How People Learn:
Brain, Mind,
Experience, and School

Part III: Teachers and Teaching



7
Effective Teaching: Examples in History, Mathematics,
and Science
The preceding chapter
explored implications of research on learning for general issues
relevant to the design of effective learning environments. We now move
to a more detailed exploration of teaching and learning in three
disciplines: history, mathematics, and science. We chose these three
areas in order to focus on the similarities and differences of
disciplines that use different methods of inquiry and analysis. A major
goal of our discussion is to explore the knowledge required to teach
effectively in a diversity of disciplines.
We noted in Chapter 2 that expertise in particular areas
involves more than a set of general problemsolving skills; it also
requires wellorganized knowledge of concepts and inquiry procedures.
Different disciplines are organized differently and have different
approaches to inquiry. For example, the evidence needed to support a
set of historical claims is different from the evidence needed to prove
a mathematical conjecture, and both of these differ from the evidence
needed to test a scientific theory. Discussion in Chapter 2 also differentiated between expertise in a
discipline and the ability to help others learn about that discipline.
To use Shulman's (1987) language, effective teachers need pedagogical
content knowledge (knowledge about how to teach in particular
disciplines) rather than only knowledge of a particular subject matter.
Pedagogical content
knowledge is different from knowledge of general teaching methods.
Expert teachers know the structure of their disciplines, and this
knowledge provides them with cognitive roadmaps that guide the
assignments they give students, the assessments they use to gauge
students' progress, and the questions they ask in the give and take of
classroom life. In short, their knowledge of the discipline and their
knowledge of pedagogy interact. But knowledge of the discipline
structure does not in itself guide the teacher. For example, expert
teachers are sensitive to those aspects of the discipline that are
especially hard or easy for new students to master. This means that new
teachers must develop the ability to "understand in a pedagogically
reflective way; they must not only know their own way around a
discipline, but must know the 'conceptual barriers' likely to hinder
others" (McDonald and Naso, 1986:8). These conceptual barriers differ
from discipline to discipline.
An emphasis on
interactions between disciplinary knowledge and pedagogical knowledge
directly contradicts common misconceptions about what teachers need to
know in order to design effective learning environments for their
students. The misconceptions are that teaching consists only of a set
of general methods, that a good teacher can teach any subject, or that
content knowledge alone is sufficient.
Some teachers are
able to teach in ways that involve a variety of disciplines.
However, their ability to do so requires more than a set of general
teaching skills. Consider the case of Barb Johnson, who has been a
sixth grade teacher for 12 years at Monroe Middle School. By
conventional standards Monroe is a good school. Standardized test
scores are about average, class size is small, the building facilities
are well maintained, the administrator is a strong instructional leader,
and there is little faculty and staff turnover. However, every year
parents sending their fifthgrade students from the local elementary
schools to Monroe jockey to get their children assigned to Barb
Johnson's classes. What happens in her classroom that gives it the
reputation of being the best of the best?
During the first week
of school Barb Johnson asks her sixth graders two questions: "What
questions do you have about yourself?" and "What questions do you have
about the world?" The students begin enumerating their questions, "Can
they be about silly, little things?" asks one student. "If they're your
questions that you really want answered, they're neither silly nor
little," replies the teacher. After the students list their individual
questions, Barb organizes the students into small groups where they
share lists and search for questions they have in common. After much
discussion each group comes up with a priority list of questions,
rankordering the questions about themselves and those about the world.
Back together in a
whole group session, Barb Johnson solicits the groups' priorities and
works toward consensus for the class's combined lists of questions.
These questions become the basis for guiding the curriculum in Barb's
class. One question, "Will I live to be 100 years old?" spawned
educational investigations into genetics, family and oral history,
actuarial science, statistics and probability, heart disease, cancer,
and hypertension. The students had the opportunity to seek out
information from family members, friends, experts in various fields,
online computer services, and books, as well as from the teacher. She
describes what they had to do as becoming part of a "learning
community." According to Barb Johnson, "We decide what are the most
compelling intellectual issues, devise ways to investigate those issues
and start off on a learning journey. Sometimes we fall short of our
goal. Sometimes we reach our goal, but most times we exceed these
goalswe learn more than we initially expected" (personal
communication).
At the end of an
investigation, Barb Johnson works with the students to help them see how
their investigations relate to conventional subjectmatter areas. They
create a chart on which they tally experiences in language and literacy,
mathematics, science, social studies and history, music, and art.
Students often are surprised at how much and how varied their learning
is. Says one student, "I just thought we were having fun. I didn't
realize we were learning, too!"
Barb Johnson's teaching
is extraordinary. It requires a wide range of disciplinary knowledge
because she begins with students' questions rather than with a fixed
curriculum. Because of her extensive knowledge, she can map students'
questions onto important principles of relevant disciplines. It would
not work to simply arm new teachers with general strategies that mirror
how she teaches and encourage them to use this approach in their
classrooms. Unless they have the relevant disciplinary knowledge, the
teachers and the classes would quickly become lost. At the same time,
disciplinary knowledge without knowledge about how students learn (i.e.,
principles consistent with developmental and learning psychology) and
how to lead the processes of learning (i.e., pedagogical knowledge)
would not yield the kind of learning seen in Barb Johnson's classes
(Anderson and Smith, 1987).
In the remainder of
this chapter, we present illustrations and discussions of exemplary
teaching in history, mathematics, and science. The three examples of
history, mathematics, and science are designed to convey a sense of the
pedagogical knowledge and content knowledge (Shulman, 1987) that
underlie expert teaching. They should help to clarify why effective
teaching requires much more than a set of "general teaching skills."




HISTORY 



Most people have had
quite similar experiences with history courses: they learned the facts
and dates that the teacher and the text deemed relevant. This view of
history is radically different from the way that historians see their
work. Students who think that history is about facts and dates miss
exciting opportunities to understand how history is a discipline that is
guided by particular rules of evidence and how particular analytical
skills can be relevant for understanding events in their lives (see
Ravitch and Finn, 1987). Unfortunately, many teachers do not present an
exciting approach to history, perhaps because they, too, were taught in
the datesfacts method.





Beyond Facts 




In Chapter 2, we discussed a study of experts in the
field of history and learned that they regard the available evidence as
more than lists of facts (Wineburg, 1991). The study contrasted a group
of gifted high school seniors with a group of working historians. Both
groups were given a test of facts about the American Revolution taken
from the chapter review section of a popular United States history
textbook. The historians who had backgrounds in American history knew
most of the items, while historians whose specialties lay elsewhere knew
only a third of the test facts. Several students scored higher than
some historians on the factual pretest. In addition to the test of
facts, however, the historians and students were presented with a set of
historical documents and asked to sort out competing claims and to
formulate reasoned interpretations. The historians excelled at this
task. Most students, on the other hand, were stymied. Despite the
volume of historical information the students possessed, they had little
sense of how to use it productively for forming interpretations of
events or for reaching conclusions.





