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6
Making Thinking Visible: Modeling and Representation
Scientists develop models and representations as ways to think about the natural
world. The kinds of models that scientists construct vary widely, both within and
across disciplines. Nevertheless, in building and testing theories, the practice of
science is governed by efforts to invent, revise, and contest models. Using models
is another important way that scientists make their thinking visible.
Representation is a predecessor to full-fledged modeling. Even very young
children can use one object to stand in for, or represent, another. But they typi-
cally do not recognize or account for the relationships and separations between
the real world and models: the features of a phenomenon that a representation
accounts for or fails to account for. The use of all forms of symbolic representa-
tion, such as graphs, tables, mathematical expressions, and diagrams, can be
developed in young children and lead to more sophisticated modeling in later
years. In “Science Class: The Nature of Gases”
(Chapter 4), we described students in an after-
school science program who were attempting
to understand air pressure. The students used
“Air Puppies” as a model to represent air mol-
ecules. They depicted Air Puppies as dots in
some scenarios and as numbers in others (see
Figures 6-1 and 6-2).
Modeling involves the construction and
testing of representations that are analogous to
systems in the real world. These representa-
FIGURE 6-1 tions can take many forms, including physical
Taylor explaining the movement
of the wall-on-wheels with Air Puppies models, computer programs, diagrams, math-
represented as dots. ematical equations, and propositions. The
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objects depicted in a model, as well
as their behavior and relationships
to each other, represent theoretically
important objects, behavior, and
relationships in the natural world.
Models allow scientists to summa-
rize and depict the known features
of a physical system and predict out-
comes using these depictions. Thus,
they are often important tools in the
development of scientific theories.
FIGURE 6-2
A key concept for students to Mitchell and Antwaune show Air Puppies in and
understand is that models are not outside a bottle as numbers (100 calm and 100
excited puppies).
meant to be exact copies. Instead,
they are deliberate simplifications of more complex systems. This means that no
model is completely accurate. For example, in modeling air molecules with Air
Puppies, certain characteristics of molecules are represented, such as the fact that
they move constantly without intention, and other characteristics are not, such as
their being composed of hydrogen and oxygen atoms. Students need guidance in
recognizing what characteristics are included in a model and how this helps fur-
ther their understanding of how a system works. When first introduced to the Air
Puppies model, students often ask, “Do Air Puppies breathe air? Do they sleep?
Do they die?” They need to figure out which aspects of Air Puppies are useful for
understanding how air molecules work.
Mathematics
For the past 200 years, science has moved toward increasing quantification, visu-
alization, and precision. Mathematics provides scientists with another system for
sharing, communicating, and understanding science concepts. Often, expressing
an idea mathematically results in the discovery of new patterns or relationships
that otherwise might not be seen.
In the grade-level representation activities that follow, third-grade children
investigating the growth of plants wondered whether the shoots (the part of the
plant growing above the ground) and the roots grow at the same rate. When they
plotted the growth on a coordinate graph that displayed millimeters of growth
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per day, students noticed immediately that the rates of growth were not the same.
However, one student pointed out that the curves for both the roots and the
shoots showed the same S-shape. This S-shape appeared again on graphs describ-
ing the growth of tobacco hornworms and populations of bacteria on a plate.
Students came to recognize this shape as a standard graph pattern that indicated
growth. This similarity in patterns would not have been noticeable without the
mathematical representation afforded by the graph.
Given the importance of mathematics in understanding science, elementary
school mathematics needs to go beyond arithmetic to include ideas regarding
space and geometry, measurement, and data and uncertainty. Measurement, for
example, is a ubiquitous part of the scientific enterprise, although its subtleties
are almost always overlooked. Students are usually taught procedures for mea-
suring but are rarely taught a theory of measure. Educators often overestimate
children’s understanding of measurement, because measuring tools—like rul-
ers and scales—resolve many of the conceptual challenges of measurement for
children. As a result, students may fail to understand that measurement entails
the use of repeated constant units and that these units can be partitioned. Even
upper elementary students who seem proficient at measuring lengths with rulers
may believe that measuring merely entails counting the units between boundar-
ies. If these students are given unconnected units (say, tiles of a constant length)
and asked to demonstrate how to measure a length, some of them almost always
place the units against the object being measured in such a way that the first and
last tiles are lined up flush with the end of the object measured, leaving spaces
among the units in between. These spaces do not trouble a student who holds
this “boundary-filling” conception of measurement.
