Professor of Education, Harvard University.
Published by North Dakota Study Group on Evaluation, Grand Forks,
is for Claire, Colette, Danielle, Evelyne, Henri, Ingrid,
Jacques, Jeanne, Lise, and Pierre, and for the scores of
other teachers who have put themselves on the line as learners
in the search for greater understanding of learning and teaching.
This is a
story about the collective creation of knowledge: its multiple
beginnings; its movement forwards, backwards, sidewards; its
intertwining pathways. The setting is a course in the educational
psychology of science teaching, at the University of Geneva.
to teaching this course is to engage the students in finding
out something for themselves through their own investigations
of everyday phenomena; and then to draw psychological and pedagogical
themes from this joint engagement. What it is that becomes
the major subject of study varies from group to group; we try
two or three kinds of activities until something catches on.
In this group,
the first one happened to catch on: why some things float and
not others. It caught on partly because of some tantalizing
phenomena that occurred in the very first session, and partly
because the members of the group (with one exception) were
willing to acknowledge that they didn't know, or couldn't remember
what they ever might have been taught about, what makes things
float. It was their willingness to be perplexed, and to struggle
publicly with their own perplexities, that created the story.
covers eight weekly sessions. In addition to the work on floating
and sinking, which took one or two of the three hours per session,
we also watched the apparent motions of the moon, trying to
fit observations with theory; and we discussed assignments
that class members carried out in their own classrooms.
in this class was twofold, but I am including only one aspect
of it here. In my work, I am interested primarily in how to
help teachers think about their teaching, but I am also interested
in how adults think and learn about experiences of the physical
world. It is my good fortune to have found a way to do the
first through the second. (See "Teachers as Learners"  and "Teaching
 further development at this point.) In many of my
classes, the most interesting developments are those which concern
the students as teachers. (See, for example, "Understanding Children's
In this class,
while the students' development as teachers was also interesting,
the story of their coming to understand more about floating
and sinking is the more cohesive and dramatic. It is a story
on its own, and on its own it sheds light on the capacities
of a group of serious, playful adults to come to their own
understanding, and on the deep complexities of an idea which
is often taken to be simple (see David Hawkins, "Critical Barriers
for Science Teaching" ). I hope it raises questions
about the teaching of science in high school and college.
sessions were not tape-recorded. I was able to spend as many
hours as it took to write down everything I remembered. I have
recreated the story from those notes.
are the following:
Participants: Claire, Danielle, and Lise, kindergarten
teachers, an inseparable trio; Colette, teacher of high-school-aged
girls in their last year of a nonacademic, not-even-vocational,
program; Evelyne and Henri, a married couple (she taught
nursery school, he usually taught fourth grade, but this
year, as part of a special project, was working in Evelyne's
classroom); Ingrid, a Dutch student, doing a master's degree
in education (having no class of her own, she did the weekly
assignments with Jacques); Jacques, teacher of fifth-grade
children in a poor neighborhood; Jeanne, second-grade teacher;
Pierre, fifth-grade teacher, the one member of the group
who considered himself experienced and knowledgeable in science.
participants: Anna, South American doctoral student in
education; Bertrand, African master's student in education;
Helen, Canadian doctoral student in education; Stuart, American
doctoral student in psychology, who carried out assignments
in the International School; Robert, French physicist, visitor
to one session; myself, teacher of the course.
Week: Early Ideas, Imaginative Explorations, Rubber Bands,
and the First Reference to the Air Hypothesis
I asked the
students to do what interested them with the following materials:
plastic dishpans and pails; water; glass, plastic, and metal
containers, with and without covers; escargot shells; nuts
and bolts; odd pieces of wood, some hard, some soft; straight
pins; corks; scrap metal; Styrofoam; rubber bands; plastic
bags; toothpicks; aluminum foil; a balance, consisting of a
piece of pegboard with a plastic pan hung from each end.
game, this group. It's not easy to create something respectable
to do with such a nondescript collection of materials; and
it is very easy to feel foolish.
group, Claire, Danielle, and Lise, started by putting things
in containers and floating them (I overheard one of them say
she was going to put some corks in a tin to make it lighter)
before settling on seeing how many bolts they could place on
a small raft. Starting with a square plastic cover, they examined
whether it mattered if the bolts were placed in the middle
or around the edges; whether the size of the raft mattered;
whether the material it was made of mattered; whether it mattered
that it had a rim or not. This group also screwed cup hooks
into corks to see how many metal washers a floating cork could
Pierre, and Henri were, by the time I saw them, holding pieces
of wood by the corner, just barely, so as to hold them upright
without affecting the depth of their floating, and trying to
determine why some of them floated deeper than others. The
pieces of wood were of highly irregular shapes, but they were
not asking about the volume of wood that was underwater--just
the linear depth. They used the balance to compare their weights,
examined them for whether they were the same kind of wood,
and compared their volumes by pushing them down in a clear
plastic bag of water, noting the height of the displaced water.
they gave an account of this work to the assembled group, they
said that they had concluded it was the volume of a piece of
wood (as assessed by the water rise in the plastic bag) which
determined how deep it floated. Among the evidence they presented
were two pieces of wood which they said had the same volume,
when it was clear to the eye that the wood in fact had different
volumes. The three were nonplussed when I questioned that,
realized that their final conclusion was undermined, became
dismayed over the difficulties of doing anything dependable,
cast aspersions on the homemade balance as an example of everything
that made the situation impossible (though when I stoutly defended
my balance, Colette acknowledged that they were only looking
for excuses), but did not think to question the wisdom of measuring
volume by water rise in a deformable container.
blew a plastic bag full of air, and added metal weights inside
it, to see how many it would take before it would sink in a
pail of water. They reached the point where it sat on the bottom
of the pail, and stuck out above the surface of the water.
I asked them if it was then sinking or floating. There were
differences of opinion. The idea occurred (though not at all
easily, and perhaps it came from me) to put more water in the
pail. They managed to cover it with water, but then could not
decide whether or not it was touching the bottom. It moved
when they stirred the water, but that was not convincing evidence.
break the whole class took up this group's problem: how to
tell whether the plastic bag was touching the bottom of the
pail, or hovering just off it. There was no consideration of
the relative likelihood of one or the other, or of whether
an object ever stays put under the water, without going either
up or down. They wanted to determine what this bag was actually
doing; and, after stirring the water, they couldn't think of
how to tell.
As a temporizing
measure, until someone came up with a better idea, I suggested
using a mirror. Somewhat to my surprise, it turned out to be
helpful: holding the mirror on the bottom of the pail and looking
down through the water, they could see, first, that the middle
of the bag was not touching, and then that the points of it
were. ("Because of the distribution of the density," explained
saw its points just barely touching, they added more water,
thinking that might float it off; it didn't. They wished they
could add a lot more water, a poolful, to see if that would
float it. I proposed making a very tiny object that just barely
sank in a cup of water, for which a pail of water would then
be a lot--close to the equivalent of a poolful for the bag.
They thought that was a good idea. I made some tiny object
of cork and clay, which sank very slowly in a cup of water,
and rested on the bottom. Predictions then were half and half
that it would float in the pail. It did not float in the pail.
that more water might support more weight then was modified--though
nobody made this connection explicit--to the idea that more
horizontal surface contact between water and object (later
referred to as the bearing surface) might support more weight.
They found two identical small, flat containers and weighted
then with equal weights, distributed so that one floated upright
and one floated flat. Both sank, but the flat one sank slower.
Everyone felt that was inconclusive.
At one point
during these experiments (which took some time to accomplish,
given the nature of our materials), Pierre said that things
floated if they had air in them, and sank if they did not.
I had expected this hypothesis at some point, since it always
comes up. I knew also that there was no very direct way of
countering it. I cut a small piece off a plastic bag, and put
it under the water: it floated up. I couldn't tell what most
of the group thought, but Pierre said, rather weakly, that
there were teeny invisible pockets of air in the plastic, which
accounted for its floating.
before, I had discovered with my students a remarkable feature
of our collection of rubber bands, to which I now turned. I
tossed in a green one, and it floated. "Capillarity,"
said Pierre (a word which is commonly used in French when we
would say "surface tension"), "like a needle." I teased him,
for seeking refuge behind a word and also pushed the rubber band
underwater to show that it rose. (It did not occur to me, however,
to show the contrast with a needle, which can be made to sit
on the surface of water, but sinks once you push it under.)
I then tossed
in a blue rubber band, which sank. They were intrigued, and
produced a color hypothesis, as a joke. (I said I liked that
better than "capillarity" and they seemed to get my point.)
They pursued the color hypothesis, and saw that indeed, the
red, green, and yellow bands systematically floated, and the
blue and white bands systematically sank. Very intrigued. Someone
tied together two of different colors, to see which won. They
also tried a plain brown one: it floated.
In the midst
of this Claire, quietly by herself, tied a knot in a yellow
band, as a better way to find out whether floating was influenced
by the amount of bearing surface. The yellow band still floated,
and she was distinctly pleased with this neat test. She caught
my eye at that point, to share her pleasure.
Week: Investigations Become More Systematic, and Explanations
early talk of moon observations and work in classrooms Colette
said that her adolescent girls had not much patience for playing
around. Above all, they wanted to know the answer. That took
me aback; I didn't know to what question they wanted to know
the answer. To all the students in this course, it was obvious:
why some things float and some sink. What had I had in mind?
one thing, not some single nugget-like answer. Rather that,
whatever people in the class thought about floating, it would
be fleshed out, called upon to account for some surprises,
attached to real-world complexities. I hoped they would come
to agree that this part of the world is more interesting and
less simple than they had thought. And I hoped they would be
able to see what it is like to figure out some of the complexities
I asked whether
there was anything from last week that they would like to pursue.
They said rubber bands. Without elaborating, Colette said that
a colleague of hers had said that the dyes must have different
specific weights. What she said seemed plausible to some, and
nobody had any other idea. But it did not seem like an answer
I then proposed
three problems as possible activities for the session, in order
to develop some surer familiarity with floating and sinking
For rubber bands, I asked them to try to quantify the various
degrees to which the different colors sank or floated;
posed the same problem more generally: do two sinking objects
of the same weight need the same amount of some floating
material to make them float?
proposed taking some object that sinks, modifying it so
it floats, and specifying ahead of time which point on
it will be on the uppermost side as it rises to the surface.
and Lise worked on the third problem, but started with a floating
object rather than a sinking one. To arrange for a given point
to be highest, while the object sinks, they found they had
to add a weight underneath, "in continuation" of the point.
them to do the problem as originally posed, and they tried
the reverse procedure. Taking a stone, they specified the point
on it that they wanted to rise, and they tied a piece of Styrofoam
on the opposite side. They found that the Styrofoam turned
the stone over, so their point rose last rather than first.