Different Views of History by Different Teachers 




Different
views of history affect how teachers teach history. For example, Wilson
and Wineburg (1993) asked two teachers of American history to read a set
of student essays on the causes of the American Revolution not as an
unbiased or complete and definitive accounts of people and events, but
to develop plans for the students' "remediation or enrichment."
Teachers were provided with a set of essays on the question, "Evaluate
the causes of the American Revolution," written by eleventhgraders for
a timed, 45minute test. Consider the different types of feedback that
Mr. Barnes and Ms. Kelsey gave a student paper; see Box 7.1.
Mr. Barnes' comments on
the actual content of the essays concentrated on the factual level. Ms.
Kelsey's comments addressed broader images of the nature of the domain,
without neglecting important errors of fact. Overall, Mr. Barnes saw
the papers as an indication of the bellshaped distribution of
abilities; Ms. Kelsey saw them as representing the misconception that
history is about memorizing a mass of information and recounting a
series of facts. These two teachers had very different ideas about the
nature of learning history. Those ideas affected how they taught and
what they wanted their students to achieve.





Studies of Outstanding History Teachers 




For expert
history teachers, their knowledge of the discipline and beliefs about
its structure interact with their teaching strategies. Rather than
simply introduce students to sets of facts to be learned, these teachers
help people to understand the problematic nature of historical
interpretation and analysis and to appreciate the relevance of history
for their everyday lives.
One example
of outstanding history teaching comes form the classroom of Bob Bain, a
public school teacher in Beechwood, Ohio. Historians, he notes, are
cursed with an abundance of datathe traces of the past threaten to
overwhelm them unless they find some way of separating what is important
from what is peripheral. The assumptions that historians hold about
significance shape how they write their histories, the data they select,
and the narrative they compose, as well as the larger schemes they bring
to organize and periodize the past. Often these assumptions about
historical significance remain unarticulated in the classroom. This
contributes to students' beliefs that their textbooks are the
history rather than a history.
Bob Bain begins his
ninthgrade high school class by having all the students create a time
capsule of what they think are the most important artifacts from the
past. The students' task, then, is to put down on paper why they chose
the items they did. In this way, the students explicitly articulate
their underlying assumptions of what constitutes historical
significance. Students' responses are pooled, and he writes them on a
large poster that he hangs on the classroom wall. This poster, which
Bob Bain calls "Rules for Determining Historical Significance," becomes
a lightening rod for class discussions throughout the year, undergoing
revisions and elaborations as students become better able to articulate
their ideas.
At first, students
apply the rules rigidly and algorithmically, with little understanding
that just as they made the rules, they can also change them. But as
students become more practiced in plying their judgments of
significance, they come to see the rules as tools for assaying the
arguments of different historians, which allows them to begin to
understand why historians disagree. In this instance, the students'
growing ability to understand the interpretative nature of history is
aided by their teacher's deep understanding of a fundamental principle
of the discipline.
Leinhardt and
Greeno (1991, 1994) spent 2 years studying a highly accomplished teacher
of advanced placement history in an urban high school in Pittsburgh.
The teacher, Ms. Sterling, a veteran of over 20 years, began her school
year by having her students ponder the meaning of the statement, "Every
true history is contemporary history." In the first week of the
semester, Sterling thrust her students into the kinds of epistemological
issues that one might find in a graduate seminar: "What is history?"
"How do we know the past?" "What is the difference between someone who
sits down to 'write history' and the artifacts that are produced as part
of ordinary experience?" The goal of this extended exercise is to help
students understand history as an evidentiary form of knowledge,
not as clusters of fixed names and dates.
One might wonder about
the advisability of spending 5 days "defining history" in a curriculum
with so much to cover. But it is precisely Sterling's framework of
subjectmatter knowledgeher overarching understanding of the
discipline as a wholethat permits students entry into the advanced
world of historical sensemaking. By the end of the course, students
moved from being passive spectators of the past to enfranchised agents
who could participate in the forms of thinking, reasoning, and
engagement that are the hallmark of skilled historical cognition. For
example, early in the school year, Ms. Sterling asked her students a
question about the Constitutional Convention and "what were men able to
do." Paul took the question literally: "Uh, I think one of the biggest
things that they did, that we talked about yesterday, was the
establishment of the first settlements in the Northwest area states."
But after 2 months of educating students into a way of thinking about
history, Paul began to catch on. By January his responses to questions
about the fall of the cottonbased economy in the South were linked to
British trade policy and colonial ventures in Asia, as well as to the
failure of Southern leaders to read public opinion accurately in Great
Britain. Ms. Sterling's own understanding of history allowed her to
create a classroom in which students not only mastered concepts and
facts, but also used them in authentic ways to craft historical
explanations.





Debating the Evidence 




Elizabeth Jensen
prepares her group of eleventh graders to debate the following
resolution:
Resolved: The British government possesses the
legitimate authority to tax the American colonies.
As her students enter
the classroom they arrange their desks into three groupson the left of
the room a group of "rebels," on the right, a group of "loyalists," and
in the front, a group of "judges." Off to the side with a spiral
notebook on her lap sits Jensen, a short woman in her late 30s with a
booming voice. But today that voice is silent as her students take up
the question of the legitimacy of British taxation in the American
colonies.
The rebels' first
speaker, a 16yearold girl with a Grateful Dead Tshirt and one
dangling earring, takes a paper from her notebook and begins:
England says she keeps troops here for our own
protection. On face value, this seems reasonable enough, but there is
really no substance to their claims. First of all, who do they think
they are protecting us from? The French? Quoting from our friend Mr.
Bailey on page 54, 'By the settlement in Paris in 1763, French power was
thrown completely off the continent of North America.' Clearly not the
French then. Maybe they need to protect us from the Spanish? Yet the
same war also subdued the Spanish, so they are no real worry either. In
fact, the only threat to our order is the Indians . . . but . . . we
have a decent militia of our own. . . . So why are they putting troops
here? The only possible reason is to keep us in line. With more and
more troops coming over, soon every freedom we hold dear will be
stripped away. The great irony is that Britain expects us to pay for
these vicious troops, these British squelchers of colonial
justice.
A loyalist responds:
We moved here, we are paying less taxes than we
did for two generations in England, and you complain? Let's look at why
we are being taxedthe main reason is probably because England has a
debt of £140,000,000. . . . This sounds a little greedy, I
mean what right do they have to take our money simply because they have
the power over us. But did you know that over onehalf of their war
debt was caused by defending us in the French and Indian War. . . .
Taxation without representation isn't fair. Indeed, it's tyranny. Yet
virtual representation makes this whining of yours an untruth. Every
British citizen, whether he had a right to vote or not, is represented
in Parliament. Why does this representation not extend to
America?
A rebel questions the
loyalist about this:
Rebel: 
What benefits do we get out of paying taxes to the
crown? 
Loyalist: 
We benefit from the protection. 
Rebel: 
(cutting in) Is that the only benefit you claim,
protection? 
Loyalist: 
Yesand all the rights of an Englishman. 
Rebel: 
Okay, then what about the Intolerable Acts . . . denying us
rights of British subjects. What about the rights we are denied? 
Loyalist: 
The Sons of Liberty tarred and feather people, pillaged
homesthey were definitely deserving of some sort of punishment. 
Rebel: 
So should all the colonies be punished for the acts of a few
colonies? 
For a moment, the room
is a cacophony of charges and countercharges. "It's the same as in
Birmingham," shouts a loyalist. A rebel snorts disparagingly, "Virtual
representation is bull." Thirtytwo students seem to be talking at
once, while the presiding judge, a wiry student with hornrimmed
glasses, bangs his gavel to no avail. The teacher, still in the corner,
still with spiral notebook in lap, issues her only command of the day.
"Hold still!" she thunders. Order is restored and the loyalists
continue their opening argument (from Wineburg and Wilson, 1991).
Another example of
Elizabeth Jensen's teaching involves her efforts to help her high school
students understand the debates between Federalists and
antiFederalists. She knows that her 15 and 16yearolds cannot begin
to grasp the complexities of the debates without first understanding
that these disagreements were rooted in fundamentally different
conceptions of human naturea point glossed over in two paragraphs in
her history textbook. Rather than beginning the year with a unit on
European discovery and exploration, as her text dictates, she begins
with a conference on the nature of man. Students in her eleventhgrade
history class read excerpts from the writings of philosophers (Hume,
Locke, Plato, and Aristotle), leaders of state and revolutionaries
(Jefferson, Lenin, Gandhi), and tyrants (Hitler, Mussolini), presenting
and advocating these views before their classmates. Six weeks later,
when it is time to study the ratification of the Constitution, these
nowfamiliar figuresPlato, Aristotle, and othersare reconvened to be
courted by impassioned groups of Federalists and antiFederalists. It
is Elizabeth Jensen's understanding of what she wants to teach and what
adolescents already know that allows her to craft an activity that helps
students get a feel for the domain that awaits them: decisions about
rebellion, the Constitution, federalism, slavery, and the nature of a
government.





