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13
Pulling Threads
M. Suzanne Donovan and John D. Bransford
What ties the chapters of this volume together are the three principles
from How People Learn (set forth in Chapter 1) that each chapter takes as its
point of departure. The collection of chapters in a sense serves as a demon-
stration of the second principle: that a solid foundation of detailed knowl-
edge and clarity about the core concepts around which that knowledge is
organized are both required to support effective learning. The three prin-
ciples themselves are the core organizing concepts, and the chapter discus-
sions that place them in information-rich contexts give those concepts greater
meaning. After visiting multiple topics in history, math, and science, we are
now poised to use those discussions to explore further the three principles
of learning.
ENGAGING RESILIENT PRECONCEPTIONS
All of the chapters in this volume address common preconceptions that
students bring to the topic of focus. Principle one from How People Learn
suggests that those preconceptions must be engaged in the learning process,
and the chapters suggest strategies for doing so. Those strategies can be
grouped into three approaches that are likely to be applicable across a broad
range of topics.
1. Draw on knowledge and experiences that students commonly bring to the class-
room but are generally not activated with regard to the topic of study.
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570 HOW STUDENTS LEARN IN THE CLASSROOM
This technique is employed by Lee, for example, in dealing with stu-
dents’ common conception that historical change happens as an event. He
points out that students bring to history class the everyday experience of
“nothing much happening” until an event changes things. Historians, on the
other hand, generally think of change in terms of the state of affairs. Change
in this sense may include, but is not equivalent to, the occurrence of events.
Yet students have many experiences in which things change gradually—
experiences in which “nothing happening” is, upon reflection, a
mischaracterization. Lee suggests, as an example, students might be asked
to “consider the change from a state of affairs in which a class does not trust
a teacher to one in which it does. There may be no event that could be
singled out as marking the change, just a long and gradual process.”
There are many such experiences on which a teacher could draw, such
as shifting alliances among friends or a gradual change in a sports team’s
status with an improvement in performance. Each of these experiences has
characteristics that support the desired conception of history. Events are
certainly not irrelevant. A teacher may do particular things that encourage
trust, such as going to bat for a student who is in a difficult situation or
postponing a quiz because students have two other tests on the same day.
Similarly, there may be an incident in a group that changes the dynamic,
such as a less popular member winning a valued prize or taking the blame
for an incident to prevent the whole group from being punished. But in
these contexts students can see, perhaps with some guided discussion, that
single events are rarely the sole explanation for the state of affairs.
It is often the case that students have experiences that can support the
conceptions we intend to teach, but instructional guidance is required to
bring these experiences to the fore. These might be thought of as “recessive”
experiences. In learning about rational number, for example, it is clear that
whole-number reasoning—the subject of study in earlier grades—is domi-
nant for most students (see Chapter 7). Yet students typically have experi-
ence with thinking about percents in the context of sale items in stores,
grades in school, or loading of programs on a computer. Moss’s approach to
teaching rational number as described in Chapter 7 uses that knowledge of
percents to which most students have easy access as an alternative path to
learning rational number. She brings students’ recessive understanding of
proportion in the context of reasoning about percents to the fore and strength-
ens their knowledge and skill by creating multiple contexts in which propor-
tional reasoning is employed (pipes and tubes, beakers, strings). As with
events in history, students do later work with fractions, and that work at
times presents them with problems that involve dividing a pizza or a pie into
discrete parts—a problem in which whole-number reasoning often domi-
nates. Because a facility with proportional reasoning is brought to bear,
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however, the division of a pie no longer leads students so easily into whole-
number traps.
Moss reinforces proportional reasoning by having students play games
in which fractions (such as 1/4) must be lined up in order of size with deci-
mals (such as .33) and percents (such as 40 percent). A theme that runs
throughout the chapters of this volume, in fact, is that students need many
opportunities to work with a new or recessive concept, especially when
doing so requires that powerful preconceptions be overturned or modified.
Bain, for example, writes about students’ tendency to see “history” and
“the past” as the same thing: “No one should think that merely pointing out
conceptual distinctions through a classroom activity equips students to make
consistent, regular, and independent use of these distinctions. Students’ hab-
its of seeing history and the past as the same do not disappear overnight.”
Bain’s equivalent of repeated comparisons of fractions, decimals, and per-
cents is the ever-present question regarding descriptions and materials: is
this “history-as-event”—the description of a past occurrence—or “history-as-
account”—an explanation of a past occurrence. Supporting conceptual change
in students requires repeated efforts to strengthen the new conception so
that it becomes dominant.
2. Provide opportunities for students to experience discrepant events that allow
them to come to terms with the shortcomings in their everyday models.
Relying on students’ existing knowledge and experiences can be diffi-
cult in some instances because everyday experiences provide little if any
opportunity to become familiar with the phenomenon of interest. This is
often true in science, for example, where the subject of study may require
specialized tools or controlled environmental conditions that students do
not commonly encounter.