They eventually placed the Styrofoam directly on the point
that they wanted to rise first, but were not satisfied with
this solution, since the point was not the first point on the
total object (stone plus Styrofoam) to get to the surface.
Later, in discussion with the whole group, I asked if it would
be possible to make an object such that the floater would not
be the topmost part. They thought then of putting some floater
on both sides, but wondered whether it would float up sideways.
We were all interested in going further with this, but put
band group put their efforts into ordering the five colors.
By racing them, ascending and descending through water, and
by putting them in salt water, they were able to tell that
the whites were greater sinkers than the blues, and the yellows
were greater floaters than the greens and reds. But they could
not yet distinguish between the reds and the greens. The colors
were size-coded, as it were: there were five different sizes,
one size per color. But the sizes did not coincide with the
floatability: the best sinkers (the whites) were the smallest,
but the best floaters (the yellows) were not the biggest. To
distinguish between the greens and the reds, their next idea
was to tie one of each of them to a white one--one red tied
to one white, one green tied to another white. When both now
sank, they tied two of each of them to a white one. The green
now sank, and the red still floated. They concluded that red
was a better floater.
that the reds were bigger didn't seem to enter into consideration--and
yet people in the class still felt pulled. I think it was because
they had established that the size of those rubber bands was
not responsible for their floatibility, and yet they were vaguely
aware that in this case the size of the red might affect how
much white it could lift. They felt they needed to know something
about the nature of the material itself. I had already been
suggesting that they cut equal-sized pieces from the green
and red bands, and use them in their experiments. Now I said
that it was precisely in order to understand better the nature
of the material that I had been suggesting that.
then to find sense in my suggestion, and carried it out. (The
green carried more white than the red.) Of course that investigation
did not tell them much about the nature of the different-colored
rubbers. But it did establish neatly the order of the colors,
and they then felt quite satisfied with what they knew--the
nature of the material was not, after all, what had been bothering
In the group
discussion, I raised the question of how they might go further,
and quantify the relationships among the colors. Several of
them were particularly sharp, as together they thought through
what they would have to do. They could, for example, use the
provided they established equal-sized pieces. Or they could
time the ascents and descents, as they had started to do, but
through greater depth; or through the relatively shallow depths
available here, but a hundred times over, taking averages.
groups worked on the third problem. They all found, with some
surprise, that different sinking objects of the same weight
might need different amounts of floating material to float.
The most striking instance was a small weighted plastic container
which needed 15 times its weight in Styrofoam to float it.
Pierre, la scientifique, was in this group, and he was again
maintaining that an object floated or not according to the "volume
of air" it contained. I tried to support any skepticism that
Henri and Jeanne might be tending to feel by exclaiming that
surely the Styrofoam did not have 15 times the amount of air.
(Pierre had not claimed it did, but I wanted to push the matter.)
Pierre's first reaction was that it could have maybe not 15
times the volume, but 15 times the quantity. (The 15 factor,
in fact, applied to weight.)
group discussion, he said maybe five times was taken up by
air, the rest was something else. I was trying to be harsh
on the air hypothesis, expecting that we could see through
it by now. I asked about the role of air in the rubber bands.
Colette suggested that perhaps one dye had more air in it.
They talked about how to know whether there's air in something.
Someone asked whether there was air in water, and Pierre said
that of course there is; how else do you think fish breathe.
Jacques countered, asking how we know it's air: maybe we should
talk of a vacuum, or maybe it's oxygen. Maybe, he said, it's
just how much space there is between the molecules. It's not
molecular, Pierre said.
surface was again considered. Claire said that couldn't be
a factor because of her experiment the week before, tying a
knot in a rubber band. Jacques agreed with her, but saw that
he wanted to contradict himself: What about those water bugs
that can stand on top of water because of their broad feet?
Stuart said that his kids had found that aluminum foil floats
in a sheet, but sinks in a small packet. Some thought that
they themselves floated more easily flat than in a ball, but
there was disagreement on this.
I tried to
summarize the factors they seemed to be considering: surface,
air, volume. All three times that I tried this summary, Pierre
broke in whenever I said "air" to specify
"volume of air." I finally paid attention, and asked why he thought
that should be specified. He did not manage to say what he was
thinking, and ended up saying the "quantity of air."
We came back
to the question of whether there was air in Styrofoam. Danielle
pointed out that you can crush Styrofoam, and proceeded to
do so. She managed to flatten it dramatically and it still
floated. She said it was still a bit spongy, by way of saying
that there was still some air in it. They compared its speed
of rise from the bottom of a pail of water with a noncrushed
piece and found no difference. Pierre, among others, said, "Of
course," or words to that effect. I said to him that I thought
he would have expected the opposite--that with less air, the
crushed piece should be less of a floater. He saw what I meant,
and responded (as the week before with respect to the balance)
by saying that you really couldn't count on these makeshift
materials for any dependable experimental findings.
At the very
end of the class it occurred to we that if the Styrofoam was
crushed underwater, bubbles should come out. About half of
us were together to try it. Oddly enough, it was not clear
whether any came out or not. We then squeezed with a vise.
Bubbles were obvious this time. (But not, it seemed to me,
with the volume one would have expected. This remains a question
Week: We Encounter Mysterious Pill Bottles, Statistical Rubber
Bands, Floating Balloons, and Sinking Wood, and We Make a
brought in two plastic pill bottles which one of her girls
had brought in to her. With no water in them, they floated.
Full of water, they still floated. Only Pierre refrained from
bring surprise. He said that of course there's air in the water.
Henri balked at this; he wasn't convinced it was due to air.
upstaged even this mystification, with a collection of colored
rubber bands which he had bought and experimented with. Apart
from the white, which systematically sank, the other four colors
had a statistical tendency to sink or float, but you couldn't
count on any one to do either. Moreover, out of a population
of 10 blue bands say, all 10 might be floating after half an
hour, only four after an hour and a half, but seven after four
hours. Total mystification. I borrowed them to study at length
a change by then. That things that are lighter than (the same
amount of) water go up and things that are heavier than (the
same amount of) water go down did not seem to be an idea that
was establishing itself easily through the collection of phenomena
we had been studying. So far, my students were tantalized and
intrigued; enough to keep pondering and to try to make sense
of what they saw. That was the best possible position from
which to proceed; but it seemed to me we needed to proceed
down a different path in order to see similar questions from
different angles. I proposed that we stop a frontal attack
for a while, and go around the edges. They agreed to that,
though they were clearly impatient to know the answer.
The end run
that I took was not, I think the best approach, although a
number of fruitful developments took place in spite of it.
It consisted of focusing on air and water (and perhaps it came
partly from my wanting to catch the bubbles squeezed out of
the Styrofoam). I know that there were surprises in the interactions
of air and water, tubes and containers, and that most of the
surprises were explained if one only kept in mind that when
put together, air goes up and water goes town. It seemed to
me that this useful, simple principle might then be helpful
in thinking about more complicated objects. For some reason,
it did not occur to me that I was highlighting the air hypothesis.
did occur, and the simple principle did emerge, but this work
in general did not contribute, as far as I could tell, to the
ideas my students were developing about floating. I shall mention
only one main pursuit that related quite explicitly to the
problem that was on their minds.
materials available that day were balloons. I did not usually
have a specific purpose in mind for materials which were available,
but the balloons had been intended to make bubbles underwater,
or to fill with air, or somehow to relate to what I had intended
as the work of the day. Jacques, Ingrid, and Anna, however,
classically turned the materials to their purposes--filling
them with water to see whether they would float! Colette's
pill bottles must have been the inspiration for this enterprise,
the only thing that would have suggested the question.
It was as
if they set out to astonish themselves, and astonish themselves
they did. The balloons did float. Half full of water, almost
full of water, and completely full of water, feeling very heavy
indeed, they did float. Ingrid thought to try a balloon by
itself, working hard to get every bit of air out of it, although
she didn't think to empty the air underwater: it floated. She
hadn't predicted that, nor the opposite. She took the outcome
with interest, and pondered it. Most of the others didn't take
much interest in it, but Claire went on to tie it in knots,
and noted that it still floated.
reporting to the rest, Anna said that the balloons floated
more or less high according to whether there was more or less
water in them. Almost everyone was astonished that they floated
at all, especially Colette: "It feels so heavy!" Pierre, once
again, was not surprised. Of course, he said again, there's
air in the water. It seemed to me that others besides Henri
were left unsatisfied by that refrain.
At the break,
I brought out a gallon jar in which there had been a piece
of wood in water since the year before. The wood was resting
on the bottom. There was great interest in this. The group
wanted to know how much it had weighed when it was dry: I didn't
know. They thought that it must have soaked up water, but did
not understand why this water-filled wood sank while a water-filled
balloon floated. As for what should be done with it, Claire
suggested taking the wood out and letting it dry. Pierre asked
that it be kept as it was for a while, so he could try to float
it, with air.
of other remarks that evening bear reporting. In the face of
our most dramatic instance of the reality and intractability
of air, Ingrid wondered what happens when a boat turns over
and sinks. For water to go in, the air has to go somewhere
else. Where does it go? No one knew.
point I heard Jacques and another wondering together whether
an object which sinks in a dishpan would do the same in a huge
amount of water. Of course, I thought we had dealt with that
the first week, with the tiny object that still sank in a bucket,
having barely sunk in a cup.
At the end,
I asked if they were still willing to pursue side-tracks, or
were they too impatient. They were impatient, but they said
they'd stick with me.
Week: We Look for "Proper Places" of Different Liquids, and
Try to Refloat the Wood
activities took place in this session. I introduced a variety
of different liquids, and most people worked with them. Henri
and Pierre, however, preferred to try to float the old piece
of wood. I shall describe their work first.
the water that the wood had been in for a year must by now
be airless, so they started by blowing bubbles into the water
through straws. When this did not float it, they removed the
wood to some fresh tap water and again blew bubbles into the
tap water. When the whole group gathered to hear about their
progress, they made a third try--replacing fresh tap water
with a solution of salt water. Henri and Pierre's idea seemed
to be the following: although the wood was placed in new water,
the old water was still what saturated it; since the old water
was airless, they would have to get it out of the wood before
new, aerated water could enter it and make it float again.