Conclusion 




These
examples provide glimpses of outstanding teaching in the discipline of
history. The examples do not come from "gifted teachers" who know how
to teach anything: they demonstrate, instead, that expert teachers have
a deep understanding of the structure and epistemologies of their
disciplines, combined with knowledge of the kinds of teaching activities
that will help students come to understand the discipline for
themselves. As we previously noted, this point sharply contradicts one
of the popularand dangerousmyths about teaching: teaching is a
generic skill and a good teacher can teach any subject. Numerous
studies demonstrate that any curriculumincluding a textbookis
mediated by a teacher's understanding of the subject domain (for
history, see Wineburg and Wilson, 1988; for math, see Ball, 1993; for
English, see Grossman et al., 1989). The uniqueness of the content
knowledge and pedagogical knowledge necessary to teach history becomes
clearer as one explores outstanding teaching in other disciplines.




MATHEMATICS 



As is the case in
history, most people believe that they know what mathematics is
aboutcomputation. Most people are familiar with only the
computational aspects of mathematics and so are likely to argue for its
place in the school curriculum and for traditional methods of
instructing children in computation. In contrast, mathematicians see
computation as merely a tool in the real stuff of mathematics, which
includes problem solving, and characterizing and understanding structure
and patterns. The current debate concerning what students should learn
in mathematics seems to set proponents of teaching computational skills
against the advocates of fostering conceptual understanding and reflects
the wide range of beliefs about what aspects of mathematics are
important to know. A growing body of research provides convincing
evidence that what teachers know and believe about mathematics is
closely linked to their instructional decisions and actions (Brown,
1985; National Council of Teachers of Mathematics, 1989; Wilson, 1990a,
b; Brophy, 1990; Thompson, 1992).
Teachers' ideas about
mathematics, mathematics teaching, and mathematics learning directly
influence their notions about what to teach and how to teach itan
interdependence of beliefs and knowledge about pedagogy and subject
matter (e.g., Gamoran, 1994; Stein et al., 1990). It shows that
teachers' goals for instruction are, to a large extent, a reflection of
what they think is important in mathematics and how they think students
best learn it. Thus, as we examine mathematics instruction, we need to
pay attention to the subjectmatter knowledge of teachers, their
pedagogical knowledge (general and content specific), and their
knowledge of children as learners of mathematics. Paying attention to
these domains of knowledge also leads us to examine teachers' goals for
instruction.
If students in
mathematics classes are to learn mathematics with understandinga goal
that is accepted by almost everyone in the current debate over the role
of computational skills in mathematics classroomsthen it is important
to examine examples of teaching for understanding and to analyze the
roles of the teacher and the knowledge that underlies the teacher's
enactments of those roles. In this section, we examine three cases of
mathematics instruction that are viewed as being close to the current
vision of exemplary instruction and discuss the knowledge base on which
the teacher is drawing, as well as the beliefs and goals which guide his
or her instructional decisions.





Multiplication with Meaning 




For teaching
multidigit multiplication, teacherresearcher Magdelene Lampert created
a series of lessons in which she taught a heterogeneous group of 28
fourthgrade students. The students ranged in computational skill from
beginning to learn the singledigit multiplication facts to being able
to accurately solve ndigit by ndigit multiplications. The lessons
were intended to give children experiences in which the important
mathematical principles of additive and multiplicative composition,
associativity, commutativity, and the distributive property of
multiplication over addition were all evident in the steps of the
procedures used to arrive at an answer (Lampert, 1986:316). It is clear
from her description of her instruction that both her deep understanding
of multiplicative structures and her knowledge of a wide range of
representations and problem situations related to multiplication were
brought to bear as she planned and taught these lessons. It is also
clear that her goals for the lessons included not only those related to
students' understanding of mathematics, but also those related to
students' development as independent, thoughtful problem solvers.
Lampert (1986:339) described her role as follows:
My role was to bring students' ideas about how to
solve or analyze problems into the public forum of the classroom, to
referee arguments about whether those ideas were reasonable, and to
sanction students' intuitive use of mathematical principles as
legitimate. I also taught new information in the form of symbolic
structures and emphasized the connection between symbols and operations
on quantities, but I made it a classroom requirement that students use
their own ways of deciding whether something was mathematically
reasonable in doing the work. If one conceives of the teacher's role in
this way, it is difficult to separate instruction in mathematics content
from building a culture of sensemaking in the classroom, wherein
teacher and students have a view of themselves as responsible for
ascertaining the legitimacy of procedures by reference to known
mathematical principles. On the part of the teacher, the principles
might be known as a more formal abstract system, whereas on the part of
the learners, they are known in relation to familiar experiential
contexts. But what seems most important is that teachers and students
together are disposed toward a particular way of viewing and doing
mathematics in the classroom.
Magdelene Lampert set
out to connect what students already knew about multidigit
multiplication with principled conceptual knowledge. She did so in
three sets of lessons. The first set used coin problems, such as "Using
only two kinds of coins, make $1.00 using 19 coins," which encouraged
children to draw on their familiarity with coins and mathematical
principles that coin trading requires. Another set of lessons used
simple stories and drawings to illustrate the ways in which large
quantities could be grouped for easier counting. Finally, the third set
of lessons used only numbers and arithmetic symbols to represent
problems. Throughout the lessons, students were challenged to explain
their answers and to rely on their arguments, rather than to rely on the
teacher or book for verification of correctness. An example serves to
highlight this approach; see Box
7.2.
Lampert (1986:337)
concludes:
. . . students used principled knowledge that was
tied to the language of groups to explain what they were seeing. They
were able to talk meaningfully about place value and order of operations
to give legitimacy to procedures and to reason about their outcomes,
even though they did not use technical terms to do so. I took their
experimentations and arguments as evidence that they had come to see
mathematics as more than a set of procedures for finding answers.
Clearly, her own deep
understanding of mathematics comes into play as she teaches these
lessons. It is worth noting that her goal of helping students see what
is mathematically legitimate shapes the way in which she designs lessons
to develop students' understanding of twodigit multiplication.