In the study of gravity, for example, students do not come to the class-
room with experiences that easily support conceptual change because grav-
ity is a constant in their world. Moreover, experiences they have with other
forces often support misconceptions about gravity. For example, students
can experience variation in friction because most have opportunities to walk
or run an object over such surfaces as ice, polished wood, carpeting, and
gravel. Likewise, movement in water or heavy winds provide experiences
with resistance that many students can easily access. Minstrell found his
students believed that these forces with which they had experience explained
why they did not float off into space (see Chapter 11). Ideas about buoyancy
and air pressure, generally not covered in units on gravity, influenced these
students’ thinking about gravity. Television images of astronauts floating in
space reinforced for the students the idea that, without air to hold things
down, they would simply float off.
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Minstrell posed to his students a question that would draw out their
thinking. He showed them a large frame from which a spring scale hung and
placed an object on the scale that weighed 10 pounds. He then asked the
students to consider a situation in which a large glass dome would be placed
over the scale and all the air forced out with a vacuum pump. He asked the
students to predict (imprecisely) what would happen to the scale reading.
Half of Minstrell’s students predicted that the scale reading would drop to
zero without air; about a third thought there would be no effect at all on the
scale reading; and the remainder thought there would be a small change.
That students made a prediction and the predictions differed stimulated en-
gagement. When the experiment was carried out, the ideas of many students
were directly challenged by the results they observed.
In teaching evolution, Stewart and colleagues found that students’ ev-
eryday observations led them to underestimate the amount of variation in
common species. In such cases, student observations are not so much “wrong”
as they are insufficiently refined. Scientists are more aware of variation be-
cause they engage in careful measurement and attend to differences at a
level of detail not commonly noticed by the lay person. Stewart and col-
leagues had students count and sort sunflower seeds by their number of
stripes as an easy route to a discrepant event of sorts. The students discov-
ered there is far more variation among seeds than they had noticed. Unless
students understand this point, it will be difficult for them to grasp that
natural selection working on natural variation can support evolutionary
change.
While discrepant events are perhaps used most commonly in science,
Bain suggests they can be used productively in history as well (see Chapter
4). To dislodge the common belief that history is simply factual accounts of
events, Bain asked students to predict how people living in the colonies
(and later in the United States) would have marked the anniversary of
Columbus’s voyage 100 years after his landing in 1492 and then each hun-
dred years after that through 1992. Students wrote their predictions in jour-
nals and were then given historical information about the changing Columbian
story over the 500-year period. That information suggests that the first two
anniversaries were not really marked at all, that the view of Columbus’s
“discovery of the new world” as important had emerged by 1792 among
former colonists and new citizens of the United States, and that by 1992 the
Smithsonian museum was making no mention of “discovery” but referred to
its exhibit as the “Columbian Exchange.” If students regard history as the
reporting of facts, the question posed by Bain will lead them to think about
how people might have celebrated Columbus’s important discovery, and not
whether people would have considered the voyage a cause for celebration
at all. The discrepancy between students’ expectation regarding the answer
to the question and the historical accounts they are given in the classroom
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lecture cannot help but jar the conception that history books simply report
events as they occurred in the past.
3. Provide students with narrative accounts of the discovery of (targeted) knowl-
edge or the development of (targeted) tools.
What we teach in schools draws on our cultural heritage—a heritage of
scientific discovery, mathematical invention, and historical reconstruction.
Narrative accounts of how this work was done provide a window into change
that can serve as a ready source of support for students who are being asked
to undergo that very change themselves. How is it that the earth was discov-
ered to be round when nothing we casually observe tells us that it is? What
is place value anyway? Is it, like the round earth, a natural phenomenon that
was discovered? Is it truth, like e = mc2, to be unlocked? There was a time, of
course, when everyday notions prevailed, or everyday problems required a
solution. If students can witness major changes through narrative, they will
be provided an opportunity to undergo conceptual change as well.
Stewart and colleagues describe the use of such an approach in teach-
ing about evolution (see Chapter 12). Darwin’s theory of natural selection
operating on random variation can be difficult for students to grasp. The
beliefs that all change represents an advance toward greater complexity and
sophistication and that changes happen in response to use (the giraffe’s
neck stretching because it reaches for high leaves, for example) are wide-
spread and resilient. And the scientific theory of evolution is challenged
today, as it was in Darwin’s time, by those who believe in intelligent de-
sign—that all organisms were made perfectly for their function by an intelli-
gent creator. To allow students to differentiate among these views and un-
derstand why Darwin’s theory is the one that is accepted scientifically, students
work with three opposing theories as they were developed, supported, and
argued in Darwin’s day: William Paley’s model of intelligent design, Jean
Baptiste de Lamarck’s model of acquired characteristics based on use, and
Darwin’s theory of natural selection. Students’ own preconceptions are gen-
erally represented somewhere in the three theories. By considering in some
depth the arguments made for each theory, the evidence that each theorist
relied upon to support his argument, and finally the course of events that led
to the scientific community’s eventually embracing Darwin’s theory, stu-
dents have an opportunity to see their own ideas argued, challenged, and
subjected to tests of evidence.