Data
Data modeling is central to a variety of scientific enterprises, including engi-
neering, medicine, and natural science. Scientists build models with an acute
awareness of the data that are required, and data are structured and recorded
as a way of making progress in articulating a scientific model or deciding
among rival models.
Students are better able to understand data if as much attention is devoted
to how they are generated as to their analysis. First and foremost, students need
to understand that data are constructed to answer questions, not provided in a
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finished form by nature. Questions are what determine the types of informa-
tion that will be gathered, and many aspects of data coding and structuring also
depend on the questions asked.
Data are inherently abstract, as they are observations that stand for con-
crete events. Data may take many forms: a linear distance may be represented by
a number of standard units, a video recording can stand in for an observation of
human interaction, or a reading on a thermometer may represent a sensation of heat.
Collection of data often requires the use of tools, and students often have
a fragile grasp of the relationship between an event of interest and the operation
or output of a tool used to capture data about the event. Whether that tool is a
microscope, a pan balance, or a simple ruler, students often need help understand-
ing the purpose of the tool and of measurement. Some students, for example,
accustomed to relying on sensory observations of “felt weight,” may find a pan
balance confusing, because they do not, at first, understand the value of using one
object to determine the weight of another.
Data do not come with an inherent structure. Rather, a structure must be
imposed on data. Scientists and students impose structure by selecting categories
with which to describe and organize the data. However, young learners often fail
to grasp this as evidenced in their tendency to believe that new questions can be
addressed only with new data. They rarely think of querying existing data sets to
explore questions that were not initially conceived when the data were collected.
For example, earlier we described a biodiversity unit in which children cataloged
a number of species in a woodlot adjacent to their school. The data generated in
this activity could later be queried to determine the spread of a given population
or which species of plants and animals tend to cluster together in certain areas of
the woodlot.
Finally, data are represented in various ways to see, understand, or com-
municate different aspects of the phenomenon being studied. For example, a bar
graph of children’s height may provide a quick visual sense of the range of heights.
In contrast, a scatterplot of children’s height by children’s age would yield a linear
relationship between height and age. An important goal for students—one that
extends over several years—is to come to understand the conventions and proper-
ties of different kinds of data displays. There are many different kinds of repre-
sentational displays, including tables, graphs of various kinds, and distributions.
Not only should students understand the procedures for generating and reading
displays, but they should also be able to critique them and to grasp the advantages
and disadvantages of different displays for a given purpose.
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Interpreting data often entails finding and confirming relationships in the
data, and these relationships can have varying levels of complexity. Simple linear
relationships are easier to spot than inverse relationships or interactions. Students
may often fail to consider that more than one type of relationship may be present.
For example, children investigating the health of a population of finches may wish
to examine the weight of birds in the population. The weight of adult finches is
likely to be a nonlinear relationship. That is, as both low weight and high weight
are disadvantageous to survival, one would expect to find a number of weights in
the middle, with fewer on both ends of the distribution.
The desire to interpret data may lead to the use of various statistical mea-
sures. These measures are a further step of abstraction beyond the objects and
events originally observed. For example, understanding the mean requires an
understanding of ratio. If students are merely taught to “average” data in a proce-
dural way, without having a well-developed sense of ratio, their performance often
degrades, mistakenly, into procedures for adding and dividing that make no sense.
However, with good instruction, middle and upper elementary students can learn
to simultaneously consider the center and the spread of the data.
Students also can generate various mathematical descriptions of error. This
is particularly true in the case of measurement: they can readily grasp the relation-
ships between their own participation in the act of measuring and the resulting
variation in measures.
Scale Models, Diagrams, and Maps
Scale models, diagrams, and maps are additional examples of modeling. Scale
models, such as a model of the solar system, are widely used in science education
so that students can visualize objects or processes that they cannot perceive or
handle directly.
The ease with which students understand these models depends on the com-
plexity of the relationship being communicated. Even preschoolers can under-
stand scale models used to depict location in a room. Elementary school students
can look beyond the appearance of a model to investigate the way it functions.