Understanding Negative Numbers 




Helping thirdgrade
students extend their understanding of numbers from the natural numbers
to the integers is a challenge undertaken by another teacherresearcher.
Deborah Ball's work provides another snapshot of teaching that draws on
extensive subject content and pedagogical content knowledge. Her goals
in instruction include "developing a practice that respects the
integrity both of mathematics as a discipline and of children as
mathematical thinkers" (Ball, 1993). That is, she not only takes into
account what the important mathematical ideas are, but also how children
think about the particular area of mathematics on which she is focusing.
She draws on both her understanding of the integers as mathematical
entities (subjectmatter knowledge) and her extensive pedagogical
content knowledge specifically about integers. Like Lampert, Ball's
goals go beyond the boundaries of what is typically considered
mathematics and include developing a culture in which students
conjecture, experiment, build arguments, and frame and solve
problemsthe work of mathematicians.
Deborah Ball's
description of work highlights the importance and difficulty of figuring
out powerful and effective ways to represent key mathematical ideas to
children (see Ball, 1993). A wealth of possible models for negative
numbers exists and she reviewed a number of themmagic peanuts, money,
game scoring, a frog on a number line, buildings with floors above and
below ground. She decided to use the building model first and money
later: she was acutely aware of the strengths and limitations of each
model as a way for representing the key properties of numbers,
particularly those of magnitude and direction. Reading Deborah Ball's
description of her deliberations, one is struck by the complexity of
selecting appropriate models for particular mathematical ideas and
processes. She hoped that the positional aspects of the building model
would help children recognize that negative numbers were not equivalent
to zero, a common misconception. She was aware that the building model
would be difficult to use for modeling subtraction of negative numbers.
Deborah Ball begins her
work with the students, using the building model by labeling its floors.
Students readily labeled the underground floors and accepted them as
"below zero." They then explored what happened as little paper people
entered an elevator at some floor and rode to another floor. This was
used to introduce the conventions of writing addition and subtraction
problems involving integers 4 — 6 = —2 and —2 + 5 = 3.
Students were presented with increasingly difficult problems. For
example, "How many ways are there for a person to get to the second
floor?" Working with the building model allowed students to generate a
number of observations. For example, one student noticed that "any
number below zero plus that same number above zero equals zero" (Ball,
1993:381). However, the model failed to allow for explorations for such
problems 5 + (—6) and Ball was concerned that students were not
developing a sense that —5 was less than —2it was lower,
but not necessarily less. Ball then used a model of money as a second
representational context for exploring negative numbers, noting that it,
too, has limitations.
Clearly, Deborah Ball's
knowledge of the possible representations of integers (pedagogical
content knowledge) and her understanding of the important mathematical
properties of integers were foundational to her planning and her
instruction. Again, her goals related to developing students'
mathematical authority, and a sense of community also came into play.
Like Lampert, Ball wanted her students to accept the responsibility of
deciding when a solution is reasonable and likely to be correct, rather
than depending on text or teacher for confirmation of correctness.





Guided Discussion 




The work of Lampert and
Ball highlights the role of a teacher's knowledge of content and
pedagogical content knowledge in planning and teaching mathematics
lessons. It also suggests the importance of the teacher's understanding
of children as learners. The concept of cognitively guided instruction
helps illustrate another important characteristic of effective
mathematics instruction: that teachers not only need knowledge of a
particular topic within mathematics and knowledge of how learners think
about the particular topic, but also need to develop knowledge about how
the individual children in their classrooms think about the topic
(Carpenter and Fennema, 1992; Carpenter et al., 1996; Fennema et al.,
1996). Teachers, it is claimed, will use their knowledge to make
appropriate instructional decisions to assist students to construct
their mathematical knowledge. In this approach, the idea of domains of
knowledge for teaching (Shulman, 1986) is extended to include teachers'
knowledge of individual learners in their classrooms.
Cognitively guided
instruction is used by Annie Keith, who teaches a combination first and
secondgrade class in an elementary school in Madison Wisconsin (Hiebert
et al., 1997). Her instructional practices are an example of what is
possible when a teacher understands children's thinking and uses that
understanding to guide her teaching. A portrait of Ms. Keith's
classroom reveals also how her knowledge of mathematics and pedagogy
influence her instructional decisions.
Word problems form the
basis for almost all instruction in Annie Keith's classroom. Students
spend a great deal of time discussing alternative strategies with each
other, in groups, and as a whole class. The teacher often participates
in these discussions but almost never demonstrates the solution to
problems. Important ideas in mathematics are developed as students
explore solutions to problems, rather than being a focus of instruction
per se. For example, placevalue concepts are developed as students use
base10 materials, such as base10 blocks and counting frames, to solve
word problems involving multidigit numbers.
Mathematics instruction
in Annie Keith's class takes place in a number of different settings.
Everyday firstgrade and secondgrade activities, such as sharing
snacks, lunch count, and attendance, regularly serve as contexts for
problemsolving tasks. Mathematics lessons frequently make use of math
centers in which the students do a variety of activities. On any given
day, children at one center may solve word problems presented by the
teacher while at another center children write word problems to present
to the class later or play a math game.
She continually
challenges her students to think and to try to make sense of what they
are doing in math. She uses the activities as opportunities for her to
learn what individual students know and understand about mathematics.
As students work in groups to solve problems, she observes the various
solutions and mentally makes notes about which students should present
their work: she wants a variety of solutions presented so that students
will have an opportunity to learn from each other. Her knowledge of the
important ideas in mathematics serves as one framework for the selection
process, but her understanding of how children think about the
mathematical ideas they are using also affects her decisions about who
should present. She might select a solution that is actually incorrect
to be presented so that she can initiate a discussion of a common
misconception. Or she may select a solution that is more sophisticated
than most students have used in order to provide an opportunity for
students to see the benefits of such a strategy. Both the presentations
of solutions and the class discussions that follow provide her with
information about what her students know and what problems she should
use with them next.
Annie Keith's strong
belief that children need to construct their understanding of
mathematical ideas by building on what they already know guides her
instructional decisions. She forms hypotheses about what her students
understand and selects instructional activities based on these
hypotheses. She modifies her instruction as she gathers additional
information about her students and compares it with the mathematics she
wants them to learn. Her instructional decisions give her clear
diagnoses of individual students' current state of understanding. Her
approach is not a freeforall without teacher guidance: rather, it is
instruction that builds on students' understandings and is carefully
orchestrated by the teacher, who is aware of what is mathematically
important and also what is important to the learner's progress.