Every scientific theory has a history that can be used to the same end.
And every scientific theory was formulated by particular people in particular
circumstances. These people had hopes, fears, and passions that drove their
work. Sometimes students can understand theories more readily if they learn
about them in the context of those hopes, fears, and passions. A narrative
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that places theory in its human context need not sacrifice any of the techni-
cal material to be learned, but can make that material more engaging and
meaningful for students.
The principle, of course, does not apply only to science and is not
restricted to discovery. In mathematics, for example, while some patterns
and relationships were discovered, conventions that form our system of
counting were invented. As the mathematics chapters suggest, the use of
mathematics with understanding—the engagement with problem solving and
strategy use displayed by the best mathematics students—is undermined
when students think of math as a rigid application of given algorithms to
problems and look for surface hints as to which algorithm applies. If stu-
dents can see the nature of the problems that mathematical conventions
were designed to solve, their conceptions of what mathematics is can be
influenced productively.
Historical accounts of the development of mathematical conventions
may not always be available. For purposes of supporting conceptual change,
however, fictional story telling may do just as well as history. In Teaching as
Story Telling, Egan1 relates a tale that can support students’ understanding of
place value:
A king wanted to count his army. He had five clueless counse-
lors and one ingenious counselor. Each of the clueless five tried to
work out a way of counting the soldiers, but came up with meth-
ods that were hopeless. One, for example, tried using tally sticks to
make a count, but the soldiers kept moving around, and the count
was confused. The ingenious counselor told the king to have the
clueless counselors pick up ten pebbles each. He then had them
stand behind a table that was set up where the army was to march
past. In front of each clueless counselor a bowl was placed. The
army then began to march past the end of the table.
As each soldier went by, the first counselor put one pebble into
his bowl. Once he had put all ten pebbles into the bowl, he scooped
them up and then continued to put one pebble down for each sol-
dier marching by the table. He had a very busy afternoon, putting
down his pebbles one by one and then scooping them up when all
were in the bowl. Each time he scooped up the ten pebbles, the
clueless counselor to his left put one pebble into her bowl [gender
equity]. When her ten pebbles were in her bowl, she too scooped
them out again, and continued to put one back into the bowl each
time the clueless counselor to her right picked his up.
The clueless counselor to her left had to watch her through the
afternoon, and he put one pebble into his bowl each time she picked
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hers up. And so on for the remaining counselors. At the end of the
afternoon, the counselor on the far left had only one pebble in his
bowl, the next counselor had two, the next had seven, the next had
six and the counselor at the other end of the table, where the sol-
diers had marched by, had three pebbles in his bowl. So we know
that the army had 12,763 soldiers. The king was delighted that his
ingenious counselor had counted the whole army with just fifty
pebbles.2
When this story is used in elementary school classrooms, Egan encourages
the teacher to follow up by having the students count the class or some
other, more numerous objects using this method.
The story illustrates nicely for students how the place-value system al-
lows the complex problem of counting large numbers to be made simpler.
Place value is portrayed not as a truth but as an invention. Students can then
change the base from 10 to other numbers to appreciate that base 10 is not
a “truth” but a “choice.” This activity supports students in understanding that
what they are learning is designed to make number problems raised in the
course of human activity manageable.
That imaginative stories can, if effectively designed, support conceptual
change as well as historical accounts is worth noting for another reason: the
fact that an historical account is an account might be viewed as cause for
excluding it from a curriculum in which the nature of the account is not the
subject of study. Historical accounts of Galileo, Newton, or Darwin written
for elementary and secondary students can be contested. One would hope
that students who study history will come to understand these as accounts,
and that they will be presented to students as such. But the purpose of the
accounts, in this case, is to allow students to experience a time when ideas
that they themselves may hold were challenged and changed, and that pur-
pose can be served even if the accounts are somewhat simplified and their
contested aspects not treated fully.
ORGANIZING KNOWLEDGE AROUND
CORE CONCEPTS
In the Fish Is Fish story discussed in Chapter 1, we understand quite
easily that when the description of a human generates an image of an up-
right fish wearing clothing, there are some key missing concepts: adapta-
tion, warm-blooded versus cold-blooded species, and the difference in mo-
bility challenges in and out of water. How do we know which concepts are
“core?” Is it always obvious?
The work of the chapter authors, as well as the committee/author dis-
cussions that supported the volume’s development, provides numerous in-
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sights about the identification of core concepts. The first is observed most
explicitly in the work of Peter Lee (see Chapter 2): that two distinct types of
core concepts must be brought to the fore simultaneously. These are con-
cepts about the nature of the discipline (what it means to engage in doing
history, math, or science) and concepts that are central to the understanding
of the subject matter (exploration of the new world, mathematical functions,
or gravity). Lee refers to these as first-order (the discipline) and second-
order (the subject) concepts. And he demonstrates very persuasively in his
work that students bring preconceptions about the discipline that are just as
powerful and difficult to change as those they bring about the specific sub-
ject matter.