However, extremely large and small-scale models often pose serious challenges for
students. For example, middle school students may struggle to work out the posi-
tional relationships of the earth, the sun, and the moon, which involves not only
reconciling different perspectives (what one sees standing on the earth, what one
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would see from a hypothetical point in space) but also visualizing how these per-
spectives would change over days and months.
Students are often expected to read or produce diagrams and integrate
information from the diagram with accompanying text. Understanding dia-
grams seems to depend less on a student’s problem-solving abilities than on
the specific design and content of the diagram. Diagrams can be difficult to
understand for many reasons. Sometimes the desired information is missing.
Sometimes a diagram does not appear in a familiar or recognizable context.
And sometimes features of a diagram can create confusion. For example, the
common misconception that the earth is closer to the sun in the summer than in
the winter may be due, in part, to the fact that two-dimensional representations
of the three-dimensional orbit make it appear as if the earth is indeed closer to
the sun at some points than at others.
Students’ understanding of maps can be particularly challenging, because
maps preserve some characteristics of the place being represented—for instance,
relative position and distance—but may omit or alter features of the actual
landscape. Recall the mapping done by Mr. Walker’s class in the case study on
biodiversity in Chapter 2, in which the students learned to develop a more sys-
tematic plan for mapping the distribution and density of common species. Young
children especially have a much easier time representing objects than representing
large-scale space. Students may also struggle with orientation, perspective (the
traditional bird’s eye view), and mathematical descriptions of space, such as polar
coordinate representations.
Modeling and Learning Progressions
In a study involving biological growth, Richard Lehrer and Leona Schauble
observed characteristic shifts in the understanding of modeling over the span
of the elementary school grades.1 They developed a learning progression that
emphasized different and increasingly complex ideas in different grade bands.
Each had a different curriculum and tasks:
• Early elementary: Growth of flowering bulbs: A focus on difference
• Middle elementary: Growth of Wisconsin Fast Plants2: A focus on ratio
• Late elementary: Growth of population: A focus on distribution
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They observed that primary grade students’ initial representations of growth
were typically focused on endpoints, for example: “How tall do plants grow?”
Students’ questions about plant height led to related concerns about identifying
the attributes of a plant that could best represent height and how those attributes
should be measured. As one might expect, students’ resolutions to these problems
varied by grade.
First-Grade Representations
First graders represented the heights of plants grown from flowering bulbs, using
green paper strips to depict the plant stems at different points in the growth cycle
(see Figure 6-3). Consistent with the claim that young children try to create mod-
els that closely resemble real or known objects, the students at first insisted that
FIGURE 6-3
A display with the paper strips be adorned with flowers.
detailed drawings However, as the teacher repeatedly focused students’ attention on succes-
of individual plants
that include flowers sive differences in the lengths of the strips, students began to make the conceptual
and colors. transition from thinking of the strips as “presenting” height to “representing”
height (see Figure 6-4). Reasoning about changes in the height differences of the
FIGURE 6-4 strips, students identified times when their plants grew “faster” and “slower.”
Displays of plant
height depicted in
Their study of the plant heights was firmly grounded in prior discussions about
bar graphs. what counted as “tall” and how to measure it reliably.
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Third-Grade Representations
In the third grade, children integrated math into their representations of Wisconsin
Fast Plants in a variety of ways. They developed “pressed plant” silhouettes
that recorded changes in plant morphology over time, coordinate graphs that
related plant height and time, sequences of rectangles representing the relationship
between plant height and canopy “width,” and various three-dimensional forms to
capture changes in plant volume.
As the diversity in types of students’ representations increased, a new
question emerged: Was the growth of roots and shoots the same or different?
Comparing the height and depth of roots and shoots, students noticed that, at
any point in a plant’s life cycle, the differences in measurement were apparent.
However, they also noted that graphs displaying the growth of roots and shoots
were characterized by similar shapes: an S-shaped logistic curve (see Figure 6-5).
Finding similarities in the shape of data describing roots and shoots but not
the measurements of roots and shoots, students began to wonder about the sig-
nificance of the similarity they observed. Why would the growth of two different
FIGURE 6-5
A display of plant
height over time
depicted in an
S-shaped curve.
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plant parts take the same form on the graph? When was the growth of the roots
and shoots the fastest, and what was the functional significance of those periods
of rapid growth?