ModelBased Reasoning 




Some attempts
to revitalize mathematics instruction have emphasized the importance of
modeling phenomena. Work on modeling can be done from kindergarten
through twelth grade (K12). Modeling involves cycles of model
construction, model evaluation, and model revision. It is central to
professional practice in many disciplines, such as mathematics and
science, but it is largely missing from school instruction. Modeling
practices are ubiquitous and diverse, ranging from the construction of
physical models, such as a planetarium or a model of the human vascular
system, to the development of abstract symbol systems, exemplified by
the mathematics of algebra, geometry, and calculus. The ubiquity and
diversity of models in these disciplines suggest that modeling can help
students develop understanding about a wide range of important ideas.
Modeling practices can and should be fostered at every age and grade
level (Clement, 1989; Hestenes, 1992; Lehrer and Romberg, 1996a, b;
Schauble et al., 1995; see Box
7.3).
Taking a modelbased
approach to a problem entails inventing (or selecting) a model,
exploring the qualities of the model, and then applying the model to
answer a question of interest. For example, the geometry of triangles
has an internal logic and also has predictive power for phenomena
ranging from optics to wayfinding (as in navigational systems) to laying
floor tile. Modeling emphasizes a need for forms of mathematics that
are typically underrepresented in the standard curriculum, such as
spatial visualization and geometry, data structure, measurement, and
uncertainty. For example, the scientific study of animal behavior, like
bird foraging, is severely limited unless one also has access to such
mathematical concepts as variability and uncertainty. Hence, the
practice of modeling introduces the further explorations of important
"big ideas" in disciplines.





Conclusion 




Increasingly,
approaches to early mathematics teaching incorporate the premises that
all learning involves extending understanding to new situations, that
young children come to school with many ideas about mathematics, that
knowledge relevant to a new setting is not always accessed
spontaneously, and that learning can be enhanced by respecting and
encouraging children to try out the ideas and strategies that they bring
to schoolbased learning in classrooms. Rather than beginning
mathematics instruction by focusing solely on computational algorithms,
such as addition and subtraction, students are encouraged to invent
their own strategies for solving problems and to discuss why those
strategies work. Teachers may also explicitly prompt students to think
about aspects of their everyday life that are potentially relevant for
further learning. For example, everyday experiences of walking and
related ideas about position and direction can serve as a springboard
for developing corresponding mathematics about the structure of
largescale space, position, and direction (Lehrer and Romberg, 1996b).
As research
continues to provide good examples of instruction that help children
learn important mathematics, there will be better understanding of the
roles that teachers' knowledge, beliefs, and goals play in their
instructional thinking and actions. The examples we have provided here
make it clear that the selection of tasks and the guidance of students'
thinking as they work through tasks is highly dependent on teachers'
knowledge of mathematics, pedagogical content knowledge, and knowledge
of students in general.




SCIENCE 



Two recent examples in
physics illustrate how research findings can be used to design
instructional strategies that promote the sort of problemsolving
behavior observed in experts. Undergraduates who had finished an
introductory physics course were asked to spend a total of 10 hours,
spread over several weeks, solving physics problems using a
computerbased tool that constrained them to perform a conceptual
analysis of the problems based on a hierarchy of principles and
procedures that could be applied to solve them (Dufresne et al., 1996).
This approach was motivated by research on expertise (discussed in Chapter 2). The reader will recall that, when asked
to state an approach to solving a problem, physicists generally discuss
principles and procedures. Novices, in contrast, tend to discuss
specific equations that could be used to manipulate variables given in
the problem (Chi et al., 1981). When compared with a group of students
who solved the same problems on their own, the students who used the
computer to carry out the hierarchical analyses performed noticeably
better in subsequent measures of expertise. For example, in problem
solving, those who performed the hierarchical analyses outperformed
those who did not, whether measured in terms of overall problemsolving
performance, ability to arrive at the correct answer, or ability to
apply appropriate principles to solve the problems; see Figure 7.1. Furthermore, similar
differences emerged in problem categorization: students who performed
the hierarchical analyses considered principles (as opposed to surface
features) more often in deciding whether or not two problems would be
solved similarly; see Figure 7.2.
(See Chapter 6 for an example of the type of
item used in the categorization task of Figure 7.2.) It is also worth noting that both Figures 7.1 and 7.2 illustrate two other issues that
we have discussed in this volume, namely that time on task is a major
indicator for learning and that deliberate practice is an efficient way
to promote expertise. In both cases, the control group made significant
improvements simply as a result of practice (time on task), but the
experimental group showed more improvements for the same amount of
training time (deliberate practice).
Introductory physics
courses have also been taught successfully with an approach for problem
solving that begins with a qualitative hierarchical analysis of the
problems (Leonard et al., 1996). Undergraduate engineering students
were instructed to write qualitative strategies for solving problems
before attempting to solve them (based on Chi et al., 1981). The
strategies consisted of a coherent verbal description of how a problem
could be solved and contained three components: the major principle to
be applied; the justification for why the principle was applicable; and
the procedures for applying the principle. That is, the what, why, and
how of solving the problem were explicitly delineated; see Box 7.4. Compared with students who
took a traditional course, students in the strategybased course
performed significantly better in their ability to categorize problems
according to the relevant principles that could be applied to solve
them; see Figure 7.3.
Hierarchical structures
are useful strategies for helping novices both recall knowledge and
solve problems. For example, physics novices who had completed and
received good grades in an introductory college physics course were
trained to generate a problem analysis called a theoretical problem
description (Heller and Reif, 1984). The analysis consists of
describing force problems in terms of concepts, principles, and
heuristics. With such an approach, novices substantially improved in
their ability to solve problems, even though the type of theoretical
problem description used in the study was not a natural one for novices.
Novices untrained in the theoretical descriptions were generally unable
to generate appropriate descriptions on their owneven given fairly
routine problems. Skills, such as the ability to describe a problem in
detail before attempting a solution, the ability to determine what
relevant information should enter the analysis of a problem, and the
ability to decide which procedures can be used to generate problem
descriptions and analyses, are tacitly used by experts but rarely taught
explicitly in physics courses.
Another approach helps
students organize knowledge by imposing a hierarchical organization on
the performance of different tasks in physics (Eylon and Reif, 1984).
Students who received a particular physics argument that was organized
in hierarchical form performed various recall and problemsolving tasks
better than subjects who received the same argument nonhierarchically.
Similarly, students who received a hierarchical organization of
problemsolving strategies performed much better than subjects who
received the same strategies organized nonhierarchically. Thus,
helping students to organize their knowledge is as important as the
knowledge itself, since knowledge organization is likely to affect
students' intellectual performance.
These examples
demonstrate the importance of deliberate practice and of having a
"coach" who provides feedback for ways of optimizing performance (see Chapter 3). If students had simply been given
problems to solve on their own (an instructional practice used in all
the sciences), it is highly unlikely that they would have spent time
efficiently. Students might get stuck for minutes, or even hours, in
attempting a solution to a problem and either give up or waste lots of
time. In Chapter 3, we discussed ways in which
learners profit from errors and that making mistakes is not always time
wasted. However, it is not efficient if a student spends most of the
problemsolving time rehearsing procedures that are not optimal for
promoting skilled performance, such as finding and manipulating
equations to solve the problem, rather than identifying the underlying
principle and procedures that apply to the problem and then constructing
the specific equations needed. In deliberate practice, a student works
under a tutor (human or computer based) to rehearse appropriate
practices that enhance performance. Through deliberate practice,
computerbased tutoring environments have been designed that reduce the
time it takes individuals to reach realworld performance criteria from
4 years to 25 hours (see Chapter 9)!