Recalling some version of osmosis, they reasoned that the salt
water would draw out the old water from the wood, and it would
then be replaced by the new water, which would be aerated.
Most of the
other class members, especially Danielle, seemed to think this
was pretty crazy--although Pierre's scientific talk of the
osmosis phenomenon made some of them take it somewhat seriously.
a kid had told him that a clothespin had floated and then sunk
because water had gone in and taken the place of the air that
was there. He proposed this as a plausible hypothesis in the
present case but was not sure it applied. Ingrid wondered (again!)
if a whole lot of water to float it in would make a difference.
There was no strong reaction to this thought; nobody took it
as a suggestion for an experiment. Ingrid also observed that
entire logs are sometimes found sunk in lakes in water not
airless. Pierre and Henri agreed, but argued that those logs
were rotten. Someone said that ours was, too, and I pointed
out the stink. The discussion was left inconclusive.
said in this discussion that oxygen is what matters, not air.
He asked if the top of the jar had any oxygen when I opened
it. I was not sure, but thought not. And I pointed out that
I didn't know whether what might have been there would have
been oxygen or air or carbon dioxide or nitrogen or what. Pierre
was sure that only oxygen would make a difference in floating.
was left in salt water so that the osmosis procedure would
have the time it needed. Ingrid asked how long it had taken
to sink. Since it had taken two or three weeks, she thought
one week too brief for this experiment. I pointed out that
wasn't necessarily the case, since this procedure was not the
reverse of the one that had brought about its sinking. The
class tended to agree.
I had brought salad oil, mineral oil, salt, liquid detergent,
alcohol, molasses, and a heavy syrup. I had also brought as
coloring agents, ink, red food coloring, and iodine; and (for
economy's sake) very skinny test tubes in which to work. In
addition to various liquids, there were various solids, in
tiny bits--plastics, toothpicks, Styrofoam, seeds, rice, rubber,
hard wood; and two large containers, one of salad oil and one
of heavy syrup, in which the class could try floating the solid
made a variety of intriguing or lovely effects with the liquids,
most of which I shall not try to describe. At the beginning,
at least, none of the work was directed toward answering any
specific question. They tried to get to know the materials.
liquids together, and watched them separate from each other.
They watched drops from one liquid rise or sink in another.
Ingrid tried to dissolve each liquid in molasses. Alcohol was
the one with which she did not experiment: all the other liquids
were similar to molasses in thickness or heaviness, and since
none of them had dissolved, she thought it unnecessary to try
In the general
discussion, we looked at Evelyne's six-layered bottle. When
she had made it earlier, she had not known what each of the
layers was, so I had asked her to try liquids two at a time.
Now, redoing some of her pairs during the discussion, we found
that alcohol tended to stay under salad oil; but when we used
a very skinny tube (nobody mentioned this necessary condition)
and put salad oil in first, alcohol stayed on top. The Claire-Danielle-Lise
group said the oil was "a barrier,"
and this was generally accepted, along with the observation that
other liquids show that characteristic--liquids went right through
them to their proper places. (I do not recall who introduced
the terminology of "proper places," but chances are that I did,
since it is probably an anglicism.) After this evening's work,
the class did have a general idea of which liquid went where.
of this notion of proper places some wondered whether one could
make alternating bands of alcohol and oil. Some wondered whether,
if one used a great deal of a liquid, it would sink down underneath
a liquid that it otherwise floated on. Some thought this would
not happen. I did not make note of their reasoning.
At the end
of the discussion, Evelyne said she did not see why weight
mattered with liquids and not with objects. This was the first
time I was aware of anyone (with the possible exception of
Ingrid in her molasses work) thinking in terms of weight. Not
thinking to ask what she meant by that--what made her think
the weight mattered, since nobody had weighed anything--I said,
instead, that hers was just the question to leave on, and we
did. Nobody, all evening, had tried to float any solid objects.
Week: We Become More Familiar with the Liquids, and We Dry
wood remained submerged --- floating neither in its salt water
nor in new water. Most of the group ---especially Daniell ---
expressed a good bit of skepticism about why on earth it would.
It seemed to me that Pierre was confused about whether he thought
it was the water in the wood or the water around the wood that
had to be changed.
said that it didn't matter what kind of water was in it, there
was water in it, and that's why it sank. Someone else, again,
compared it to wood on the bottom of rivers and lakes. Pierre
again observed that that wood is rotten; and said it does matter
what kind of water is in the wood. If the water was oxygenated,
he said, our piece of nonrotten wood would float. Then, Lise
said, this piece of wood, once dried out and floating again,
would stay afloat provided its water was changed every day,
or stirred, or had air blown into it. Pierre and many others
agreed that this would be a test of his ideas. So they set
out, now, to dry it, after first weighing it wet, out of general
interest in the comparison of its wet and dry weights. Claire
and Henri did the weighing.
In the interests
of having the class develop a yet more solid feeling for the
buoyancy of different liquids, I directed their attention to
eight different questions:
the proper places of these liquids with respect to each other
is the lowest possible place for salt water--that is, when
it is as salty as it can get?
liquids form barriers and in what conditions?
liquids mix with each other?
liquids do the various solid bits float on?
the amount of a liquid influence its proper place?
you make alternating stripes with two liquids?
question: How come we talk about weight with liquids, when
we ruled it out with solids?
arose, of course, both in individual work and in the discussion.
These involved the difficulty in separating certain liquids
from oil, if they had been thoroughly shaken; characteristic
ways different liquids move; curving interfaces; whether colors
leave one liquid and enter another; whether molasses is after
all a liquid or not. What about jam? What about sugar? What
remains of various of the liquids after evaporation?
At one point,
Evelyne had a container in which some drops went up and some
went down, through a middle liquid. She thought that what went
up were air bubbles surrounded by alcohol and what went down
were air bubbles surrounded by syrup. Pierre, working beside
her, was maintaining that they were solid drops-without air
in the middle. His major reason was that if it's air, it can't
that he had worked on trying to alternate layers, working with
syrup and salad oil. While on the whole the syrup settled under
the salad oil, a slim amount of syrup stayed on top. He had
tried to thicken that layer, but it never got thicker; as he
added syrup, it went down through the oil, leaving the small
amount on top. It was unclear to inspection whether this was
a thin layer, or simply a ring around the edge of the tube.
He had concluded that it was a layer, because when he posed
a grain of rice in the middle, it was braked, before it went
rushing down through the oil. In a tube of oil alone, the rice
grain went immediately rushing down.
to Henri, the grain of rice went through the oil and came to
rest on the syrup. As we passed the tube around, however, it
went on down through the syrup to the bottom; Colette spent
some effort trying to get it back to rest on the top of the
and Lise tried floating different objects in different liquids.
They prepared a set of plastic cups, each with about an inch
of one liquid. They were disappointed, in general, not to find
more differences from one liquid to another. Perhaps due to
this dull outcome of their well-planned investigation they
began, after a while, to use grains of salt as objects, and
drops of liquid, also. They seemed to acknowledge at the end
that they were changing the liquid when salt or other liquids
dissolved in them, but as they worked they were giving no thought
to that fact.
that salt water could be made to be heavier than liquid detergent,
but not heavier than syrup. In her tube which contained, from
bottom to top, syrup, salt water, detergent, and fresh water,
she dropped grains of salt, attempting to make the top layer
slightly salty. It failed to work; the grains went right down
through all the layers.
working to establish the liquids' proper places, first poured
alcohol followed by mineral oil and then poured mineral oil
followed by alcohol. She convinced herself that alcohol stays
on top. She then poured equal quantities of six liquids-syrup,
ink, water, alcohol, salad oil, and mineral oil. She didn't
give an explicit reason for comparing equal quantities, but
it was a first, even if tiny, step in a direction that led
to important consequences later. There was a visual elegance
to the array, even though three of the liquids mixed with each
other. She had, from bottom up, a layer of syrup, an obviously
triple layer containing a mixture of ink, water, and alcohol,
a layer of mineral oil, and a layer of salad oil. Its orderliness
was very appealing.
leaving for the night, Pierre tried to float the waterlogged
wood, which seemed dry. It sank, in tap water, but slower than
it did when they first tried it, he said. Claire had weighed
it partway through the session. It seemed dry then also, but
there had been no difference in the weight.
As we left,
Jacques said we should get a physicist some day, to answer
all this. I said-with considerable hubris, considering that
I have no physics training-that I thought we could answer most
of it ourselves.
Week: We Get a Physicist and Narrowly Escape Disaster
Robert a tout fichu en l'air tout ce que j'essayais de faira
avec Pierre."(Translation: "Zut, alors! Robert totally wrecked
everything I was trying to do with Pierre.")
a physicist friend and colleague from a French university who
was himself interested and engaged in working with teachers
on the teaching of science from a Piaget perspective. We had
often talked about our pedagogy. Specifically, I had been telling
him about the investigations of this group, week by week. I
knew our pedagogical approaches were different, but I had not
realized how little he understood of what I tried to do. It
did not occur to me to give him special instructions about
how to behave this evening, especially since I had told him
about it each week. As I introduced him to the group, I said
I was asking him to promise not to explain anything, but that
was the extent of my instructions.
the whole of the experimental period glued to the side of Robert.
I do not know what he asked nor what Robert said, but there
was a lot of questioning and explaining taking place. My attempts
to break them up became rather comic. Once I said: Don't sit
here next to Robert, he'll lead you into conversation. Robert
said, "I didn't say a word, it was him." I said, "I know, but
he's my student. I want to be polite to him." Jacques found
that very funny. I think it was he who most appreciated the
comedy of the evening.
In this session
I wanted them to work more systematically with solid objects.
I had prepared a tube with liquid detergent, mineral oil, and
alcohol--three liquid layers, with solid bits at both interfaces,
as well as on the top and on the bottom. They liked it a lot.
Rather than asking them to try to do something similar, I gave
them much more focused instructions. Most of them had not yet
paid attention to the fact that a given solid object might
float in one liquid and sink in another. I knew what complications
arise with more than one liquid at a time, and judged that
the previous two sessions had been amply characterized by such
complications. Now I wanted the activity to be simple enough
to provide some regularities. I asked them each, or in each
group, to take one liquid, and classify the solids as to whether
they floated or sank in their liquid. Halfway through, I asked
them to do the same with a second liquid.