Students became competent at using a variety of representational forms as
models. For example, students noted that growth over time x,y-coordinate graphs
of two different plants looked similar in that they were equally “steep.” Yet the
graphs actually represented different rates of growth, because the students who
generated the graphs used different scales to represent the height of their plants.
The discovery that graphs might look the same and yet represent different rates of
growth influenced the students’ interpretations of other graphs in this and other
contexts throughout the year.
Fifth-Grade Representations
In the fifth grade, children again investigated growth, this time in tobacco horn-
worms (Manduca), but their mathematical resources now included ideas about
distribution and sample. Students explored relationships between growth factors:
for example, different food sources and the relative dispersion of characteristics in
the population at different points in the life cycle of the hornworms.
Questions posed by the fifth graders focused on the diversity of charac-
teristics within populations—for example, length, circumference, weight, and
days to pupation—rather than simply shifts in central tendencies of attributes
(see Figure 6-6 on page 120). As the students’ ability to use different forms of
representation grew, so, too, did their consideration of what might be worthy
of investigation.
Shifts in Understanding
In sum, over the span of the elementary school grades, these researchers observed
characteristic shifts from an early emphasis on models that used literal depic-
tion toward representations that were progressively more symbolic in charac-
ter. Increased competence in using a wider range of representational types both
accompanied and helped promote conceptual change.
As students developed and used new mathematical means for character-
izing growth, they understood biological change in increasingly dynamic ways.
For example, once students understood the mathematics of changing ratios, they
began to conceive of growth not as a simple linear increase but as a patterned
rate of change. These shifts in both conceptual understanding and forms of
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written or graphic representation appeared to support each other, opening up
new paths of inquiry.
Students noticed similarities and differences among graphs and wondered
whether plant growth was similar to animal growth and whether the growth of
yeast and bacteria on a Petri dish was similar to that of a single plant. Students
studying the growth of such organisms as plants, tobacco hornworms, and
populations of bacteria noted that when they graphed changes in heights over
a life span, all the organisms studied produced an S-shaped curve on the graph.
However, making this connection required a prior understanding of a Cartesian
coordinate system. In this case and in others, explanatory models and data mod-
els worked together to further conceptual development. At the same time, growing
understanding of concepts led to increased sophistication and diversity of repre-
sentational resources.
Current instruction often underestimates the difficulty of connecting repre-
sentations with reasoning about the scientific phenomena they represent. Students
need support in both interpreting and creating data representations that carry
meaning. Students learn to use representations that are progressively more sym-
bolic and mathematically powerful.
Teachers need to encourage this pro-
cess over multiple grades.
Let’s take a closer look at how
children develop scientific represen-
tations. In the following case, also
taken from the work of Lehrer and
Schauble, we examine a group of fifth
graders working on an investigation of
plant growth. They are challenged to
develop representations of their data in
order to reach particular goals in com-
municating.
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Science Class
REPRESENTING DATA3
Students need opportunities to build models and representations that suit particular explanatory and communica-
tive purposes. They need experience refining and improving models and representations, experience that can be
facilitated by critically examining the qualities of multiple models or representations for a given purpose.
In the following example we visit a fifth-grade classroom in which students are studying species variation. Having
tracked the growth of Wisconsin Fast Plants over a period of 19 days, they are grappling with the best way to rep-
resent their data. Hubert Rohling, the teacher, has posted a list of unordered measures that the students had taken
over the previous 18 days on chart paper at the front of the class. He has asked them to consider two questions:
(1) how they might organize the data in a way that would help them consider typical height on the 19th day and
(2) how to characterize how spread out the heights were on this day. He chose to have the students focus on these
qualities of their representation in order to draw their attention to critical aspects of representing data sets.
Mr. Rohling understood that his students would need to grapple with how best to portray data and to practice
doing so as a purposeful activity. Rather than assigning children particular data displays to use in capturing data,
he asked them to invent displays. He introduced additional uncertainty into the assignment by asking students to
identify typical values. Often the approach to learning about typical values is to teach children different measures
of central tendency and to assign children to calculate means, or identify the modal or median values in a data set.
Mr. Rohling’s interest, however, was to push children to wrestle with the notion of typicality and articulate their
understanding through creating and critiquing data displays.
In the process students would be forced to grapple with the value of maintaining regular intervals between data
points (thus providing a visual cue as to the quantitative relationship among points) and sampling distribution.