Conceptual Change 




Before
students can really learn new scientific concepts, they often need to
reconceptualize deeply rooted misconceptions that interfere with the
learning. As reviewed above (see Chapters 3 and
4), people spend considerable time and effort
constructing a view of the physical world through experiences and
observations, and they may cling tenaciously to those viewshowever
much they conflict with scientific conceptsbecause they help them
explain phenomena and make predictions about the world (e.g., why a rock
falls faster than a leaf).
One instructional
strategy, termed "bridging," has been successful in helping students
overcome persistent misconceptions (Brown, 1992; Brown and Clement,
1989; Clement, 1993). The bridging strategy attempts to bridge from
students' correct beliefs (called anchoring conceptions) to their
misconceptions through a series of intermediate analogous situations.
Starting with the anchoring intuition that a spring exerts an upward
force on the book resting on it, the student might be asked if a book
resting on the middle of a long, "springy" board supported at its two
ends experiences an upward force from the board. The fact that the bent
board looks as if it is serving the same function as the spring helps
many students agree that both the spring and the board exert upward
forces on the book. For a student who may not agree that the bent board
exerts an upward force on the book, the instructor may ask a student to
place her hand on top of a vertical spring and push down and to place
her hand on the middle of the springy board and push down. She would
then be asked if she experienced an upward force that resisted her push
in both cases. Through this type of dynamic probing of students'
beliefs, and by helping them come up with ways to resolve conflicting
views, students can be guided into constructing a coherent view that is
applicable across a wide range of contexts.
Another effective
strategy for helping students overcome persistent erroneous beliefs are
interactive lecture demonstrations (Sokoloff and Thornton, 1997;
Thornton and Sokoloff, 1997). This strategy, which has been used very
effectively in large introductory college physics classes, begins with
an introduction to a demonstration that the instructor is about to
perform, such as a collision between two air carts on an air track, one
a stationary light cart, the other a heavy cart moving toward the
stationary cart. Each cart has an electronic "force probe" connected to
it which displays on a large screen and in realtime the force acting on
it during the collision. The teacher first asks the students to discuss
the situation with their neighbors and then record a prediction as to
whether one of the carts would exert a bigger force on the other during
impact or whether the carts would exert equal forces.
The vast majority of
students incorrectly predict that the heavier, moving cart exerts a
larger force on the lighter, stationary cart. Again, this prediction
seems quite reasonable based on experiencestudents know that a moving
Mack truck colliding with a stationary Volkswagen beetle will result in
much more damage done to the Volkswagen, and this is interpreted to mean
that the Mack truck must have exerted a larger force on the Volkswagen.
Yet, notwithstanding the major damage to the Volkswagen, Newton's Third
Law states that two interacting bodies exert equal and opposite forces
on each other.
After the students make
and record their predictions, the instructor performs the demonstration,
and the students see on the screen that the force probes record forces
of equal magnitude but oppositely directed during the collision.
Several other situations are discussed in the same way: What if the two
carts had been moving toward each other at the same speed? What if the
situation is reversed so that the heavy cart is stationary and the light
cart is moving toward it? Students make predictions and then see the
actual forces between the carts displayed as they collide. In all
cases, students see that the carts exert equal and opposite forces on
each other, and with the help of a discussion moderated by the
instructor, the students begin to build a consistent view of Newton's
Third Law that incorporates their observations and experiences.
Consistent with the
research on providing feedback (see Chapter 3),
there is other research that suggests that students' witnessing the
force displayed in realtime as the two carts collide helps them
overcome their misconceptions; delays of as little as 2030 minutes in
displaying graphic data of an event occurring in realtime significantly
inhibits the learning of the underlying concept (Brasell, 1987).
Both bridging and the
interactive demonstration strategies have been shown to be effective at
helping students permanently overcome misconceptions. This finding is a
major breakthrough in teaching science, since so much research indicates
that students often can parrot back correct answers on a test that might
be erroneously interpreted as displaying the eradication of a
misconception, but the same misconception often resurfaces when students
are probed weeks or months later (see Mestre, 1994, for a review).





Teaching as Coaching 




One of the best
examples of translating research into practice is Minstrell's (1982,
1989, 1992) work with high school physics students. Minstrell uses many
researchbased instructional techniques (e.g., bridging, making
students' thinking visible, facilitating students' ability to
restructure their own knowledge) to teach physics for understanding. He
does this through classroom discussions in which students construct
understanding by making sense of physics concepts, with Minstrell
playing a coaching role. The following quote exemplifies his innovative
and effective instructional strategies (Minstrell, 1989:130131):
Students' initial ideas about mechanics are like strands
of yarn, some unconnected, some loosely interwoven. The act of
instruction can be viewed as helping the students unravel individual
strands of belief, label them, and then weave them into a fabric of more
complete understanding. An important point is that later understanding
can be constructed, to a considerable extent, from earlier beliefs.
Sometimes new strands of belief are introduced, but rarely is an earlier
belief pulled out and replaced. Rather than denying the relevancy of a
belief, teachers might do better by helping students differentiate their
present ideas from and integrate them into conceptual beliefs more like
those of scientists.
Describing a lesson on
force, Minstrell (1989:130131) begins by introducing the topic in
general terms:
Today we are going to try to explain some rather
ordinary events that you might see any day. You will find that you
already have many good ideas that will help explain those events. We
will find that some of our ideas are similar to those of the scientist,
but in other cases our ideas might be different. When we are finished
with this unit, I expect that we will have a much clearer idea of how
scientists explain those events, and I know that you will feel more
comfortable about your explanations . . . A key idea we are going to use
is the idea of force. What does the idea of force mean to you?
Many views emerge from
the ensuing classroom discussion, from the typical "push or pull" to
descriptions that include sophisticated terms, such as energy and
momentum. At some point Minstrell guides the discussion to a specific
example: he drops a rock and asks students how the event can be
explained using their ideas about force. He asks students to
individually formulate their ideas and to draw a diagram showing the
major forces on the rock as arrows, with labels to denote the cause of
each force. A lengthy discussion follows in which students present
their views, views that contain many irrelevant (e.g., nuclear forces)
or fictitious forces (e.g., the spin of the earth, air). In his
coaching, Minstrell asks students to justify their choices by asking
questions, such as "How do you know?" "How did you decide?" "Why do you
believe that?"
With this approach,
Minstrell has been able to identify many erroneous beliefs of students
that stand in the way of conceptual understanding. One example is the
belief that only active agents (e.g., people) can exert forces, that
passive agents (e.g., a table) cannot. Minstrell (1992) has developed a
framework that helps both to make sense of students' reasoning and to
design instructional strategies. (For a related theoretical framework
for classifying and explaining student reasoning, see the discussion of
"phenomenological primitives" in DiSessa, 1988, 1993.) Minstrell
describes identifiable pieces of students' knowledge as "facets," a
facet being a convenient unit of thought, a piece of knowledge, or a
strategy seemingly used by the student in addressing a particular
situation. Facets may relate to conceptual knowledge (e.g., passive
objects do not exert force), to strategic knowledge (e.g., average
velocity can be determined by adding the initial and final velocities
and dividing by two), or generic reasoning (e.g., the more the X, the
more the Y). Identifying students' facets, what cues them in different
contexts, and how students use them in reasoning are all helpful in
devising instructional strategies.