For teachers, knowing the core concepts of the discipline itself—the
standards of evidence, what constitutes proof and disproof, and modes of
reasoning and engaging in inquiry—is clearly required. This requirement is
undoubtedly at the root of arguments in support of teachers’ course work in
the discipline in which they will teach. But that course work will be a blunt
instrument if it focuses only on second-order knowledge (of subject) but not
on first-order knowledge (of the discipline). Clarity about the core concepts
of the discipline is required if students are to grasp what the discipline—
history, math, or science—is about.
For identifying both first- and second-order concepts, the obvious place
to turn initially is to those with deep expertise in the discipline. The con-
cepts that organize experts’ knowledge, structure what they see, and guide
their problem solving are clearly core. But in many cases, exploring expert
knowledge directly will not be sufficient. Often experts have such facility
with a concept that it does not even enter their consciousness. These “expert
blind spots” require that “knowledge packages”3 —sets of related concepts
and skills that support expert knowledge—become a matter for study.
A striking example can be found in Chapter 7 on elementary mathemat-
ics. For those with expertise in mathematics, there may appear to be no
“core concept” in whole-number counting because it is done so automati-
cally. How one first masters that ability may not be accessible to those who
did so long ago. Building on the work of numerous researchers on how
children come to acquire whole-number knowledge, Griffin and Case’s4
research conducted over many years suggests a core conceptual structure
that supports the development of the critical concept of quantity. Similar
work has been done by Moss and Case5 (on the core conceptual structure
for rational number) and by Kalchman, Moss, and Case6 (on the core con-
ceptual structure for functions). The work of Case and his colleagues sug-
gests the important role cognitive and developmental psychologists can play
in extending understanding of the network of concepts that are “core” and
might be framed in less detail by mathematicians (and other disciplinary
experts).
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The work of Stewart and his colleagues described in Chapter 12 is an-
other case in which observations of student efforts to learn help reshape
understanding of the package of related core concepts. The critical role of
natural selection in understanding evolution would certainly be identified as
a core concept by any expert in biology. But in the course of teaching about
natural selection, these researchers’ realization that students underestimated
the variation in populations led them to recognize the importance of this
concept that they had not previously identified as core. Again, experts in
evolutionary biology may not identify population variation as an important
concept because they understand and use the concept routinely—perhaps
without conscious attention to it. Knowledge gleaned from classroom teach-
ing, then, can be critical in defining the connected concepts that help sup-
port core understandings.
But just as concepts defined by disciplinary experts can be incomplete
without the study of student thinking and learning, so, too, the concepts as
defined by teachers can fall short if the mastery of disciplinary concepts is
shallow. Liping Ma’s study of teachers’ understanding of the mathematics of
subtraction with regrouping provides a compelling example. Some teachers
had little conceptual understanding, emphasizing procedure only. But as
Box 13-1 suggests, others attempted to provide conceptual understanding
without adequate mastery of the core concepts themselves. Ma’s work pro-
vides many examples (in the teaching of multidigit multiplication, division
of fractions, and calculation of perimeter and area) in which efforts to teach
for understanding without a solid grasp of disciplinary concepts falls short.
SUPPORTING METACOGNITION
A prominent feature of all of the chapters in this volume is the extent to
which the teaching described emphasizes the development of metacognitive
skills in students. Strengthening metacognitive skills, as discussed in Chapter
1, improves the performance of all students, but has a particularly large
impact on students who are lower-achieving.7
Perhaps the most striking consistency in pedagogical approach across
the chapters is the ample use of classroom discussion. At times students
discuss in small groups and at times as a whole class; at times the teacher
leads the discussion; and at times the students take responsibility for ques-
tioning. A primary goal of classroom discussion is that by observing and
engaging in questioning, students become better at monitoring and ques-
tioning their own thinking.
In Chapter 5 by Fuson, Kalchman, and Bransford, for example, students
solve problems on the board and then discuss alternative approaches to
solving the same problem. The classroom dialogue, reproduced in Box 13-2,
supports the kind of careful thinking about why a particular problem-solv-
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Conceptual Explanation Without Conceptual Understanding
BOX 13-1
Liping Ma explored approaches to teaching subtraction with regrouping (problems
like 52 – 25, in which subtraction of the 5 ones from the 2 ones requires that the
number be regrouped). She found that some teachers took a very procedural ap-
proach that emphasized the order of the steps, while others emphasized the con-
cept of composing a number (in this case into 5 tens and 2 ones) and decomposing
a number (into 4 tens and 12 ones). Between these two approaches, however,
were those of teachers whose intentions were to go beyond procedural teaching,
but who did not themselves fully grasp the concepts at issue. Ma8 describes one
such teacher as follows:
Tr. Barry, another experienced teacher in the procedurally directed
group, mentioned using manipulatives to get across the idea that
“you need to borrow something.” He said he would bring in quarters
and let students change a quarter into two dimes and one nickel: “a
good idea might be coins, using money because kids like money. . . .
The idea of taking a quarter even, and changing it to two dimes and
a nickel so you can borrow a dime, getting across that idea that you
need to borrow something.”