I had told
them to use yogurt containers and everyone did but Pierre.
He used a test tube--I presume because it seemed more scientific.
As a result, the small objects had no room to pass each other;
everything stuck together in the oil and his findings turned
the beginning of the experimenting, he put a piece of chalk
into his liquid, and while it sunk a trail of bubbles rose
to the surface. I asked him where he thought they came from
and with a characteristic little smile, which indicated his
unhappiness with not knowing for sure, and addressing his words
to Robert alone, not to me, he said, "It's a chemical reaction
of some sort, isn't it?"
beginning of our summarizing discussion Robert intervened to
ask what I meant by proper places. Was it Aristotelian? I said
it was our terminology. He said, yes, but for what? I produced
an account which satisfied him, but I was rather short with
him, which I did not feel good about in the context of the
class. In general, instead of trying to explain to him what
we were about while we were about it, I tried to proceed, cutting
him short in order to minimize the damage, intent on telling
him afterwards what a nuisance he had been and why.
of kinds of solid bits had behaved inconsistently, and we had
difficulty deciding where they belonged--matches, straws, wood,
aluminum foil. The best discussion concerned the question of
what might make an object not go to its proper place: it might
be pushed by another object on its way to a different place,
and then caught there; there might be bubbles attached to it;
it might have a coating--une carapace--around it of some other
liquid it had traveled through; sometimes they stopped at the
surface of a liquid, unless you pushed them through it.
I asked them
to go back to working out the four troublesome kinds of solids,
with these factors in mind. They worked for some time, but
without conclusiveness. Many of them worked with aluminum foil,
and it is the hardest material of all to understand. A sheet
is so thin, it does not break through the surface of the water,
and a ball is full of air pockets--so in both these forms it
stays on top of water, though the proper place of aluminum
is on the bottom. Some class members were led to speak again
of the bearing surface.
drew a connection between the aluminum foil, floating needles,
water bugs walking on water, and their observation that sometimes
solid bits stayed at the surface of a liquid unless you pushed
them through. With hindsight, I now wish I had drawn more attention
and thought to that phenomenon right there. We never came back
this close to surface tension.
of the evening's work then did manage to establish, in spite
of a few confusing contradictions, the idea that both liquids
and solids have their proper places in the floating order.
We then moved
on to the now-dried piece of wood. Henri and Claire weighed
it, as we all watched, putting the dry wood on one side and
its wet weight (measured in unit pieces of steel) on the other.
The equal arm balance thudded down on the side of the wet weight.
Whereupon Robert said that he had changed the hole in which
the balance arm was suspended--just to give us pause about
taking a balance for granted! I put it back, saying curtly
that we would study balances another time. (In a parallel class,
in fact, during these same weeks, balances were the subject
of an equally lengthy and perplexing study.) The corrected
balance still showed a clear difference in weight between the
wet and dry states of the wood. Henri had the nice idea of
finding the amount of water that corresponded to the weight
of the steel; it amounted to about half a yogurt container.
Some had thought it would be a great deal more than that.
It was my
suggestion then to attach the weights to the wood, to see if
it would sink. It did, but not with a thump, as one of the
class pointed out. Someone said that that was how it had sunk
when they first took it out of the water, but I am not sure
that was true.
I had the
idea of floating a fresh piece of wood in the old water, which
we had kept. Everyone, including Pierre, agreed that that would
be a test of Pierre's hypothesis that the wood had sunk because
the water that it was in lacked air. We found a piece of the
same kind of wood, and it floated. Nonetheless, great lengthy
discussion ensued about various kinds of water and their role
in floating wood! And that became the very worst part of the
evening, from my point of view.
is any basic principle in my teaching this way, it is that
people are to feel free to express their thoughts about what
is going on and why, and that those thoughts are to be taken
seriously. Now a number of the ideas that had been put forth
throughout these weeks were quite fanciful, but we had always
given them due attention, and tried collectively to devise
experiments that would check them out. Robert, on the other
hand, referred to several ideas as harebrained--de la fouthèse.
At the same time, Pierre's ideas, which were the most muddled
but the most couched in scientific-sounding words, he reinterpreted
until they made sense; and then he gave them his support. I
was too distressed to follow, let alone remember, what was
I did make
some efforts to redirect the discussion, which included asking
the group what they thought of as the proper place of this
wood--floating or sinking. This approach, however, led to no
great insight. They compared the situation, quite properly,
with chalk, which floats for a while and then sinks, and said
that one would have to know the time scale involved to answer
the question. When they reached this point, Pierre seized the
occasion to say that time was working against his hypothesis;
no failures could be taken as conclusive because they were
being carried out in a short time scale.
end, Robert asked to say two things. Hoping that by then he
had seen enough of my approach to be judicious in his remarks,
I gave him the floor: I was wrong. First he lectured the group
on how it was all very well to do experiments, but not good
enough if that's all one did. The point of experiments, he
went on, is to check a hypothesis; Pierre has hypotheses, which
makes the experiments they give rise to worth doing; but most
of the rest of the class. . . .
wished he would go away. I did not try to develop my view that
before you can form a hypothesis you have to explore, developing
the familiarity out of which hypotheses can grow; that the
trouble with the science most of us have learned is that it
is made far too neat--neat hypotheses, neat formulas, neat
answers--before we have any sense of what the questions and
perplexities are, so the science we learn never touches what
we think about the world around us.
Nor did I
point out that Pierre's hypothesis was his greatest liability!
In contrast to the other members of this group (but in common
with large numbers of people), Pierre didn't want to acknowledge
that he didn't know, and he had a hypothesis wonderfully suited
to that purpose: if an object floats it is because it has enough
air in it; if an object sinks it is because it doesn't have
enough air in it. He could explain anything. And, at the same
time, he could hold himself back from any perplexities that
might lead him to figure out something new. My hardest job
of the year was to move Pierre beyond that hypothesis. I wished
Robert would go away.
however. As his second point, he described a procedure for
finding the density of flour. Now nobody had mentioned density
all evening--neither the word nor the notion figured yet in
our talking about the issues. But he hadn't noticed this. He
understood that a key to our questions was the notion of density,
and he wanted to tell about a neat technique that involved
that idea--whether or not it had any relation to floating and
sinking. (It didn't.)
I was, again,
beside myself at this intrusion. This group of intelligent
and willing adults had spent six evenings working towards an
understanding of floating and sinking. It would make all the
difference in their feelings about the accessibility of science
if it were their own ideas and explorations that took them
to this understanding. If Robert, discussing his fancy technique
for establishing the density of flour, took that possibility
from them, convinced them that they should have known all along
that it was a simple matter of density, they would lose most
of the benefit of their struggle.
I was also
beside myself when, in the course of his account, he described
how wood, with its fibers and open tubes, differed in structure
from Styrofoam, with its closed air pockets. (I had told him,
in week-to-week discussions, about the attempts to squeeze
the air out of the Styrofoam.) "Merde, merde, merde," is the
close of this paragraph in my notes. And I would not know until
the next session what the effects of these lectures had been.
But a better
moment arrived even before the evening ended. Robert based
part of his explanation on the formula plvl=p2v2. All of the
class (I expect Pierre was an exception, but I did not make
note of this) had the confidence--which I like to think came
at least in part from their experience in this course--to admit
good-naturedly, if with some small embarrassment, that the
formula meant nothing to them. Robert was astonished. Hadn't
they taken physics? Of course they had! The formula nonetheless
meant nothing to them. Robert explained it and continued his
account but it was not clear that any of them followed it.
the evening, during the part of the class that dealt with their
work with children, Henri gave a long statement on the virtues
of "this method" compared with "the other method,"
and he referred to Robert's formula as something which they had
all learned at some time by lecture and memory.
At some other
point in the evening Jacques made a comment to the effect that
experimenting was better than lectures and Robert responded
sympathetically to this criticism of lecturing, saying that
what one remembered, in spoken discourse, was not what one
heard but what one said. We all liked that. To me it seemed
it could have led Robert directly away from the kind of explaining
he was doing all evening.
Week: I Start My Journal; "Super Class, Super Great Class
(In My Opinion)"
by telling the class how troubled I had been by Robert's visit.
They were very sympathetic. Jacques said Robert was very like
a Frenchman, having to show what he knew. Henri and Evelyne
said something to the effect that he was always explaining,
and why couldn't he not explain. Ingrid, who had left early,
asked: Oh, did he finally get a chance to talk; I thought he
must have been very frustrated.
I said that
I felt especially bad about having, essentially, withheld from
them the chance to talk to a physicist--as Jacques in particular
had said he would like to be able to do. Jacques, however,
hardly seemed to remember; he more or less shrugged it off.
I also said that I had gotten angry at Robert afterwards for
a few things; they laughed and said they weren't surprised.
I said we had often talked about our pedagogy, and I had not
realized that I should have explained more to him before his
visit; they said it's hard to understand. Pierre unfortunately
was absent this evening. (So were Danielle and Lise.)
discussion, I felt much better. My students seemed above all
amused by the clash of fundamental views of how to help people
learn science. I never would know for sure how Robert's interventions
influenced their ideas about floating and sinking, but it seems
to me the net result was more confusion than clarity. He certainly
hadn't made them feel they had nothing more to do.
everyone worked together, standing around one high table. Colette
said she wanted to do a final class with her students this
week on floating and sinking, and she wanted to be ready. So
we rolled up our sleeves. Henri said quite firmly, to start
us off, that he believed it was air that made things float.
Ingrid demurred right away; she did not think there was air
in the plastic pill bottles that Colette floated. She spoke
of "specific weight" (the French equivalent of specific gravity),
but could not quite say what that meant.
I asked about
the liquids: did they think there was more air in alcohol than
in syrup. Led by Henri they thought not. Henri said he thought
that for liquids it was weight that mattered. I took a large
amount of oil and a small amount of water, and pointed out
that this oil would weigh more than the water. Henri then said
he meant the weight for the same quantity. Colette had mentioned
weighing liquids in class with her students. We never had done
that in our class (she had layered equal quantities, but not
weighed them), so she brought out their results. The girls
had weighed one decilitre of each liquid. The order of the
weights was the same as the order in which they floated on
one another (with one exception). Colette was surprised; she
had not noticed when her students had done this that there
had been a correlation between these two orderings. The group
as a whole took the girls' work as convincing evidence that
weight-for-a-given-amount was the relevant factor for liquids.