Interactive Instruction in Large Classes 




One of the obstacles to
instructional innovation in large introductory science courses at the
college level is the sheer number of students who are taught at one
time. How does an instructor provide an active learning experience,
provide feedback, accommodate different learning styles, make students'
thinking visible, and provide scaffolding and tailored instruction to
meet specific student needs when facing more than 100 students at a
time? Classroom communication systems can help the instructor of a
large class accomplish these objectives. One such system, called
Classtalk, consists of both hardware and software that allows up to four
students to share an input device (e.g., a fairly inexpensive graphing
calculator) to "sign on" to a classroom communication network that
permits the instructor to send questions for students to work on and
permits students to enter answers through their input device. Answers
can then be displayed anonymously in histogram form to the class, and a
permanent record of each student's response is recorded to help evaluate
progress as well as the effectiveness of instruction.
This technology has
been used successfully at the University of MassachusettsAmherst to
teach physics to a range of students, from nonscience majors to
engineering and science majors (Dufresne et al., 1996; Wenk et al.,
1997; Mestre et al., 1997). The technology creates an interactive
learning environment in the lectures: students work collaboratively on
conceptual questions, and the histogram of students' answers is used as
a visual springboard for classwide discussions when students defend the
reasoning they used to arrive at their answers. This technology makes
students' thinking visible and promotes critical listening, evaluation,
and argumentation in the class. The teacher is a coach, providing
scaffolding where needed, tailoring "minilectures" to clear up points
of confusion, or, if things are going well, simply moderating the
discussion and allowing students to figure out things and reach
consensus on their own. The technology is also a natural mechanism to
support formative assessment during instruction, providing both the
teacher and students with feedback on how well the class is grasping the
concepts under study. The approach accommodates a wider variety of
learning styles than is possible by lectures and helps to foster a
community of learners focused on common objectives and goals.





Science for All Children 




The examples
above present some effective strategies for teaching and learning
science for high school and college students. We drew some general
principles of learning from these examples and stressed that the
findings consistently point to the strong effect of knowledge structures
on learning. These studies also emphasize the importance of class
discussions for developing a language for talking about scientific
ideas, for making students' thinking explicit to the teacher and to the
rest of the class, and for learning to develop a line of argumentation
that uses what one has learned to solve problems and explain phenomena
and observations.
The question that
immediately occurs is how to teach science to younger children or to
students who are considered to be educationally "at risk." One approach
that has been especially useful in science teaching was developed with
languageminority gradeschool children: Chèche Konnen, which in
Haitian Creole means search for knowledge (Rosebery et al., 1992). The
approach stresses how discourse is a primary means for the search for
knowledge and scientific sensemaking. It also illustrates how
scientific ideas are constructed. In this way it mirrors science, in
the words of Nobel Laureate Sir Peter Medawar (1982:111):
Like other exploratory processes, [the scientific
method] can be resolved into a dialogue between fact and fancy, the
actual and the possible; between what could be true and what is in fact
the case. The purpose of scientific enquiry is not to compile an
inventory of factual information, nor to build up a totalitarian world
picture of Natural Laws in which every event that is not compulsory is
forbidden. We should think of it rather as a logically articulated
structure of justifiable beliefs about a Possible Worlda story which
we invent and criticize and modify as we go along, so that it ends by
being, as nearly as we can make it, a story about real life.
The Chèche
Konnen approach to teaching began by creating "communities of scientific
practice" in languageminority classrooms in a few Boston and Cambridge,
MA public schools. "Curriculum" emerges in these classrooms from the
students' questions and beliefs and is shaped in ongoing interactions
that include both the teacher and students. Students explore their own
questions, much as we described above in Barb Johnson's class. In
addition, students design studies, collect information, analyze data and
construct evidence, and they then debate the conclusions that they
derive from their evidence. In effect, the students build and argue
about theories; see Box 7.5.
Students constructed
scientific understandings through an iterative process of theory
building, criticism, and refinement based on their own questions,
hypotheses, and data analysis activities. Question posing, theorizing,
and argumentation formed the structure of the students' scientific
activity. Within this structure, students explored the implications of
the theories they held, examined underlying assumptions, formulated and
tested hypotheses, developed evidence, negotiated conflicts in belief
and evidence, argued alternative interpretations, provided warrants for
conclusions, and so forth. The process as a whole provided a richer,
more scientifically grounded experience than the conventional focus on
textbooks or laboratory demonstrations.
The emphasis on
establishing communities of scientific practice builds on the fact that
robust knowledge and understandings are socially constructed through
talk, activity, and interaction around meaningful problems and tools
(Vygotsky, 1978). The teacher guides and supports students as they
explore problems and define questions that are of interest to them. A
community of practice also provides direct cognitive and social support
for the efforts of the group's individual members. Students share the
responsibility for thinking and doing: they distribute their
intellectual activity so that the burden of managing the whole process
does not fall to any one individual. In addition, a community of
practice can be a powerful context for constructing scientific meanings.
In challenging one another's thoughts and beliefs, students must be
explicit about their meanings; they must negotiate conflicts in belief
or evidence; and they must share and synthesize their knowledge to
achieve understanding (Brown and Palincsar, 1989; Inagaki and Hatano,
1987).
What do students learn
from participating in a scientific sensemaking community? Individual
interviews with students before and after the water taste test
investigation (see Box 7.4),
first in September and again the following June, showed how the
students' knowledge and reasoning changed. In the interviews (conducted
in Haitian Creole), the students were asked to think aloud about two
openended realworld problemspollution in the Boston Harbor and a
sudden illness in an elementary school. The researchers were interested
in changes in students' conceptual knowledge about aquatic ecosystems
and in students' uses of hypotheses, experiments, and explanations to
organize their reasoning (for a complete discussion, see Rosebery et
al., 1992).
Conceptual Knowledge
Not surprisingly, the
students knew more about water pollution and aquatic ecosystems in June
than they did in September. They were also able to use this knowledge
generatively. One student explained how she would clean the water in
Boston Harbor (Rosebery et al., 1992:86).
Like you look for the things, take the garbage out of
the water, you put a screen to block all the paper and stuff, then you
clean the water; you put chemical products in it to clean the water, and
you'd take all the microscopic life out. Chlorine and alum, you put in
the water. They'd gather the little stuff, the little stuff would stick
to the chemical products, and they would clean the water.
Note that this
explanation contains misconceptions. By confusing the cleaning of
drinking water with the cleaning of sea water, the student suggests
adding chemicals to take all microscopic life from the water (good for
drinking water, but bad for the ecosystem of Boston Harbor). This
example illustrates the difficulties in transferring knowledge
appropriately from one context to another (see Chapter 3). Despite these shortcomings, it is clear
that this student is starting on the path to scientific thinking,
leaving behind the more superficial "I'd take all the bad stuff out of
the water" type of explanation. It is also clear that by making the
student's thinking visible, the teacher is in an excellent position to
refine her (and perhaps the class's) understanding.