(What aspect of the data provides a fair sense of the overall shape of the data set?) Students would confront the
same kinds of problems that scientists do in the course of their work. They must find meaningful ways to organize
information to reveal particular characteristics of the data.
The students had previously been assigned to seven working teams of three to four students each. The students
in each group worked to construct a data display that they believed would support answers to Mr. Rohling’s two
questions. Mr. Rohling encouraged each group to come up with its own way to arrange the data, explaining that
it was important that the display, standing alone, make apparent the answer to the two questions about typicality
and spread of heights.
The students’ solutions were surprisingly varied. From the seven groups, five substantively different representa-
tional designs were produced. Over the next two days, students debated the advantages and trade-offs of their
representational choices; their preferences shifted as the discussion unfolded. To encourage broad participation in
critical discussion of displays, Mr. Rohling assigned pairs of students to present displays that their classmates had
developed. And following this he facilitated discussions which drew in display authors, presenters, and other class-
mates. Despite the opportunity to exchange ideas with their peers, students did not easily or simply adopt conven-
tions suggested by others. Instead, there was a long process of negotiation, tuning, and eventually convergence
toward a shared way of inscribing what students came to refer to as the shape of the data.
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The first display discussed is shown in Figure
6-6. One of the students, Will, and his team-
mate presented this graph on large, easel-
sized graph paper. As the figure shows, the
authors first developed a scale (along the left
side of the graph) to include all the observed
heights of the plants. Then they simply drew
lines to that scale, representing the height of
each plant, ordered from the shortest to the
tallest. As the class considered this display,
Will tried to explain how this graph could be
used to answer Question 1: “What is a typical
height on Day 19?”
Will: “The tops of the lines represent
height, and you have to see which lines
stop and go along on one level. It’s . . . it’s
the same number.” [He points toward a
space in the middle of the graph where
all the lines appear to be about the same
height.]
Mr. Rohling: “So you’re looking for a flat FIGURE 6-6
line to tell you what typical is?” A data display representing individual specimen height
with a vertical line.
Will: “Yes, then you can tell how many of
those there are.” difficult to read, especially from the back of the
room. Will volunteered that the authors might
Mr. Rohling: “What about Question 2: How
consider alternating colors for different values, to
spread out are the plants on Day 19?”
make it easier to discern small changes in contigu-
Will: “You can look at the graph and see that ous values.
it starts low down here on the left and goes The authors of the second display (Figure 6-7)
up on the right.“ simply ordered the values from lowest to high-
est and then wrote them along the bottom of the
Mr. Rohling: “If the data weren’t spread out,
paper, stacking the values that occurred multiple
what would it look like?”
times. The chart makers apparently ran out of room
Will: “One flat horizontal line.” along the bottom of the page and, to avoid start-
ing over, placed the remaining four values (200, 205,
This exchange shows that Will understood that 212, 255) on the upper left, surrounded by a box.
“plateaus” on the graph denote clumps in the data. Although the values are separated by commas, this
However, he went on to admit that the graph was display, like the display previously discussed, fails to
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preserve interval. That is, the authors did not use Matt: “I think this part on the top shouldn’t
spaces to indicate missing values. Therefore, linear be there [pointing to the “leftover” num-
distance does not accurately represent spread in the bers in the box]. It’s kind of confusing.
data. Keith and Matt interpreted this graph. Those numbers on the top, they ran out of
room.”
Keith: “The typical number is, like, the one
that goes higher than the others. You can just The third display (Figure 6-8) presented values
tell. The most common one is the highest col- stacked in “bins” of 10. This display preserves each
umn [the typical value.] The next question, case value as well as the interval (the bin) as each
for how spread out the data is . . . we just plant height is written above its “bin” in ascending
took the lowest number here . . . it was 30 . . . order. This form of display was used the previous year
and subtracted it from 255. We got 225.” in a rocket investigation, and the students may have
had at least a vague memory of its form.
Mr. Rohling: “So does the graph itself help
Looking at the display, Julia and Angelique
you see that? Or do you have to do some-
identified the mode as the “typical value,” pointing
thing with the numbers?”
out that most of the values were in the 160s col-
Keith: “You can tell the graph is pretty umn. However, one student found the graph con-
spread out from 30 to 255.” [He sweeps his fusing. She asked, “How come it’s all grouped by
hand across the line.] tens?” Julia replied, “That’s just how they did it.”