I went back
to whether air was a necessary hypothesis for objects. Bertrand
thought that for objects it was a question of the material
they were made of. I asked what it was about the material that
mattered; someone suggested the amount of air it contained.
tentatively and very quietly wondered if it was the same for
objects as for liquids--an object would float on a liquid if
it was lighter. I asked her (since no one had heard it) to
say that again. She did, but nobody took it up.
A short while
later, Claire said she thought Colette was right, and I tried
to pursue the idea. I brought out a balloon full of water,
floated it, and asked with what they would compare its weight.
This proved more confusing than helpful. Colette suggested
making the comparison with the weight of the water in the pan.
Others said the amount of water in the pan didn't matter.
My idea with
the balloons had been vague. I was as much groping to make
a connection with my students' thoughts as they were, and I
moved as stumblingly toward the helpful questions as they moved
stumblingly toward the helpful ideas. In any event, what I
thought we needed next were two objects of the same volume
and different weights. The objects I came up with were two
identical and capped bottles. I left one filled with air, and
put enough water in the other until it just barely floated.
I was now
thinking of these two bottles as objects themselves, with equal
volumes and different weights. The water in one of them was
meant to make it heavier than the other, but it just happened
to be a liquid. I hadn't meant to draw attention to the contents
in their own right. However, when I asked again, as with the
water-filled balloons, with what we should compare the bottle
that just barely floated, it was soon clear that nobody thought
of it as bottle-with-contents-as-object. It did serve a different
it up and looked at it closely--clearly looking at the water,
not at the bottle-with-contents-as-object. She started to say
that we should now fill this bottle with the various different
liquids, one after the other--and then she stopped in confusion.
It was a germ of an idea, but not clear to her. Claire took
it the next step. She said they should now try to see how much
syrup they would have to put in the other bottle to make it
also just barely float, and then how much alcohol, in yet another
identical bottle. A few were about to try that, when some others
said they already knew it would take less syrup than alcohol.
about to let that drop, when I said--offhand, and really as
a joke--that there would be more air in the one with the syrup.
They laughed at my devilry, found this thought totally perplexing,
and decided they had better do it. Lo and behold, when they
did it, they both just barely floated, but one had more air!
That was the single greatest blow to the air hypothesis!
weighing the two liquids, which led Claire to point out that
this was already a balance. They had taken great pains to see
that they floated at the same level (namely--just barely),
and since the bottles were identical, that meant the amounts
of liquid weighed the same.
work was going on, it became clear to me that to pursue Colette's
idea that maybe it's the same for objects as for liquids, what
we needed was not a just-barely-floating object, but a sinking
object, an object whose weight-volume characteristics would
be different from the water around it. I tossed a rubber stopper
into the water, and it sank.
up on Colette's idea, I asked what should we compare it with?
Once again, their ideas first took off in a quite different
direction from what I had had in mind. Claire remarked, as
she looked at the stopper, that objects don't weigh the same
in water. There was considerable discussion of this phenomenon,
the nicest comment being Stuart's explanation of why a pail
of water is easier to lift under water: Under water you're
only lifting the pail, while out of the water, you're lifting
the water, too. Jacques, during this discussion, lifted the
stopper from the bottom of the pail, into the air, and said
he could feel a slight difference, though he said you would
need to do it with something bigger (or did he say heavier?)
to be sure.
Colette, during this time, had been struggling together to
remember what they had once learned of Archimedes's principle.
Jointly, haltingly, they produced it: a body in water loses
in weight the weight of the amount of water it displaces. I
said to myself: Well, there it is, they've done it by remembering.
But it led absolutely nowhere! Everyone heard it, laughed,
and went back to work on their problem.
the first suggestion about what to compare the rubber stopper
with, building on her interest in weight, and ways of weighing
things. She suggested finding its equivalent weight of water,
then seeing whether it floated in syrup, and, if so, finding
its equivalent weight of syrup.
I was privately thrilled by this suggestion, it was not received
with much interest by the group. Some said they knew it would
take less syrup than water. Others recalled that Colette had
reported that her students had done something similar with
three rolls of scotch tape (admittedly an unusual unit of weight):
they had found out how much of each liquid it took to make
the weight of these three rolls. Though Claire was hesitant
about her idea, she stuck with it. Her one response to the
girls' experiment was: we don't know whether the three rolls
floated. It turned out that the stopper did float in syrup,
and a couple of other people became interested in Claire's
clear why, but, I think, with some sense it might be a good
thing to do, they decided to do it. Claire did the weighing,
and we ended up with syrup in one plastic cup, water in another,
and the rubber stopper in a third, all weighing the same thing.
People seemed to recognize that it was a neat little setup
but nobody knew what to do with it.
As they were
talking about it, my attention was caught by Jacques, fingering
some materials hidden from most of us behind a bucket on the
table, and muttering to himself. After some time, he described
a plan. In three identical goblets, put equal amounts of alcohol,
and put these equal weights of syrup, rubber stopper, and water
into these three equal amounts of alcohol. All three will sink
in the alcohol, we know, and then we can see how high the alcohol
will rise in each case.
Few, if any,
followed Jacques's thinking. They asked: Why alcohol? Someone
said: They'll mix anyway. They said: We already can see that
the water will make it go higher than the syrup. What will
that tell us; we already know they all have the same weight.
Jacques finally convinced most of them that it was true we
knew the water would push the alcohol level higher than the
syrup would, but we didn't know about the rubber stopper; he
thought that since it sinks in one and floats in the other,
it would push the alcohol to a level somewhere between the
levels of the other two. He barely mentioned volume, and nobody
ever talked about a relationship between weight and volume.
But he certainly knew what he wanted to do and why.
It came out
very close--especially between the rubber stopper and the water--but
they read it as confirming Jacques's prediction. It was a good
I noted that
Evelyne was still very perplexed. I did not quite follow her,
but it concerned trying to keep straight the idea that, of
weight and volume, one was kept constant across these three
items (syrup, rubber stopper, water), and the other went in
an ordered series, from less to more.
time given to considering the significance of what Jacques
had done, someone (it might have been myself, but perhaps it
was Claire) wondered about another object, which would float
on water: Would it make the alcohol rise higher? We decided
to make such an object--weighing the same as the rubber stopper--and
proceeded to make it from cork and plasticene.
It took some
time, of course, assembling it and checking until it weighed
just what the stopper weighed, and then checking to see that
it floated on water. A fourth goblet with the same amount of
alcohol was readied. Jacques knew that this object, made to
float on water, could not be counted on to sink in alcohol,
and was prepared to hold it down with the point of a needle,
if necessary. The moment was even more intense than Jacques's
original experiment, because by now almost everybody at least
knew what they were looking for, if not why. It did sink. And
when the alcohol level this time was the highest of all, the
pleasure was great.
was done, Anna said that they hadn't needed to put the last
object in the alcohol, because they knew its volume was bigger.
How did they know, I asked. By looking, she replied. We took
them out then, the rubber stopper and the invented object,
to compare them, and few thought it obvious. I had actually
asked them, before they put it in the alcohol, whether it looked
larger, and nobody had paid any attention, including Anna.
I believe she had become so convinced of the idea that she
simply convinced herself you could see it.
perplexed: The prediction worked for that object, but why should
we assume it would work for another? We should make another
object the same weight as the stopper, that floats in water,
but that is smaller than the stopper. That idea bore considerable
interest for the group. But we had worked hard and it was time
for a break.
I noted that
nobody ever spelled out what these experiments meant. There
was hardly a mention of different volumes for the same weight,
and certainly no mention of trying different weights of the
same volume. There was just a greatly satisfying sense for
most of them that they could produce something systematic and
to that extent, comprehensible. A couple mentioned that they
still didn't know why the experiments turned out that way.
It took another whole week before most of them could relate
this elegant experiment to the other phenomena they had been
trying to understand.
At the end
of the class, Jacques and Colette lingered, along with Helen
and Anna. I asked Colette if she was ready to put things together
with her girls. She said she thought the answer was specific
weight. I asked her what that meant to her. She buried her
face in her hands to think; what she came up with was not clear
(and I could not remember it). I asked how she now understood
her pill bottles. She said something to the effect that they
were light, but did not articulate light in relation to what.
If you melted them down and made a cube, that cube would float.
He said it was like the balloons--full of water or not, they
float, so they floated. Colette said: So the plastic floats,
whether full of air or full of water. I brought out the balloons,
which were still in the cupboard--one full of water, and one
two-thirds full of water, but we did not get out a pan of water
to float them in. Anna remembered that the amount of water
in the balloons made a difference to how high they floated.
Jacques said they floated at about the level of the water inside
them. And when I asked Colette if she remembered how high her
pill bottles floated, full and empty, she said she would have
to do it again, but she thought that full of water they floated
just at their caps, while full of air they sat on top.
now once again to say what specific weight meant to her: you
have to state the weight for a cubic meter, or something. It
was still quite vague and, notably, seeking to repeat a school
formula. In an attempt to turn her attention to what we had
been doing, I muttered something about weight for anything,
but she did not pick it up. I said that float or sink was a
pretty crude measure--and also arbitrary. There was some acknowledgment
of this point, but it was not a striking idea for anybody.
I was annoyed with myself for not having drawn attention any
earlier to how high or low an object floats.
two other nice insights in the course of the evening's work.
They seem related, but they were separate insights coming at
separate times in the discussion. At one point, in considering
the ordering of liquids that float on each other, Colette said,
"And air is the lightest of all." And in the discussion of proper
places, Jacques said that a balloon filled with a gas lighter
than air will float up until it gets to a place where the air
is not heavier than it, and then stop, having found its proper
by the way, was the last time the wood was mentioned. Colette,
who had been absent last time, asked what had happened as it
dried. They told her that it now floated, and that the loss
of weight represented so much water. Jacques explained that
they had learned about long openings in it that were initially
full of air, which water replaced over time. Nobody went to
Week: Lingering Confusion, Consolidation, and Further Questions
In this session,
I was, to begin with, greatly disappointed. In spite of my
knowing that structuring complex ideas is never straightforward,
I had expected that last week's breakthrough would have left
clear marks of excitement and understanding. Excitement and
understanding did emerge as the evening went on, but things
were far from clear.
