources for parents to follow up:
Wikipedia article on the square-cube law: this repeats much of what I said about volume vs area, but has links to related articles in more depth. In particular, if you like to think about animals, you might want to check out....
Wikipedia article on allometry: allometry is the fact that most larger animals are NOT scaled-up smaller animals, but have changes in body plan.
Wikipedia article on Meganeura: this is the 2.5-foot dragonfly. It has a link to the 1999 Nature article linking insect gigantism to oxygen supply.
Friday, September 23, 2011
Friday, September 16, 2011
The Sift Hits the Sand
Today was my first day with the 4-6 year olds. I generally try to
think of an activity which builds on or is related to what the kids
are doing the rest of the week, so that my visits are not put in a
pigeonhole marked "Science" which has nothing to do with the rest of
their lives or studies. (A big point I want to get across is that
Everything is Connected. At the university level this might mean
emphasizing the unity of knowledge---students tend to see different
chapters of a textbook or different lectures as unrelated pigeonholes,
and must be prodded to think about the connections, which are actually
the important part! But for these kids, it's enough to make
connections between science and their everyday lives.)
But this being the start of the school year, the emphasis so far has
been on community-building, and there wasn't an obvious hook into
physical science. Teacher Jessica said that the kids had been
fascinated with some aspects of sand, so I thought of a way to build
on that. I had them separate big, medium, and small particles from
the sandpile, and used that to discuss solids, liquids, and molecules,
as well as engineering.
Before class, I built seven sets (seven is the maximum number of kids
per group) of coarse and fine sifters at low cost as follows. I took
a 4" diameter PVC pipe and sliced it into short segments to form the
frames of the sifters. For the mesh, I bought screen material. I
wanted a variety of mesh sizes, but this was difficult at the hardware
store. I ended up using what is basically window screen material. I
also had on hand a much coarser wire mesh designed to form a skeleton
for papier mache constructions. So I had two sizes, although I would
have liked even more and I will keep my eye out for different
materials in the future. (A baker's sifter has a finer mesh, but mine
had no walls so it was too easy to spill the sane rather than sift the
sand.) I cut the meshes into circles and duct-taped them onto the PVC
frames. I also brought some small cardboard boxes, some paper coffee
filters, and 21 (3 for each child) 44-oz plastic cups, which happen to
have mouths which fit well with the 4" PVC pipe. I wanted to bring
tweezers as well, but I forgot it.
I showed each group that I had been able (before class) to obtain one
cup of big stones and woodchips, one of medium stones, and one of fine
sand, and I gave them 10 minutes or so to experiment with any and all
of these tools to see if they could do it. They all pretty much got
it, usually with some guidance (as much to keep them focused as to
show them how to do it), and no one found it so easy as to be boring.
One of the girls found an advanced way to do it: stack the coarse
filter on top of the fine filter on top of a cup, load the top with
sand, and shake the whole thing to do it all at once. Like an oil
refinery, but with the heavy stuff staying on top! This is why I
mentioned engineering: although I often emphasize the cognitive value
in being able to understand or accomplish something in more than one
way, there is often great practical value in finding the most
efficient way!
We then talked about alternative ways to do the separation. Some had
wanted tweezers to separate the particles one by one; I forgot to
bring tweezers, but that's a valid---even if very
time-consuming!---way to do it. No one thought of using the box, but
when I asked how they would use the box about one kid in each group
guessed that if I just shake a box full of this mixture, the bigger
pieces come to the top. I even brought a cereal box to make the
connection to every kid's experience of the small pieces of cereal
always being on the bottom. This is because only the small particles
are able to fall into the small gaps which open up when the box is
shaken, very much like a sifter.
Next, I asked them if we could figure out a way to separate the fine
sand into even finer particles. We tried a coffee filter, but the
holes in the coffee filter were too small to let any sand through.