Instead of letting this answer stand, Mr. Rohling
Mr. Rohling: “What could you do to show pushed the discussion further. He wanted the stu-
typical and spread better?” dents to think about why “binning” the values might
produce different views (shapes of
data) of typicality and spread. To raise
these issues, he asked the class to think
about a contrast, between the simple,
ordered list (Figure 6-7) and the display
currently under consideration (Figure
6-8). Of Figure 6-7, he asked, “How did
this group bin them?”
A student replied, “One value per
bin.”
Another student asked, referring to
Figure 6-8, “Why did you select bins of
10?” Tanner and Erica, the authors of
the graph, explained their reasoning:
FIGURE 6-7
A display featuring ordered values of
plant heights.
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Tanner: “We wanted to make our
numbers bigger and easier to see,
so we didn’t want to waste a bunch
of room.”
Erica: “We also thought it would be
easier to answer the two questions
this way.”
Mr. Rohling: “So you’re saying that
binning them helps you see what’s
typical?”
Erica: “Yes, and how spread out
they are.”
Mr. Rohling: “How does binning
help you do that?”
Tanner: “Typical is from 160 to
169. It’s not that there is a typical FIGURE 6-8
number; it’s the typical group, I A data display using “bins” of ten.
would say.
This idea of a typical group or
typical region would come to play an
increasingly central role over the subse-
quent weeks of instruction, especially as
the class began to discuss sampling. For
the time being, Mr. Rohling decided to
go on to the next display (Figure 6-9).
This display listed values in ascending
order from left to right, starting at the
top left and moving down the page
in rows, with repeated values stacked
together. Katie and Greg, the present-
ers, noted that the authors had writ-
ten their proposed typical value on the
lower right of the display and that they
had also marked out the 160s in their
FIGURE 6-9
display, presumably to indicate that these were the A data display with rows of ascending values and repeated
values selected as typical. However, Katie and Greg values stacked.
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thought that this graph made it difficult to answer Mr. Rohling: “So, Julia, do you think if I wrote
the question about “spread.” the number 555 right here [he appends the
value 555 immediately at the end of the
Katie: “This graph is a little more clumped
ordered list on Figure 6-7], it would be the eas-
up than the others (Figures 6-6 and 6-8 for
iest graph to see that this has a lot of spread?”
example). It’s not in a line, so it’s a little
harder to see. They were doing it in rows, but Katie: “I think probably this graph [Figure
they did columns, too. That was kind of hard 6-8] would probably be better for spread
to figure out.” because they still leave the spaces there, even
if there’s nothing there. So you can really see
Mr. Rohling: “So you’re saying if you just had
how spread out it is. You can see how much
to use the graph data. . .”
space there is.”
Keith: “We’d be way off.”
Mr. Rohling: “You’re saying if it was 555,
Mr. Rohling then returned to the previous dis- we’d figure 555 would be out here? [He
plays to juxtapose two different approaches to indicates a space way off the right edge of
spread, one focusing on ordered cases (Figure 6-7) the graph.] Then the graph would actually
and the other on interval (Figure 6-8). He employed look like it’s spread? What helps you see the
an imagined value (555) to highlight the difference spread, then?”
between interval and order.
Isaac: “Not just the numbers that we actu-
Mr. Rohling: “I’m wondering which graph ally measured that are in between, but
would show the spread better? Let’s ignore empty spaces in all the numbers that are in
255 for a minute [the highest value on both between.”
graphs] and assume that the highest value is
At this point, the students appeared to reach
more like 555 [he opens his hands wider]. Does
agreement that if a display is to show the spread in
that feel quite a bit different than 255? If we
the data, it is necessary to scale the graph in a way
include that number, that would become a
that preserves intervals, even intervals for which no
much bigger spread. So let’s pretend that the
values have been observed. Although few of the
high value is 555. Which graph would help us
original displays met this criterion, all of the displays
see that it’s more spread out? What about the
made after the discussion did so.
one with the bins [Figure 6-8]? Is there a graph
Other displays were also presented (see Figures
up there that would help?”