There are two difficulties with this idea. First of all, the mathemati-
cal problem in Tr. Barry’s representation was 25 – 10, which is not a
subtraction with regrouping. Second, Tr. Barry confused borrowing
in everyday life—borrowing a dime from a person who has a
quarter—with the “borrowing” process in subtraction with regroup-
ing—to regroup the minuend by rearranging within place values. In
fact, Tr. Barry’s manipulative would not convey any conceptual
understanding of the mathematical topic he was supposed to teach.
Another teacher who grasps the core concept comments on the idea of “bor-
rowing” as follows:9
Some of my students may have learned from their parents that you
“borrow one unit form the tens and regard it as 10 ones”. . . . I will
explain to them that we are not borrowing a 10, but decomposing a
10. “Borrowing” can’t explain why you can take a 10 to the ones
place. But “decomposing” can. When you say decomposing, it
implies that the digits in higher places are actually composed of
those at lower places. They are exchangeable . . . borrowing one unit
and turning it into 10 sounds arbitrary. My students may ask me how
can we borrow from the tens? If we borrow something, we should
return it later on.
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ing strategy does or does not work, as well as the relative benefits of differ-
ent strategies, that can support skilled mathematics performance.
Similarly, in the science chapters students typically work in groups, and
the groups question each other and explain their reasoning. Box 13-3 repro-
duces a dialogue at the high school level that is a more sophisticated version
of that among young mathematics students just described. One group of
students explains to another not only what they concluded about the evolu-
tionary purpose of different coloration, but also the thinking that led them to
that conclusion and the background knowledge from an earlier example
that supported their thinking. The practice of bringing other knowledge to
bear in the reasoning process is at the heart of effective problem solving, but
can be difficult to teach directly. It involves a search through one’s mental
files for what is relevant. If teachers simply give students the knowledge to
incorporate, the practice and skill development of doing one’s own mental
search is shortchanged. Group work and discussions encourage students to
engage actively in the mental search; they also provide examples from other
students’ thinking of different searches and search results. The monitoring of
consistency between explanation and theory that we see in this group dis-
cussion (e.g., even if the male dies, the genes have already been passed
along) is preparation for the kind of self-monitoring that biologists do rou-
tinely.
Having emphasized the benefits of classroom discussion, however, we
offer two cautionary notes. First, the discussion cited in the chapters is guided
by teachers to achieve the desired learning. Using classroom discussion well
places a substantial burden on the teacher to support skilled discussion,
respond flexibly to the direction the discussion is taking, and steer it produc-
tively. Guiding discussion can be a challenging instructional task. Not all
questions are good ones, and the art of questioning requires learning on the
part of both students and teachers.10 Even at the high school level, Bain (see
Chapter 4) notes the challenge a teacher faces in supporting good student
questioning:
Sarena Does anyone notice the years that these were
written? About how old are these accounts?
Andrew?
Andrew They were written in 1889 and 1836. So some
of them are about 112 years old and others are
about 165 years old.
Teacher Why did you ask, Sarena?
Sarena I’m supposed to ask questions about when the
source was written and who wrote it. So, I’m
just doing my job.
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Supporting Skilled Questioning and Explaining in
BOX 13-2
Mathematics Problem Solving
In the dialogue below, young children are learning to explain their thinking
and to ask questions of each other—skills that help students guide their
own learning when those skills are eventually internalized as self-ques-
tioning and self-explaining.
Teacher Maria, can you please explain to your friends in
the class how you solved the problem?
Maria Six is bigger than 4, so I can’t subtract here
[pointing] in the ones. So I have to get more
ones. But I have to be fair when I get more
ones, so I add ten to both my numbers. I add a
ten here in the top [pointing] to change the 4 to
a 14, and I add a ten here in the bottom in the
tens place, so I write another ten by my 5. So
now I count up from 6 to 14, and I get 8 ones
(demonstrating by counting “6, 7, 8, 9, 10, 11,
12, 13, 14” while raising a finger for each word
from 7 to 14). And I know my doubles, so 6 plus
6 is 12, so I have 6 tens left. [She thought, “1 +
5 = 6 and 6 + ? = 12 tens. Oh, I know 6 + 6 = 12,
so my answer is 6 tens.”]
Jorge I don’t see the other 6 in your tens. I only see
one 6 in your answer.
Maria The other 6 is from adding my 1 ten to the 5
tens to get 6 tens. I didn’t write it down.
Andy But you’re changing the problem. How do you
get the right answer?
Maria If I make both numbers bigger by the same
amount, the difference will stay the same.
Remember we looked at that on drawings last
week and on the meter stick.
Michelle Why did you count up?
Palincsar11 has documented the progress of students as they move be-
yond early, unskilled efforts at questioning. Initially, students often parrot
the questions of a teacher regardless of their appropriateness or develop
questions from a written text that repeat a line of the text verbatim, leaving
a blank to be filled in. With experience, however, students become produc-
tive questioners, learning to attend to content and ask genuine questions.
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Maria Counting down is too hard, and my mother
taught me to count up to subtract in first
grade.