Here I made the connection to the atomic theory of matter: water does
go through the holes in the coffee filter, and so must be made up of
very small particles, too small to see. The same with air; air is able to push things because it is made up of small particles, even though we can't see them.
Finally, we talked about solids vs liquids. I can pour sand from a
cup, so is it a liquid? Most didn't want to say it's a liquid but
couldn't say why. Again, it's useful to point out the progression of
sizes. The bigger stones could be poured out of a cup but look
nothing like the flow of a liquid. The finest sand flows more like a
liquid, but not quite. The liquid has invisibly small particles, so
flows perfectly smoothly as far as we can see. You can pour sand and
make a pile, but you cannot pour water and make a pile of water!
At the start of each group, I promised them that we would experiment
with quicksand if they made good choices during the main experiment.
The night before, I whipped up a batch of water-soaked sand, which,
with some imagination, could be quicksand. (Quicksand is water-soaked
sand, but apparently not quite the kind of sand we have in our
sandbox!) This mixture of a liquid plus small solid particles has
interesting properties which are between those of a solid and those of
a liquid. They had fun with this, but I plan to someday make better
quicksand, perhaps with corn starch.
All in all, I think this 20-minute activity worked very well for the
4-6 year-olds, and I think it will be something they will continue to
experiment with even after my visit. I limited it to 20 minutes
because we had to get four groups through, but a longer time would be
fine too because many kids wanted to do more sifting.
think of an activity which builds on or is related to what the kids
are doing the rest of the week, so that my visits are not put in a
pigeonhole marked "Science" which has nothing to do with the rest of
their lives or studies. (A big point I want to get across is that
Everything is Connected. At the university level this might mean
emphasizing the unity of knowledge---students tend to see different
chapters of a textbook or different lectures as unrelated pigeonholes,
and must be prodded to think about the connections, which are actually
the important part! But for these kids, it's enough to make
connections between science and their everyday lives.)
But this being the start of the school year, the emphasis so far has
been on community-building, and there wasn't an obvious hook into
physical science. Teacher Jessica said that the kids had been
fascinated with some aspects of sand, so I thought of a way to build
on that. I had them separate big, medium, and small particles from
the sandpile, and used that to discuss solids, liquids, and molecules,
as well as engineering.
 |
| Simple materials. |
Before class, I built seven sets (seven is the maximum number of kids
per group) of coarse and fine sifters at low cost as follows. I took
a 4" diameter PVC pipe and sliced it into short segments to form the
frames of the sifters. For the mesh, I bought screen material. I
wanted a variety of mesh sizes, but this was difficult at the hardware
store. I ended up using what is basically window screen material. I
also had on hand a much coarser wire mesh designed to form a skeleton
for papier mache constructions. So I had two sizes, although I would
have liked even more and I will keep my eye out for different
materials in the future. (A baker's sifter has a finer mesh, but mine
had no walls so it was too easy to spill the sane rather than sift the
sand.) I cut the meshes into circles and duct-taped them onto the PVC
frames. I also brought some small cardboard boxes, some paper coffee
filters, and 21 (3 for each child) 44-oz plastic cups, which happen to
have mouths which fit well with the 4" PVC pipe. I wanted to bring
tweezers as well, but I forgot it.
I showed each group that I had been able (before class) to obtain one
cup of big stones and woodchips, one of medium stones, and one of fine
sand, and I gave them 10 minutes or so to experiment with any and all
of these tools to see if they could do it. They all pretty much got
it, usually with some guidance (as much to keep them focused as to
show them how to do it), and no one found it so easy as to be boring.