Scientific Thinking 




Striking changes
appeared in students' scientific reasoning. In September, there were
three ways in which the students showed little familiarity with
scientific forms of reasoning. First, the students did not understand
the function of hypotheses or experiments in scientific inquiry. When
asked for their ideas about what could be making the children sick, the
students tended, with few exceptions, to respond with short,
unelaborated, often untestable "hypotheses" that simply restated the
phenomena described in the problem: "That's a thing . . . . Ah, I
could say a person, some person that gave them something . . . .
Anything, like give poison to make his stomach hurt" (Rosebery et al.,
1992:81).
Second, the students
conceptualized evidence as information they already knew, either through
personal experience or secondhand sources, rather than data produced
through experimentation or observation. When asked to generate an
experiment to justify an hypothesis"How would you find out?"they
typically offered declarations: "Because the garbage is a poison for
them . . . . The garbage made the fish die" (Rosebery et al., 1992:78).
Third, the students
interpreted an elicitation for an experiment"How would you be
sure?"as a text comprehension question for which there was a "right"
answer. They frequently responded with an explanation or assertion of
knowledge and consistently marked their responses as explanatory
("because"): "Because fish don't eat garbage. They eat plants under
the water" (page 78).
In the June interviews,
the students showed that they had become familiar with the function of
hypotheses and experiments and with reasoning within larger explanatory
frameworks. Elinor had developed a model of an integrated water system
in which an action or event in one part of the system had consequences
for other parts (Rosebery et al., 1992:87):
You can't leave [the bad stuff] on the ground. If you
leave it on the ground, the water that, the earth has water underground,
it will still spoil the water underground. Or when it rains it will
just take it and, when it rains, the water runs, it will take it and
leave it in the river, in where the water goes in. Those things, poison
things, you aren't supposed to leave it on the ground.
In June, the students
no longer invoked anonymous agents, but put forward chains of hypotheses
to explain phenomena, such as why children were getting sick (page 88):
Like, you could test what the kids ate and, like, test
the water, too; it could be the water that isn't good, that has
microbes, that might have microscopic animals in it to make them
sick.
The June interviews
also showed that students had begun to develop a sense of the function
and form of experimentation. They no longer depended on personal
experience as evidence, but proposed experiments to test specific
hypotheses. In response to a question about sick fish, Laure clearly
understands how to find a scientific answer (page 91):
I'd put a fish in fresh water and one fish in a water
full of garbage. I'd give the fresh water fish food to eat and the
other one in the nasty water, I'd give it food to eat to see if the
fresh water, if the one in the fresh water would die with the food I
gave it, if the one in the dirty water would die with the food I gave
it. . . . I would give them the same food to see if the things they eat
in the water and the things I give them now, which will make them
healthy and which wouldn't make them healthy.





Conclusion 




Teaching and learning
in science have been influenced very directly by research studies on
expertise (see Chapter 2). The examples
discussed in this chapter focus on two areas of science teaching:
physics and junior high school biology. Several of the teaching
strategies illustrated ways to help students think about the general
principles or "big" ideas in physics before jumping to formulas and
equations. Others illustrate ways to help students engage in deliberate
practice (see Chapter 3) and to monitor their
progress.
Learning the strategies
for scientific thinking have another objective: to develop thinking
acumen needed to promote conceptual change. Often, the barrier to
achieving insights to new solutions is rooted in a fundamental
misconception about the subject matter. One strategy for helping
students in physics begins with an "anchoring intuition" about a
phenomenon and then gradually bridging it to related phenomena that are
less intuitive to the student but involve the same physics principles.
Another strategy involves the use of interactive lecture demonstrations
to encourage students to make predictions, consider feedback, and then
reconceptualize phenomena.
The example of
Chèche Konnen demonstrates the power of a sensemaking approach
to science learning that builds on the knowledge that students bring
with them to school from their home cultures, including their familiar
discourse practices. Students learned to think, talk, and act
scientifically, and their first and second languages mediated their
learning in powerful ways. Using Haitian Creole, they designed their
studies, interpreted data, and argued theories; using English, they
collected data from their mainstream peers, read standards to interpret
their scientific test results, reported their findings, and consulted
with experts at the local water treatment facility.




CONCLUSION 



Outstanding teaching
requires teachers to have a deep understanding of the subject matter and
its structure, as well as an equally thorough understanding of the kinds
of teaching activities that help students understand the subject matter
in order to be capable of asking probing questions.
Numerous studies
demonstrate that the curriculum and its tools, including textbooks, need
to be dissected and discussed in the larger contexts and framework of a
discipline. In order to be able to provide such guidance, teachers
themselves need a thorough understanding of the subject domain and the
epistemology that guides the discipline (for history, see Wineburg and
Wilson, 1988; for math and English, see Ball, 1993; Grossman et al.,
1989; for science, see Rosebery et al., 1992).
The examples in this
chapter illustrate the principles for the design of learning
environments that were discussed in Chapter 6:
they are learner, knowledge, assessment, and community centered. They
are learner centered in the sense that teachers build on the knowledge
students bring to the learning situation. They are knowledge centered
in the sense that the teachers attempt to help students develop an
organized understanding of important concepts in each discipline. They
are assessment centered in the sense that the teachers attempt to make
students' thinking visible so that ideas can be discussed and clarified,
such as having students (1) present their arguments in debates, (2)
discuss their solutions to problems at a qualitative level, and (3) make
predictions about various phenomena. They are community centered in the
sense that the teachers establish classroom norms that learning with
understanding is valued and students feel free to explore what they do
not understand.
These examples
illustrate the importance of pedagogical content knowledge to guide
teachers. Expert teachers have a firm understanding of their respective
disciplines, knowledge of the conceptual barriers that students face in
learning about the discipline, and knowledge of effective strategies for
working with students. Teachers' knowledge of their disciplines
provides a cognitive roadmap to guide their assignments to students, to
gauge student progress, and to support the questions students ask. The
teachers focus on understanding rather than memorization and routine
procedures to follow, and they engage students in activities that help
students reflect on their own learning and understanding.
The interplay between
content knowledge and pedagogical knowledge illustrated in this chapter
contradicts a commonly held misconception about teachingthat effective
teaching consists of a set of general teaching strategies that apply to
all content areas. This notion is erroneous, just as is the idea that
expertise in a discipline is a general set of problemsolving skills
that lack a content knowledge base to support them (see Chapter 2).
The outcomes of new
approaches to teaching as reflected in the results of summative
assessments are encouraging. Studies of students' discussions in
classrooms indicate that they learn to use the tools of systematic
inquiry to think historically, mathematically, and scientifically. How
these kinds of teaching strategies reveal themselves on typical
standardized tests is another matter. In some cases there is evidence
that teaching for understanding can increase scores on standardized
measures (e.g., Resnick et al., 1991); in other cases, scores on
standardized tests are unaffected, but the students show sizable
advantages on assessments that are sensitive to their comprehension and
understanding rather than reflecting sheer memorization (e.g., Carpenter
et al., 1996; Secules et al., 1997).
It is noteworthy that
none of the teachers discussed in this chapter felt that he or she was
finished learning. Many discussed their work as involving a lifelong
and continuing struggle to understand and improve. What opportunities
do teachers have to improve their practice? The next chapter explores
teachers' chances to improve and advance their knowledge in order to
function as effective professionals.