Teacher How many of you remember how confused we
were when we first saw Maria’s method last
week? Some of us could not figure out what
she was doing even though Elena and Juan
and Elba did it the same way. What did we do?
Rafael We made drawings with our ten-sticks and
dots to see what those numbers meant. And
we figured out they were both tens. Even
though the 5 looked like a 15, it was really just
6. And we went home to see if any of
our parents could explain it to us, but we had
to figure it out ourselves and it took us 2 days.
Teacher Yes, I was asking other teachers, too. We
worked on other methods too, but we kept
trying to understand what this method was
and why it worked.
And Elena and Juan decided it was clearer if
they crossed out the 5 and wrote a 6, but Elba
and Maria liked to do it the way they learned at
home. Any other questions or comments for
Maria? No? Ok, Peter, can you explain your
method?
Peter Yes, I like to ungroup my top number when I
don’t have enough to subtract everywhere. So
here I ungrouped 1 ten and gave it to the 4
ones to make 14 ones, so I had 1 ten left here.
So 6 up to 10 is 4 and 4 more up to 14 is 8, so
14 minus 6 is 8 ones. And 5 tens up to 11 tens
is 6 tens. So my answer is 68.
Carmen How did you know it was 11 tens?
Peter Because it is 1 hundred and 1 ten and that is
11 tens.
Similarly, students’ answers often cannot serve the purpose of clarifying
their thinking for classmates, teachers, or themselves without substantial
support from teachers. The dialogue in Box 13-4 provides an example of a
student becoming clearer about the meaning of what he observed as the
teacher helped structure the articulation.
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582 HOW STUDENTS LEARN IN THE CLASSROOM
Questioning and Explaining in High School Science
BOX 13-3
The teacher passes out eight pages of case materials and asks the stu-
dents to get to work. Each group receives a file folder containing the task
description and information about the natural history of the ring-necked
pheasant. There are color pictures that show adult males, adult females,
and young. Some of the pages contain information about predators, mat-
ing behavior, and mating success. The three students spend the remain-
der of the period looking over and discussing various aspects of the case.
By the middle of the period on Tuesday, this group is just finalizing their
explanation when Casey, a member of another group, asks if she can talk
to them.
Casey What have you guys come up with? Our group
was wondering if we could talk over our ideas
with you.
Grace Sure, come over and we can each read our
explanations.
These two groups have very different explanations. Hillary’s group is
thinking that the males’ bright coloration distracts predators from the nest,
while Casey’s group has decided that the bright coloration confers an
advantage on the males by helping them attract more mates. A lively
discussion ensues.
Ed But wait, I don’t understand. How can dying be
a good thing?
Jerome Well, you have to think beyond just survival of
the male himself. We think that the key is the
survival of the kids. If the male can protect his
Group work and group or classroom discussions have another potential
pitfall that requires teacher attention: some students may dominate the dis-
cussion and the group decisions, while others may participate little if at all.
Having a classmate take charge is no more effective at promoting
metacognitive development—or supporting conceptual change—than hav-
ing a teacher take charge. In either case, active engagement becomes unnec-
essary. One approach to tackling this problem is to have students rate their
group effort in terms not only of their product, but also of their group dy-
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young and give them a better chance of
surviving then he has an advantage.
Claire Even if he dies doing it?
Grace Yeah, because he will have already passed on
his genes and stuff to his kids before he dies.
Casey How did you come up with this? Did you see
something in the packets that we didn’t see?
Grace One reason we thought of it had to do with the
last case with the monarchs and viceroy.
Hillary Yeah, we were thinking that the advantage isn’t
always obvious and sometimes what is good
for the whole group might not seem like it is
good for one bird or butterfly or whatever.
Jerome We also looked at the data in our packets on
the number of offspring fathered by brighter
versus duller males. We saw that the brighter
males had a longer bar.
Grace See, look on page 5, right here.
Jerome So they had more kids, right?
Casey We saw that table too, but we thought that it
could back up our idea that the brighter males
were able to attract more females as mates.
The groups agree to disagree on their interpretation of this piece of
data and continue to compare their explanations on other points. While it
may take the involvement of a teacher to consider further merits of each
explanation given the data, the students’ group work and dialogue pro-
vide the opportunity for constructing, articulating, and questioning a sci-
entific hypothesis.
namics.12 Another approach, suggested by Bain (Chapter 4), is to have stu-
dents pause during class discussion to think and write individually. As stu-
dents discussed the kind of person Columbus was, Bain asked them to write
a 2-minute essay before discussing further. Such an exercise ensures that
students who do not engage in the public discussion nonetheless formulate
their ideas.
Group work is certainly not the only approach to supporting the devel-
opment of metacognitive skills. And given the potential hazard of group
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584 HOW STUDENTS LEARN IN THE CLASSROOM
Guiding Student Observation and Articulation
BOX 13-4
In an elementary classroom in which students were studying the behav-
ior of light, one group of students observed that light could be both re-
flected and transmitted by a single object. But students needed consider-
able support from teachers to be able to articulate this observation in a
way that was meaningful to them and to others in the class:
Ms. Lacey I’m wondering. I know you have a lot of see-
through things, a lot of reflect things. I’m
wondering how you knew it was see-through.