One of the girls found an advanced way to do it: stack the coarse
filter on top of the fine filter on top of a cup, load the top with
sand, and shake the whole thing to do it all at once. Like an oil
refinery, but with the heavy stuff staying on top! This is why I
mentioned engineering: although I often emphasize the cognitive value
in being able to understand or accomplish something in more than one
way, there is often great practical value in finding the most
efficient way!
 |
| The oil-refinery configuration with the finer mesh in the middle. If we had more types of mesh, we could separate into many different sizes all at once. Chaining together many separation devices to derive an ultrapure sample is a principle used in a variety of contexts I did not discuss with these kids, such as uranium enrichment. They might be able to see that a coin-sorting machine might be built this way, though. |
We then talked about alternative ways to do the separation. Some had
wanted tweezers to separate the particles one by one; I forgot to
bring tweezers, but that's a valid---even if very
time-consuming!---way to do it. No one thought of using the box, but
when I asked how they would use the box about one kid in each group
guessed that if I just shake a box full of this mixture, the bigger
pieces come to the top. I even brought a cereal box to make the
connection to every kid's experience of the small pieces of cereal
always being on the bottom. This is because only the small particles
are able to fall into the small gaps which open up when the box is
shaken, very much like a sifter.
 |
| Pub mix after a light shake: the trend from small things at the bottom to big things at the top is pretty clear. |
sand into even finer particles. We tried a coffee filter, but the
holes in the coffee filter were too small to let any sand through.
Here I made the connection to the atomic theory of matter: water does
go through the holes in the coffee filter, and so must be made up of
very small particles, too small to see. The same with air; air is able to push things because it is made up of small particles, even though we can't see them.
Finally, we talked about solids vs liquids. I can pour sand from a
cup, so is it a liquid? Most didn't want to say it's a liquid but
couldn't say why. Again, it's useful to point out the progression of
sizes. The bigger stones could be poured out of a cup but look
nothing like the flow of a liquid. The finest sand flows more like a
liquid, but not quite. The liquid has invisibly small particles, so
flows perfectly smoothly as far as we can see. You can pour sand and
make a pile, but you cannot pour water and make a pile of water!
At the start of each group, I promised them that we would experiment
with quicksand if they made good choices during the main experiment.
The night before, I whipped up a batch of water-soaked sand, which,
with some imagination, could be quicksand. (Quicksand is water-soaked
sand, but apparently not quite the kind of sand we have in our
sandbox!) This mixture of a liquid plus small solid particles has
interesting properties which are between those of a solid and those of
a liquid. They had fun with this, but I plan to someday make better
quicksand, perhaps with corn starch.
All in all, I think this 20-minute activity worked very well for the
4-6 year-olds, and I think it will be something they will continue to
experiment with even after my visit. I limited it to 20 minutes
because we had to get four groups through, but a longer time would be
fine too because many kids wanted to do more sifting.
Friday, September 9, 2011
Icebreaking Activity: Mystery Tubes
Today was my first day with the elementary kids. This is a brand-new school with about 22 students total in grades 1-6, and there is flexibility to work with age-segregated or mixed-age groups. I did the mystery tube activity (with extension #1) because it's a good icebreaker, and it naturally comes first because it addresses the nature of science. (By the way, I discovered this activity when the folks from http://undsci.berkeley.edu/ came to UC Davis and conducted a workshop on science outreach. Their website is worth a look, especially the diagram showing the real process of science, which is the exact opposite of the cookbook 5-step procedure you see in most textbooks. But maybe that's another post.)
I chose mixed-age groups because I was afraid the younger kids would struggle with it, and could use assistance from the older ones (the activity is recommended for grades 6-16, but I was pretty confident that grades 4-6 could handle it well). There was a fair amount of awkwardness because everybody was new to the school, and there was no established pattern of working in groups; some students still didn't know some other students' names! Considering that, it seemed to go fairly well. While some younger kids did struggle, a few other younger kids just nailed it. So while I still wouldn't recommend it for a group younger than 4th grade, it was eye-opening to see some really good results from individual 2nd-graders. At the same time, I have to admit that there wasn't much discussion of concepts like "Test results sometimes cause scientists to revise their hypotheses." We were doing those concepts, but it was hard to discuss them in these mixed-age groups. In the future, if I have mixed-age activities I might think about how to "debrief" the older kids separately afterward, to discuss how they can take what they learned in the activity and generalize it to make it useful in other parts of their studies and their lives.