Evelyne remarked that the last had been a tough session--exhausting.
Jeanne was more confused at the end of it, she said, than before.
A number of people had tried to look things up in books in
the meantime, or talk to more knowledgeable friends. Colette,
for example, had looked up plastic in a chemistry book, and
found it contained oxygen--though she was not sure that meant
it had air in it. Helen had included discussion of this problem
in her weekly telephone call to Canada. She thought they were
so close now that people had to know. (And I had gone away
thinking that now for the most part they did know.)
his understanding. It had three parts: (1) one liquid floats
on another when it weighs less, given the same amount; (2)
for solid objects, the relationship between how big it is and
how heavy it is matters; (3) the amount of air matters.
discussion followed about the role of air before Jacques said
he thought he could account for everything with one hypothesis.
He said it clearly--if two things weighed the same and one
took up more space, it would float on the other--but he said
it hesitatingly, so some did not follow him. Most did follow
the second time he said it, but did not leap to agree.
the meantime had also looked up specific weight in a book,
and said that she saw a relationship between what she had read
there and what Jacques had just said. She was able to say what
she meant by specific weight this time; but of course in the
standard definition, on which she based what she said, it is
volume which is held constant, and weight which varies. Henri
pointed that out, that it was in that way different from Jacques's
hypothesis. Someone else said that it didn't matter, that it
amounted to the same thing. Henri repeated his point, but agreed
that it came to the same thing, and nobody disagreed.
during this discussion, said her colleague had claimed air
to be important only for its effect on the specific weight
of an object. She said this unclearly and nobody followed up
who had been absent the week before, I drew Jacques's experiment
on the blackboard. It went slowly, but we finally arrived at
Ingrid's question: Could you make an object of the same weight
as the rubber stopper, which floated on water, and which was
smaller than the rubber stopper? At this point everybody went
to work, some on that problem, and some on more general ways
they wished to check Jacques's theory.
and Lise, with Helen watching, carried out an investigation
based on one that Claire had read in a book. They had a plastic
ball which they could open to fill with plasticene. This they
did, then weighed the ball, put it in a container brimming
with water, gathered the water it caused to overflow, and compared
the weight of that water with the weight of the ball: the ball
weighed more. Then they removed some of the plasticene, until
the ball just barely floated, and found that it now weighed
the same as the overflow water.
phase of the experiment was a little confused, I think. When
they removed the rest of the plasticene, and the ball floated
high on the water, it displaced less than its whole volume.
The book talked of weighing the amount of water that the ball
now displaced, but this had, so far, no basis in anything that
they had done, nor in any of the questions that they had had.
That led to some confusion on their part about what they were
doing, why they were doing it, and what they might expect to
find. I believe they were weighing the amount of water now
displaced by the floating ball and expecting this water to
weigh more than the ball, confounding the book experiment and
Jacques's experiment. And I believe that they convinced themselves
that that was the case--not hard to do with the difficulties
of accuracy presented by a small high-floating object and a
to check his theory with a ball of aluminum foil. I was annoyed
at his choosing this oddball idea, and it was problematic.
But using a vise to squeeze out enough air so it sank, and
using a narrow test tube so a water rise would be most easily
visible, it did confirm his predictions.
more interested in talking out ideas than in experimenting.
He also seemed to feel a bit lost. He engaged Stuart and me
in a discussion of the nature of density. He said: We are 15
people in this room. That's the density of people in this room.
I said: That's the number. He said: But take another room,
with a different number of people, that would be a different
density--the number of people for the surface, that's what
density is, isn't it? So we can say the same for objects, can't
we? That's why there's more weight for the surface. That's
why they can have the same size and one be heavier--it's denser.
At one point, when he took a hollow plastic ball and spoke
of the amount of matter for the surface, I finally exclaimed, "For
the volume!" He said, "Yes, surface or volume." "It's not the
same thing," I said, and he replied, "Yes, volume." I was not
sure what that meant to him.
working hard here, not trying to keep uncertainty at bay by
using good words. But I was extremely tired during this class,
feeling neither patient nor inventive, and unable to find the
ways to help him think through the ideas he was finally grappling
also concerned with experimental issues. He said you can never
do a proper experiment, never check on what makes it float,
because all the factors are intertwined; you can't keep both
weight and density constant and change only volume, for example,
so you can't do a well-controlled experiment. He and Stuart
kept up this discussion as I left, Stuart developing the view
that you could nonetheless do a well-controlled experiment.
intriguing set of experiments was the work carried out by Evelyne,
Henri, and Jeanne on Ingrid's question: Could we make an object
which floats on water and which takes up less room than its
equivalent weight of water? First they used a snail shell,
stopped with plasticene, and floating. It turned out to be
bigger than its equivalent weight of water. Then they tried
to make many objects smaller than the snail shell, which would
also float. They tried cork with plasticene, Styrofoam with
coins, cork with coins. All of them, at the same weight as
the snail shell, and smaller, sank.
many statements that started, "What we need now is . . .," and
then responses of two sorts--either no, then it would be too
big, or no, then it would sink. Henri concluded that it was
impossible. Jeanne and Evelyne held the opposite. Jeanne exclaimed, "Everything's
possible in science." She considered that they just did not
have the right materials; they needed the right kind of plastic--like
Colette's pill bottles.
working time we assembled again, and this second discussion
was rather more encouraging. At the beginning of the discussion
several people said things were clearer now. Jeanne volunteered
that she now understood Jacques's experiment. Ingrid said very
slowly and hesitatingly approximately what Jacques had said
at the beginning, adding that it depends on the liquid. Again,
she had to repeat it for people to follow what she was saying,
but then many indicated agreement. She herself was very surprised
that it depended on the liquid. She had not seen the point
of doing different liquids until last week; she seemed quite
moved by the depth of her present understanding of this matter.
this point, said that it does not depend on the liquid, and
a very long discussion ensued in order to clear up that disagreement.
It turned out that Lise was referring to the fact--which came
to her group as a surprise--that the plastic ball, submerged,
displaced the same amount of liquid in syrup or detergent as
it did in water.
In the course
of that discussion, in listening to them describe their experiment,
Ingrid predicted that an object which just barely floated would
weigh the same as its volume of water. She was delighted that
their experiment confirmed that.
and Henri described their attempts to make an object that floated
on water, and which took up less room than its weight of water.
I asked the whole group who among them thought it was possible:
only Jeanne and Evelyne did. They were doubtful about it, given
everyone else's views, but they did not see why it couldn't
I asked Henri
what he thought now about his three hypotheses. He still thought
they were right, and he repeated then. This time, the weight-volume
relationship was stated clearly, and the air hypothesis became, "And
then if there is more or less air, that makes a difference
to the weight-volume." He thought, too, he could take any object
now and predict whether it would float or not: weigh it against
water, then put it and the water in equal amounts of another
liquid and see which takes more room . . . and so on. There
was general agreement.
Stuart both seemed clear that the air hypothesis was unnecessary.
Jacques said it quite clearly: It was taken care of in the
weight-volume relationship. I did not sense a universal rallying
round, but a general sense that he was probably right. We kept
coming back to it though.
said that until now she had never understood the balloons--they
were heavy, so why didn't they sink--came back to them in this
final discussion. She quoted a girl in Jacques's class, who
had said, "Water goes with water and the balloon floats." She
had not understood at the time what the child meant, but she
did now, and liked it a lot. Jeanne admitted she did not understand
that, so we brought out the balloons again. Watching them float
at different heights, according to the amounts of water and
air in them, everybody came to understand what the child meant.
They noticed again that the water levels were even, inside
and outside of the balloon. They noticed also--it was the first
time it was mentioned--that the air was always at the top;
Stuart said, "Air floats," which they liked. Ingrid said that
if the balloons were filled with syrup, instead of water, they
would sink. And full of oil? They would float, but higher.
if there were a void inside, Ingrid now asked. I took a capped
jar, full of air, which floated, and asked what it would do
if we removed the air. Most people thought it would sink. They
mainly seemed to agree it would weigh less, but thought it
would sink anyway. Claire gave the major clue to this reasoning:
she thought that perhaps the volume would now be different.
I did not
find out for sure what she meant by that, but my sense is that
it is some view of volume that takes into account how much
of the volume is filled with matter. If a jar has air in it,
then the whole system is full of matter, and each part of the
system takes up space. If it doesn't, then the inside, left
to its own devices, would not take up any space and the only
volume is that of the glass itself, and of its cover. It is
a notion of volume partially contaminated by a notion of density.
(This interpretation seems to be consistent with the way everyone
looked at the bottle I prepared in the previous class--the
bottle weighted with some water, which I had intended to be
seen as a single object, bottle-with-contents. Nobody thought
to look at that as a single object, having a weight and a volume--the
contents demanded too much attention in their own right.
to think that making a void would be changing conditions enough
so that a different theory might be needed; after all, theories
hold in certain conditions, and often you find that outside
those conditions they must be modified.
that it would still float; he fit the situation easily into
his general one. But apart from him, and perhaps Stuart, there
was no certainty, no way to think about this question.
point we had a small glass bottle which, filled with air, sank.
Someone suggested filling it with Styrofoam instead, but everyone
agreed--I noted that Jeanne was among the most adamant--that
it would not float then either; it would be still heavier.
to struggle with the air hypothesis: Couldn't you say that
air had something to do with it, more air floated more? Jacques
made a very clear statement to the effect that it simply wasn't
necessary, with which I finally allied myself. Pierre turned
out, however, not to have been swayed.
In one of
the last points of discussion, we came back to boats. Claire
asked how come plasticene sank, in a ball, but floated when
posed on the water shaped like a boat. Then she answered her
own question: It's now taking up more space in the water. Everyone
seemed to understand and agree, and she was very pleased with
herself. It seems to me now that this would have been a good
time to ask her again what she thought about the volume of
the vacuum jar, but I missed it at the time.
It is only
now--while working on revisions of this account--that I have
noticed that, with the exception of Robert's lecture and Pierre's
solo discussion, the word density never appeared. Specific
weight did. But whenever anyone referred to Jacques's experiment
and the relationship between weight and volume, the term they
used was weight-volume. I do not even know whether Pierre knew
that weight-volume was density, and wanted to understand that
better; or whether density represented something different
for him. I have the same question about what Lise wrote in
the postscript that follows.