Kevin It would shine just, straight through it.
Ms. Lacey What did you see happening?
Kevin We saw light going through the . . .
Derek Like if we put light . . .
Kevin Wherever we tried the flashlight, like right
here, it would show on the board.
Derek And then I looked at the screen [in front of and
to the side of the object], and then it showed a
light on the screen. Then he said, come here,
and look at the back. And I saw the back, and it
had another [spot].
Ms. Lacey Did you see anything else happening at the
material?
Kevin We saw sort of a little reflection, but we, it had
mostly just see-through.
Derek We put, on our paper we put reflect, but we
had to decide which one to put it in. Because it
had more of this than more of that.
Ms. Lacey Oh. So you’re saying that some materials . . .
Derek Had more than others . . .
dynamics, using some individual approaches to supporting self-monitoring
and evaluation may be important. For example, in two experiments with
students using a cognitive tutor, Aleven and Koedinger13 asked one group to
explain the problem-solving steps to themselves as they worked. They found
that students who were asked to self-explain outperformed those who spent
the same amount of time on task but did not engage in self-explanation on
transfer problems. This was true even though the common time limitation
meant that the self-explainers solved fewer problems.
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Ms. Lacey . . . are doing, could be in two different
categories.
Derek Yeah, because some through were really
reflection and see-through together, but we
had to decide which.
[Intervening discussion takes place about
other data presented by this group that had to
do with seeing light reflected or transmitted as
a particular color, and how that color com-
pared with the color of the object.]
[at the end of this group’s reporting, and after
the students had been encouraged to identify
several claims that their data supported
among those that had been presented previ-
ously by other groups of students]
Ms. Lacey There was something else I was kinda con-
vinced of. And that was that light can do two
different things. Didn’t you tell me it went both
see-through and reflected?
Kevin & Derek Yeah. Mm-hmm.
Ms. Lacey So do you think you might have another claim
there?
Derek Yeah.
Kevin Light can do two things with one object.
Ms. Lacey More than one thing?
Kevin Yeah.
Ms. Lacey Okay. What did you say?
Kevin & Derek Light can do two things with one object.
See Chapter 10 for the context of this dialogue.
Another individual approach to supporting metacognition is suggested
by Stewart (Chapter 12). Students record their thinking early in the treatment
of a new topic and refer back to it at the unit’s end to see how it has
changed. This brings conscious attention to the change in a student’s own
thinking. Similarly, the reflective assessment aspect of the ThinkerTools cur-
riculum described in Chapter 1 shifts students from group inquiry work to
evaluating their group’s inquiry individually. The results in the ThinkerTools
case suggest that the combination of group work and individual reflective
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586 HOW STUDENTS LEARN IN THE CLASSROOM
assessment is more powerful that the group work alone (see Box 9-5 in
Chapter 9).
PRINCIPLES OF LEARNING AND CLASSROOM
ENVIRONMENTS
The principles that shaped these chapters are based on efforts by re-
searchers to uncover the rules of the learning game. Those rules as we
understand them today do not tell us how to play the best instructional
game. They can, however, point to the strengths and weakness of instruc-
tional strategies and the classroom environments that support those strate-
gies. In Chapter 1, we describe effective classroom environments as learner-
centered, knowledge-centered, assessment-centered, and community-
centered. Each of these characteristics suggests a somewhat different focus.
But at the same time they are interrelated, and the balance among them will
help determine the effectiveness of instruction.
A community-centered classroom that relies extensively on classroom
discussion, for example, can facilitate learning for several reasons (in addi-
tion to supporting metacognition as discussed above):
• It allows students’ thinking to be made transparent—an outcome that
is critical to a learner-centered classroom. Teachers can become familiar
with student ideas—for example, the idea in Chapter 7 that two-thirds of a
pie is about the same as three-fourths of a pie because both are missing one
piece. Teachers can also monitor the change in those ideas with learning
opportunities, the pace at which students are prepared to move, and the
ideas that require further work—key features of an assessment-centered class-
room.
• It requires that students explain their thinking to others. In the course
of explanation, students develop a disposition toward productive interchange
with others (community-centered) and develop their thinking more fully
(learner-centered). In many of the examples of student discussion through-
out this volume—for example, the discussion in Chapter 2 of students exam-
ining the role of Hitler in World War II—one sees individual students becom-
ing clearer about their own thinking as the discussion develops.
• Conceptual change can be supported when students’ thinking is chal-
lenged, as when one group points out a phenomenon that another group’s
model cannot explain (knowledge-centered). This happens, for example, in
a dialogue in Chapter 12 when Delia explains to Scott that a flap might
prevent more detergent from pouring out, but cannot explain why the amount
of detergent would always be the same.