A few tips for those wishing to do this:
Most discussions of the process of science focus on the mechanics of it. Students pose a question ("How does this thing work?"), suggest hypotheses (saying "I think there's a knot inside" and drawing a diagram of where and what kind of knot), and then test their hypotheses ("If I pull here it should..."). This is all great, but teachers usually present it in a context where the correct answer is already known, or revealed at the end. If the answer is already known ("today we will measure the density of water"), the activity turns into a dry, dull exercise. If the answer is revealed at the end, the whole idea of science as an ongoing process of inquiry is subverted.
I chose mixed-age groups because I was afraid the younger kids would struggle with it, and could use assistance from the older ones (the activity is recommended for grades 6-16, but I was pretty confident that grades 4-6 could handle it well). There was a fair amount of awkwardness because everybody was new to the school, and there was no established pattern of working in groups; some students still didn't know some other students' names! Considering that, it seemed to go fairly well. While some younger kids did struggle, a few other younger kids just nailed it. So while I still wouldn't recommend it for a group younger than 4th grade, it was eye-opening to see some really good results from individual 2nd-graders. At the same time, I have to admit that there wasn't much discussion of concepts like "Test results sometimes cause scientists to revise their hypotheses." We were doing those concepts, but it was hard to discuss them in these mixed-age groups. In the future, if I have mixed-age activities I might think about how to "debrief" the older kids separately afterward, to discuss how they can take what they learned in the activity and generalize it to make it useful in other parts of their studies and their lives.
A few tips for those wishing to do this:
- have a bucket of threadable beads ready. These are handy for tying onto the strings in the models so they don't slip through the holes in the toilet paper tubes, as well as for connecting the strings in the interior (in any way they wish; I don't hint in any way that they should use the beads to connect the strings, but they get used because they're handy).
- I made tubes with different types of connections in the interior, because often when two groups of scientists think they're doing the same experiment, they're not really, due to some confounding variable. So I think having all tubes identical subverts the process-of-science aspect of the lesson, and this came in really handy when students begged me for the answer (I honestly didn't know the answer for each individual tube) or thought they figured out the answer and tried to tell everyone else rather than let the others experiment more.
- If you suspect groups might not function well as a group, it's ok to forget about "sharing findings" and the like. I wish I had more toilet paper tubes because many students wanted to make their own model, and I think that would have been better than forcing students to build models in groups. It's hard to wait for your turn at improving the model!
- We did it in 3 rotations of 20-25 minutes each. I think it needs a bit more time than that, like 30 minutes.
Most discussions of the process of science focus on the mechanics of it. Students pose a question ("How does this thing work?"), suggest hypotheses (saying "I think there's a knot inside" and drawing a diagram of where and what kind of knot), and then test their hypotheses ("If I pull here it should..."). This is all great, but teachers usually present it in a context where the correct answer is already known, or revealed at the end. If the answer is already known ("today we will measure the density of water"), the activity turns into a dry, dull exercise. If the answer is revealed at the end, the whole idea of science as an ongoing process of inquiry is subverted.
The World in a Grain of Sand: Science Through Beginners' Eyes
To see a world in a grain of sand,--William Blake
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.
This blog will be about my adventure in bringing science to preschool and elementary-school children over the coming year. Although Blake is known as a mystic rather than a scientist, this quote captures how I feel about bringing science to kids: too many people have never seen that curiosity and a sense of wonder are a big part of science. Everyday objects like grains of sand and flowers, if looked at with a genuine sense of curiosity, lead us to think about bigger and bigger ideas, until we practically hold infinity in the palms of our minds.
The main purpose of this blog is to record something about my activity each week: what I did with the kids, how well it worked, how it could be made to work better, etc. This is for my own benefit so that I have a journal to refer back to if I do this again next year. If others happen to look at this blog and use it to help bring science to kids, that would be fantastic.
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