So the group "invented
but gave it their own name! I expect that at least some of them
did not know that what they had invented was the idea of density;
and that the word density still represented for them something
else, that they may or may not have thought they knew something
about. I wish I had caught onto that in time to say simply,
"What we've been calling weight-volume is what is ordinarily
referred to as density."
A few weeks
later each of the 10 regular participants wrote a brief paper
about their learning about floating and sinking. I used these
to try to state where each of them now stood.
and Lise all seemed to have internalized Jacques's experiment
(to which, of course, Claire had made a major contribution)
and to feel they had gone one step further with their experiments
the last day.
Danielle wrote of her understanding why a small jar sank while
a large jar floated:
found that strange, and was sure that it was because there
was a lot of air in the big one, and less in the little one,
but I couldn't go any further. Then we left that, to work on
the "proper weights" (sic) of liquids: we came to realize that
the weights of the liquids, in relation to the volume they
occupy are different according to the liquids, which implies
that they can float on each other. So, in alcohol, the lightest
of all the liquids that we saw, objects float, but less well
than in water, which is heavier in relation to the volume it
occupies. I thought then maybe, in spite of the air in it,
the big jar would sink in alcohol, thus, that there was a parallel
to make between the weight in relation to the volume of the
object and the weight in relation to the volume of the liquid.
a wonderful account of the ups and downs of her coming, once
and for all, to the conclusion that the bearing surface is
not a factor in floating. Lise wrote of the same issue (referring
to the same experiments they had done together) in slightly
retrospect I realize . . . that in fact the shape doesn't change
anything about the property of the object. If the volume is
the same, the shape can be as different as can be and that
doesn't change anything; it's the law of conservation of quantities.
For a given volume, we can give it any shape, and that will
not change anything about its property, if it floats, it will
float, even with another shape, etc.
Here is what
Lise wrote about density:
syrup is heavier than water, we know that it has a greater
density than water . . . but in fact why is it so and what
exactly is density. It seems to me that we never talked together
about what density represents.
four stages. (1) "Fog: what are we looking for? What is expected
of us?" (2) "Clearing: enumeration of different factors . .
. and verification of their relevance." (She wrote,
"I was able . . . to assure myself that none of these factors
alone was the cause. I confess, however, that for a long time
I attributed a predominant role to air.") (3) "Confusion: the
appearance of terms and principles from physics, and the impossibility
of relating them to the factors we had been working on."
(4) "Comprehension: but I must admit that I am not sure that
I would have reached it without the theoretical explanations
I was given."
comprehension did not come. Here is part of what she wrote:
went well until the second last lesson. The more I manipulated,
the more I was satisfied, and the more I had the impression
that all these different approaches would soon lead me to understand
the problem of floating. Yes, I would have liked to consult
books and good old Archimedes was in my mind. But each time
I decided against it because I was afraid I would lose my ardour
for manipulating, and would fall back into book knowledge.
This concern was accentuated when our visitor chose to dazzle
us with p1v1=p2v2.
time in December I felt generally that I was becoming less
receptive, in the class and elsewhere. And when Jacques
had a wonderful idea, it certainly was that for him and
maybe for others (c.f. Henri), but for me it was the coup
de grace. In an instant, everything collapsed and everything
I had brought to bear in the preceding sessions was wiped
out. I had the impression that all was but wind.
disorder, Henri tried to help me out, but no matter how
much he explained to me, made me drawings, and repeated
it to me in different forms, nothing was internalized.
I seem to understand for the moment, but when I have to
re-explain it, I get muddled, and nothing is coherent any
more . . . .
not try to analyze whether or not the situation causes
me anguish. Personally, it doesn't discourage me at all.
It is perhaps the best proof that I still need time to
continue to manipulate, while I try to form hypotheses.
Because while in the first session I was able to play without
any clear goal, now I feel very deeply the need to know
what I want to know. So now I have to begin to form hypotheses,
so I can verify them. All this will make me relate my different
manipulations to each other, so as to find the link among
right about Henri. He wrote:
first time, it seemed to me, that I began to feel an answer
to the floating problem was when Jacques Bonnard did his experiment
with the stopper, syrup, water, and alcohol, and he observed,
at equal weights, the volume of the first three in alcohol.
This clicked for me. From then on I only tried to confirm or
invalidate what appeared to me.
He gave an
extremely clear account, with no mention of air, of how to
predict whether a given object will float in a given liquid.
little about what she now felt she understood of floating and
sinking (she wrote more of pedagogical insights). But she did
write the following:
discussions, above all the ones before Christmas [the two
last], were for me steps forward. My mind was sometimes
completely muddled and then the next week glimmers appeared.
For example, after Jacques's experiment I was disequalibriated
and the discussion the next week was helpful. My mind became
clearer and it seemed to me that I took a step forward in my
understanding. I have the impression of having understood .
. . why one object sinks and another floats.
had the following interesting comment:
have opened my eyes to a lot of notions that hadn't interested
me before. For example: why in a mountain chalet does the condensation
form on the outside window while in my Geneva apartment it
forms on the inside window? One question leads to another and
another. You start asking about everything.
think that alone I never would have taken the next step. I
do not remember how we came to the idea of weighing the object
and taking a quantity of liquid that has the same weight of
the object and seeing whether the object has a greater or smaller
volume than the quantity of liquid and thus predicting whether
the object floats or sinks.
She was the
only one who referred to the aluminum foil, and acknowledged
not understanding phenomena which in fact, though she did not
know it, depended on surface tension.
Here is some of what he wrote:
the beginning I had ideas stemming from previous personal experiences
and physics courses in high school. I have to say that these
ideas have not changed: rather they were refined, reinforced
by becoming more real as a result of the manipulations during
sessions went by, I was more and more convinced that the
phenomenon . . . results from a set of intertwined factors
and that it was just about impossible to isolate some factors
without bringing in others. This was also confirmed for
me by our physicist-visitor.
as far as the experiments carried out in this course are
concerned, it seems to me that the factors involved in
floating and sinking are the material the object is made
of, the air it contains, the form and the mass of the object.
When I speak of the form, I am thinking of a very thin
needle . . . .
back on everything I've done, I note that I observed many
things, but I don't have the impression of gaining an understanding
of fundamental aspects of the phenomenon of floating.
have saved for the end. He wrote at length about the phenomena
which particularly caught his interest (starting with the rubber
bands), and how they affected him, before moving on to consider
the effects of working as a group:
experiments of the others, their discoveries, their conclusions,
the discussions that we had for each session were for me the
stimulus to keep going with the search, the opener of new tracks
to follow, the confirmation or disconfirmation of my own findings.
of details of time and place, as he closes this section, makes
it sound like a moment of historic importance:
I think that if, the 13th of December, 19__, in the Dacha (our
small classroom building) between 7 and 8 p.m., everyone present
(at least I hope so) came to grasp the principle of floating,
we can attribute a part of the success to each of us.
considering many of the elements which went into the class
and his involvement in it, his final paragraph expresses again
his own excitement:
Tuesday evening, all propositions, good or bad, right or wrong,
could be made, and never was any value judgment brought to
bear on them. Everyone had the occasion to have wonderful ideas
and to be proud of them. To the point (please forgive me if
this appears somewhat presumptuous) that I currently have the
very validating feeling of having myself discovered the principle
tells how much there is to learn about knowledge from people
who are in the throes of learning, developing their ideas in
public and out loud.
- It is
significant to me that the work was characterized as much
by feeling as by ideas. Playfulness, mystification, laughter,
excitement, frustration, trust, confusion, fascination, determination,
aesthetic appreciation, enjoyment of each other--all of these
were involved in keeping the work going.
- I was
impressed that the knowledge was clearly a collective construction,
extending even as far as to include some of the group members'
students. It was not simply a matter of pooling results.
It was far more a matter of intellectual exchange in the
search for understanding--an exchange in which one person's
idea stirs another's. The seventh session is the most striking
example of this kind of reciprocal stirring of ideas, but
it characterized each session, and Jacques's closing comments
speak to it eloquently. Often the contributions are made
backwards or sidewards: one person's idea is not immediately
taken up as such, but suggests something else to another.
Piaget speaks of tatonnement, groping towards an understanding,
with stabs in a general direction, guided on the one hand
by some feeling about where things aren't quite working right,
and on the other hand where there is some familiarity with
something you know. Transferring back and forth among different
minds, the process is more visible here, and there are more
different sources of tentative ideas to try out, but I think
it may be very similar to what takes place within one mind.
- I am just
as interested in following an individual's particular ways
of getting a handle on the ideas. Not knowing week by week
what would turn out to be significant, I missed many individual
continuities, I am sure. Two of which I took some note both
concern Claire. There was her preoccupation with the shape
of things, and whether the bearing surface played a role.
Without comment in the first class she tied a knot in a floating
rubber band. Again without comment in the third class, she
tied a knot in a balloon. And in the last class, she was
the one who questioned, and then figured out for herself,
cases in which a change in the shape does make a difference--when
a ball of plasticene is reshaped into a boat, for example.
continuity was expressed again when she gave particular
attention to ways of weighing things, and circumstances
when weighing might be an important thing to do, starting,
I think, with wanting to compare the weight of the wood,
wet and dry.
- I am impressed
that right ideas are so meaningless and hard to recognize
before the frame has been built into which they can fit.
There was Colette's "maybe it's the same for objects,"
to which hardly anyone paid any attention; there was the recital
of Archimedes's principle which made no connection whatsoever
with anything they were thinking or doing; there was Claire's
idea of making equal weights of water, rubber stopper, and
syrup, which nobody found interesting; there was Robert's entire
lecture about density; there were the various things that they
read in books, or that friends and colleagues explained to
them. A special case is that of critical experiments that turn
out not to be critical. In the first class, we conducted a
critical experiment to see whether a larger container of water
could float an object that sank in very little water. The object
sank, but the question kept coming up for weeks. In the sixth
session, everybody agreed that a certain experiment would test
Pierre's hypothesis that the wood had sunk because there was
no air in the water in which we tried to float it. But when
the experiment was done, the results gave rise to endless interpretations.
the most dramatic instance of a critical experiment not
being critical was the supplement to Jacques's experiment,
in the seventh and eighth sessions. The prediction was
that an object of the same weight as the rubber stopper,
that floated on water, would take up more room than the
stopper, and more room than its equivalent weight of
water. For those who followed the ideas, and had some
way of understanding the point of his experiment, it
was an exciting confirmation. For those who did not,
it proved nothing: It worked for this object, why should
we believe that it will work for any other? The same
reasoning applies, I think, to every critical experiment
for which the ideas are not ready, including the two
other examples above: It worked this time, but why should
it work another time? And Jeanne exclaimed, "Everything's
possible in science!"