6-10 and 6-11). And, as the discussion progressed, it
Julia: “I think this one [Figure 6-9] might be was clear that there were two competing value sys-
harder to read from far away. They put the tems in the air that were driving the students’ display
data in a square instead of a line.” preferences. On one hand, students’ own designs or
those made by close friends were especially favored,
At this point, one of the authors of that graph and novelty and creativity were also highly prized.
protested, “We wanted people to be able to see the For example, as the presenters explained Figure 6-10,
numbers. If they’re small, they’re hard to read. If we murmurs of “Oh, that’s cool!” and “You guys are so
had more paper, we’d have done it on a line.”
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cool!” were heard from about half the class.
On the other hand, about half the students
expressed concerns that the “cool” solution
did not seem to provide an illustration of
either typicality or spread. The display depict-
ed in Figure 6-11 was deemed even “cooler”
but, as more than one classmate noted, did
not surrender its design logic readily. It took
two full days of discussion before students
finally surrendered their focus on novelty of
design and gravitated instead toward criteria
favoring clarity of the mathematical ideas.
FIGURE 6-10
A data display on a two-
dimensional coordinate grid.
FIGURE 6-11
A data display showing median
at the apex of a pyramid.
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As this case illustrates, elementary school students can create representations
that have clear communicative features. The representations themselves and the
rich discussions they support offer an important window into how students are
thinking about representation and about the phenomena being studied. Generating
multiple representations and critiquing their utility for a particular goal can com-
pel elementary school students to develop a clearer sense of the considerations that
go into developing representations.
In addition to supporting students’ skill at creating and using representation,
modeling data through displays is fertile ground for advancing all four strands of
science learning. In the case above, for example, children developed their substan-
tive understanding of plant growth and population as they discussed and critiqued
the data representations (Strand 1). They developed facility with graphing and
making sense of data as they constructed representations of plant heights that
conveyed information about the data spread and typical values (Strand 2). They
embraced science as a dynamic undertaking and reflected on the adequacy of their
representations. Over time their ideas changed—favoring “cool” displays slowly
gave way to favoring displays that communicated clearly. Students whose previ-
ous displays did not retain intervals used intervals in subsequent displays, building
on the cumulative insight of the group (Strand 3). Finally, their arguments and
approaches to revising their models were governed by the goals and norms of sci-
ence. As they analyzed and discussed the data displays, they practiced scientific
norms by critically appraising each other’s displays and explicitly reasoning about
how well the displays accomplished the intended communicative goals (Strand 4).
Importantly, learning in each of the strands did not take place in isolation.
Rather, advances in one strand supported and were catalyzed by advances in the
other strands. This underscores a key point established in previous chapters: sci-
ence is complex and learning science takes time and practice. The sophistication of
students in the case above is the result of engaging in a rich investigative task, but
also of many months and even years of science instruction that supported their
knowledge and skill across all four strands.
Some important generalizations can be drawn from the examples of represen-
tation discussed in this chapter. Graphs, tables, computer-based tools, and math-
ematical expressions are examples of important symbolic and communication tools
used in modeling. Scientists, as well as students of science, use representations to
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convey complex ideas, patterns, trends, or proposed explanations of phenomena
in compressed, accessible formats. These tools require expertise to understand
and use. Teachers can help students reflect on the features and purposes of rep-
resentations by asking them to generate and critique their own representational
solutions to problems, by encouraging them to interpret the representations
developed by other students, and by asking them to consider what a representa-
tion shows and hides so that they come to understand representational choices as
trade-offs. Although working with representations poses challenges for learners, it
also can help bridge between the knowledge and skills they bring to the classroom
and more sophisticated scientific practices.
For Further Reading
Lehrer, R., and Schauble, L. (2004). Modeling natural variation through distribution.
American Educational Research Journal, 41(3), 635-679.
McNeill, K.L., Lizotte, D.J., Krajcik, J., and Marx, R.W. (2006). Supporting students’
construction of scientific explanations by fading scaffolds in instructional materials.
Journal of the Learning Sciences, 15(2), 153-191.
National Research Council. (2007). Teaching science as practice. Chapter 9 in
Committee on Science Learning, Kindergarten Through Eighth Grade, Taking science to
school: Learning and teaching science in grades K-8 (pp. 251-295). R.A. Duschl, H.A.
Schweingruber, and A.W. Shouse (Eds.). Center for Education, Division of Behavioral
and Social Sciences and Education. Washington, DC: The National Academies Press.
126 Ready, Set, SCIENCE!