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At the same time, emphasizing the benefits of classroom discussion in
supporting effective learning does not imply that lectures cannot be excel-
lent pedagogical devices. Who among us have not been witness to a lecture
from which we have come away having learned something new and impor-
tant? The Feynman lectures on introductory physics mentioned in Chapter 1,
for example, are well designed to support learning. That design incorpo-
rates a strategy for accomplishing the learning goals described throughout
this volume.14 Feynman anticipates and addresses the points at which stu-
dents’ preconceptions may be a problem. Knowing that students will likely
have had no experiences that support grasping the size of an atom, he
spends time on this issue, using familiar references for relative size that
allow students to envision just how tiny an atom is.
But to achieve effective learning by means of lectures alone places a
major burden on the teacher to anticipate student thinking and address prob-
lems effectively. To be applied well, this approach is likely to require both a
great deal of insight and much experience on the part of the teacher. With-
out such insight and experience, it will be difficult for teachers to anticipate
the full range of conceptions students bring and the points at which they
may stumble.15 While one can see that Feynman made deliberate efforts to
anticipate student misconceptions, he himself commented that the major
difficulty in the lecture series was the lack of opportunity for student ques-
tions and discussion, so that he had no way of really knowing how effective
the lectures were. In a learner-centered classroom, discussion is a powerful
tool for eliciting and monitoring student thinking and learning.
In a knowledge-centered classroom, however, lectures can be an impor-
tant accompaniment to classroom discussion—an efficient means of consoli-
dating learning or presenting a set of concepts coherently. In Chapter 4, for
example, Bain describes how, once students have spent some time working
on competing accounts of the significance of Columbus’s voyage and struggled
with the question of how the anniversaries of the voyage were celebrated,
he delivers a lecture that presents students with a description of current
thinking on the topic among historians. At the point at which this lecture is
delivered, student conceptions have already been elicited and explored.
Because lectures can play an important role in instruction, we stress once
again that the emphasis in this volume on the use of discussion to elicit
students’ thinking, monitor understanding, and support metacognitive de-
velopment—all critical elements of effective teaching—should not be mis-
taken for a pedagogical recommendation of a single approach to instruction.
Indeed, inquiry-based learning may fall short of its target of providing stu-
dents with deep conceptual understanding if the teacher places the full bur-
den of learning on the activities. As Box 1-3 in Chapter 1 suggests, a lecture
that consolidates the lessons of an activity and places the activity in the
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588 HOW STUDENTS LEARN IN THE CLASSROOM
conceptual framework of the discipline explicitly can play a critical role in
supporting student understanding.
How the balance is struck in creating a classroom that functions as a
learning community attentive to the learners’ needs, the knowledge to be
mastered, and assessments that support and guide instruction will certain
vary from one teacher and classroom to the next. Our hope for this volume,
then, is that its presentations of instructional approaches to addressing the
key principles from How People Learn will support the efforts of teachers to
play their own instructional game well. This volume is a first effort to elabo-
rate those findings with regard to specific topics, but we hope it is the first of
many such efforts. As teachers and researchers become more familiar with
some common aspects of student thinking about a topic, their attention may
begin to shift to other aspects that have previously attracted little notice. And
as insights about one topic become commonplace, they may be applied to
new topics.
Beyond extending the reach of the treatment of the learning principles
of How People Learn within and across topics, we hope that efforts to incor-
porate those principles into teaching and learning will help strengthen and
reshape our understanding of the rules of the learning game. With physics
as his topic of concern, Feynman16 talks about just such a process: “For a
long time we will have a rule that works excellently in an overall way, even
when we cannot follow the details, and then some time we may discover a
new rule. From the point of view of basic physics, the most interesting
phenomena are of course in the new places, the places where the rules do
not work—not the places where they do work! That is the way in which we
discover new rules.”
We look forward to the opportunities created for the evolution of the
science of learning and the professional practice of teaching as the prin-
ciples of learning on which this volume focuses are incorporated into class-
room teaching.
NOTES
1. Egan, 1986.
2. Story summarized by Kieran Egan, personal communication, March 7, 2003.
3. Liping Ma’s work, described in Chapter 1, refers to the set of core concepts and
the connected concepts and knowledge that support them as “knowledge
packages.”
4. Griffin and Case, 1995.
5. Moss and Case, 1999.
6. Kalchman et al., 2001.
7. Palincsar, 1986; White and Fredrickson, 1998.
8. Ma, 1999, p. 5.
9. Ma, 1999, p. 9.
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10. Palincsar, 1986.
11. Palincsar, 1986.
12. National Research Council, 2005 (Stewart et al., 2005, Chapter 12).
13. Aleven and Koedinger, 2002.
14. For example, he highlights core concepts conspicuously. In his first lecture, he
asks, “If, in some cataclysm, all of scientific knowledge were to be destroyed,
and only one sentence passed on to the next generation of creatures, what
statement would contain the most information in the fewest words? I believe it
is the atomic hypothesis that all things are made of atoms—little particles that
move around in perpetual motion, attracting each other when they are a little
distance apart, but repelling upon being squeezed into one another.
15. Even with experience, the thinking of individual students may be unantici-
pated by the teacher.
16. Feynman, 1995, p. 25.
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