This has some essential similarities with a five-year-old's
search for reasons why some objects float: this one floats
because it is big and strong enough; this one floats because
it is small and light enough. Each instance is a case apart,
when the unifying ideas are beyond you. 
in these examples, is powerful evidence of the inadequacy
of empiricism as a theory of knowing. The critical experiments
themselves cannot impose their own meanings. One has
to have done a major part of the work already, one has
to have developed a network of ideas, in which to embed
the experiment. Jacques himself is the best example of
this--he did the major part of the work in his mind,
assembling the ideas which his experiment then could
confirm (and I wager that if the experiment had not confirmed
it, he would have redone it until it did)--and he was
the only person for whom, when he first did it, it meant
- What I
have described so far are some general phenomena, not specific
to this topic. I am also intrigued by specific ideas about
floating, sinking, and density that came up in this study.
I am intrigued, for example, that Jacques's invention took
the form of holding weights constant and comparing volumes.
Was that largely due to the circumstances of the activities
we pursued, or is the notion somehow more readily apprehensible
in that form than in the standard, "measure out equal amounts
and then weigh them"? For the liquids, measuring equal amounts
is pretty simple, and this procedure did occur both to this
group and to Colette's girls. For solid objects of random
forms, measuring out equal amounts is no easy job. The liquids
served as an intermediary, first for finding equal weights,
and then, after one more step, for finding equal volumes.
started when I chanced to put different liquids in identical
pill bottles to make two objects of the same volume and
different weights--starting from the classic way of thinking
about density. The liquids in the bottles weren't meant
to be the object of attention; they were there only to
influence the weights of the bottles. Evelyne then had
the idea of comparing liquids in some way by using them
in these identical bottles. Claire had the idea of putting,
in each of several different bottles, just enough liquid,
a different one in each bottle, to make it float. Then
another person questioned the weights of these different
amounts of different liquids, and it was agreed that
these different amounts weighed the same.
this way, with liquids, the idea first arose of making
equal weights with different amounts. Then, later, in
the presence of the rubber stopper, Claire had the idea
of making its equal weight in different liquids--a big
step, which perhaps she was able to take because, at
the beginning of the evening, she noticed Colette's wondering
whether it might work the same for objects as it did
for liquids. It was after all this had been done, with
the two liquids and one object of equal weight lined
up in front of him, that it occurred to Jacques to look
at how much room the object took up compared to the room
the liquids took up.
of course, also intrigued that Jacques's experiment was
so difficult to grasp, for those who had not come up
with it--that is, for everyone but Jacques. What is it
that is hard about putting those particular ideas together?
Evelyne gives us the clue that the difficulty is in developing
two systematic sets of relationships, and both separating
them and integrating them at the same time. But I would
like to know more about what is hard about that, and
what is invoked in coordinating them together.
Piaget, especially the sensorimotor books, has given
me a strong sense of how coordinating ideas can be understood
to be very like coordinating physical acts. Anyone who
has taken a gym class or a dance class knows that getting
each part of a large action going smoothly itself, and
integrated with each other smoothly-going part, is no
easy matter. How it all manages to come together is also
rather mysterious; there does not seem to be any way
to guarantee that the coordination will take place. Coordinating
ideas seems to me to be just like that.
both cases there is differentiation as well as integration.
In the case of physical coordination, one of the major
steps is differentiating movements from another. For
example, two limbs are likely to move together, globally,
when a new movement is attempted for the first time,
but they move in ways they have moved before, not in
the ways desired for this new movement. Elements of the
original global movement must be differentiated from
one another so that some of them can be integrated again
in a new way.
our case here, a global idea of amount must be differentiated
into weight and room-taken-up. Each must be developed
systematically, and each object has its rank order on
each of these scales, which have no necessary relationship
to each other. Then they must be integrated back together
into a totally different idea from the original undifferentiated
- I am also
intrigued with the question of the glass jar with nothing
in it, not even air, and I would like to be able to check
out my understanding of why they thought the volume might
now be different (see the section about the eighth week),
and to find out what is involved in coming to think about
that situation differently.
should be said about my own role. I consider that I did two
main things. One was to capture the class's interest in trying
to find out about something and to find ways to keep the
interest going over time. The second was to encourage them
to say what they thought, to express ideas in their own terms,
in ways that made sense for them, to listen to each other,
to take their ideas seriously and see where they led.
and maintaining the class's interest often meant taking
the lead with suggestions of activities to do. I did
not by any means leave to my students all decisions about
what they were to do, I did not, in the first class,
ask them what they wanted to learn about. I gave them
materials and instructions (loose though these were)
that I judged would keep them engaged (for an hour, I
thought--two months was far more than I anticipated).
I chose for discussion those of their activities that
I thought had a chance to engage them further. In the
second class, and subsequent ones, I proposed specific
activities for them of my devising, though often they
were based on questions that had arisen from them.
the point of the activities that I devised was not at
all that they discover anything in particular, but simply
that they should keep at it, doing something, until they
encountered some phenomenon that captured them again,
when they would again take off on their own. Some of
the activities I proposed did not work very well from
this point of view; others did.
them to say what they thought had two main purposes.
One was that trying to be clear about what you think
about something often helps you get clearer. The second
was to encourage them to take their own ideas seriously.
I have found that most adults are nervous about expressing
their own ideas, especially in a realm where they are
quite sure there is some appointed wisdom to which they
are not privy. They think, then, that their own ideas
are not worth attending to; they should drop them, and
catch on to the real, authenticated knowledge.
on the contrary, that one's own knowledge is all one
ever has, and that the only way to develop it further
is to pay attention to it, figure out what needs to be
further thought about, modify it, keep striving to make
it more adequate to one's experiences. But it takes work
on the part of a teacher, as I was in this case, to convey
that view of the importance of one's own knowledge. The
best way to do it is to get to the point where they are
willing to say what they think (mainly by making it playful
in the beginning), and then to make it absolutely clear
that what they think is O.K. with you; that you will
attend to it carefully and help them to do the same.
My ways of doing this are fairly muted in this account,
since what I found interesting enough to record in my
notes was what they said, rather than what I said. And
I do not always do it well. But it is always what I strive
- I think
both Pierre and Evelyne merit special mention, since in a
sense one could say that this experience failed with them.
To start with Pierre, he was the one member of the class
who did not, at least during the first six classes, make
himself vulnerable by acknowledging how much he did not know,
putting forth ideas he was not sure of, asking questions
which he might believe anybody else present would be able
to answer, expressing his surprise. I am glad that all the
other members of the class managed to resist his seeming
authority, to come to their own position on his air hypothesis
and the experiments he devised to support it, and to resist
his use of words in lieu of explanations. I take their reactions
to Robert's p1v1=p2v2 lecture as evidence that they had come
to realize that technical language did not necessarily represent
knowledge that made sense to them.
in the process, I lost Pierre himself. I think Robert's
visit probably did influence Pierre to feel that his
ideas and ways of going about things were good science.
Then Pierre missed the seventh class, when most of the
hard thinking was done. And in the final class, where
he was finally willing to struggle with ideas that he
acknowledged not to be clear, I did not do a good job:
on the one hand, as I have said, it did not occur to
me that he (or the others) might not have made the connection
between what they referred to as weight-volume, and what
the rest of the world called density; on the other hand,
I did not follow through with him, either in our one-to-one
conversation, or in the context of the class discussion,
on the ideas he was grappling with. He came a long way
to get to this place, I believe, and I wish I had managed
to help him realize that.
for Evelyne, I felt very badly to read in her retrospective
journal that "everything I had brought to bear in the
preceding sessions was wiped out," but much relieved
to read a few paragraphs later that "it doesn't discourage
me at all." I am fascinated, and feel encouraged in the
way I went about it, by the fact that Henri's explanations
helped her not at all. I have no way of knowing what
Henri put into his explanations, of course, but I imagine
that they were a fairly adequate account of what he by
own account is deeply interesting, I find. She had a
wonderful time finding out about liquids and floating
phenomena, convinced that at some point everything would
fall into place. She did not herself work at how they
might fall into place, developing her own hypotheses
as she went, but I share her conviction that when she
does, the ideas will come and the coordinations will
 I would have loved to be there when it happened
- I have
one final comment. In this commentary I have pointed to some
general themes which I think can be found in my story. I
believe that the themes hold for teaching situations in general.
But I find them far less engrossing than the specifics of
the story. I do not consider such themes to be the outcome,
or the results, or the findings. These are to be found in
the details. General themes are easy to come by and easy
to state. Specifics are far rarer. In giving this account,
as detailed as my 12-hour memory allowed, I hope to make
such situations, charged with learning potential, easier
to recognize and to create.
"Teachers as Learners," The Genetic Epistemologist, 13, 2, 1984.
 ______. "Teaching
as Research," Harvard Educational Review, 56, 4 1986.
 ______. "Understanding
Children's Understanding," in V. Windley, M. Dorn, and L. Weber,
eds., Building on the Strengths of Children, City College of
the City of New York, 1981.
"Critical Barriers to Science Learning," Outlook, 29, 1978.
Inholder and Jean Piaget, The Growth of Logical Thinking, New
York: Basic Books, Inc., 1958.
fact, Evelyne's first-person account gives support to the findings
of A. Karmiloff-Smith and B. Inholder in their paper, "If You
Want to Get Ahead, Get a Theory," Cognition, 1975, Vol. 3,
Duckworth, Professor of Education, Harvard University; published
by North Dakota Study Group on Evaluation, Grand Forks, N.D.,
1986. Reprinted with permission of the author.
grew up in Montreal and Halifax, Nova Scotia; studied with
Piaget and served as his interpreter; and was a member of the
staff of the Elementary Science Study for four years. She is
currently Associate Professor of Education at the Harvard Graduate
School of Education. Her book of essays, The Having of Wonderful
Ideas and Other Essays on Teaching and Learning, will be published
by Teachers College Press (New York) in 1987.
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