Cornelia C. Tierney
Ricardo Nemirovsky
In this TERC study we investigated how fourth-grade children spontaneously represent changes over time in the motion of cars and in the number of people in a place. Typically, children are introduced to graphing in school through a systematic format for graphs. Rather than teach such graphical systems -- bar graphs in the early grades, line graphs later -- we want to design learning ramps that connect children’s natural representations with more formal graphing techniques. In this study, we found that when no particular format was required, children created idiosyncratic representational systems to express actual data they collected and the information in the situation that they found relevant to communicate.
We asked a group of nine-year-old children1 to make "pictures or graphs or charts" to show changing events. The situations we asked them to represent were:
- Changing population in a neighborhood restaurant and in their classroom over a day
- Changing number of people in their homes over a day
- Sequence of changes (adding and taking away) of objects in a bag
- Changes in the speed of a car described by a story
- Motions of a toy car moving across a table, as presented in a video
Students worked alone or with a partner to design representations. Then they discussed their representations in small groups and tested their understanding of each other’s representations by acting them out. To act out the graphs of people in their houses, students moved small blocks "into" or "out of" a drawing of a house. To act out the motion graphs, students moved a toy car along the floor.
Patterns in the Children’s Approaches to Making Graphs to Communicate
One of the main attributes of a representational system is to delineate a universe of possibilities regardless of their "actuality." However, the children generally limited their representations to what they actually perceived. The children’s inventions incorporated some conventional graph features, but rather than allow for a range of possibilities, they often represented only the actual data. They also often included illustrative features that were external to the graphing system.
Thinking of change
discretely
The changing speed of a
car is a continuous phenomenon.
The car cannot go from
motion to a stop without
going through every speed
in between. However, the
children reflected the
data they actually had,
and these data were always
discrete. Instead of showing
a continuum of speeds,
they made categories such
as "slow," "stop,"
and "fast." When
they watched a video of
a toy car moving, they
jotted down their observations
in terms of these categories.
Similarly, for the population
of a restaurant and of
their classroom, they used
categories such as "nobody,"
"few people,"
and "many people."
When data fell between
these categories, students
did not know how to include
them. For example, one
student, in trying to represent
a situation when one third
of the students were out
of the classroom, said,
"I don’t know
how to put lots [of students
in the room] but not as
many as lots."
Including figural
elements
The children made keys
for their different categories
-- for example, "ft"
or a zigzag line for fast,
"s" or a curved
line for slow, and "st"
or a picture of a stoplight
for stop. These keys were
a special example of the
children's use of figural,
or illustrative, elements.
In representing motion,
the students often included
drawings of perceptual
elements of the situation
(a toy car, a table, a
street) even if the illustrations
did not provide information
about the movement itself.
See Figure 1.
Figure 1
For changes in population,
drawings (usually stick
figures) were used more
informatively, for instance,
as headings in a chart
or icons in a pictogram.
Marking zero values
with special symbols
Many of the children explicitly
marked the zero, usually
in a manner distinctive
from the markings for other
values. More often than
not, children who made
bar graphs to show population
shifts throughout the day
drew a block for zero people
home. One boy put one block
for both zero and one person;
he then used a color key
to distinguish them. Another
child used one block to
show no one home, two blocks
to show one person home,
and so on. One girl used
one block for zero persons
home and two blocks for
two persons home; she was
happy with this "because
there is never one person
home." See Figure
2.
Figure 2
Students whose representations showed some continuity of motion treated a stop as a special kind of situation, not as a particular value on the continuum. Thus a child who increased the frequency of a wavy line to show slowing down, drew a dot to show a stop. Others used spaces, vertical lines, or illustrations of stoplights to depict a stop. See Figure 3 for two examples.
Figure 3
Omitting categories
that had no members
One child who made charts
had trouble symbolizing
"never." Because
she specified times in
intervals when certain
numbers of people were
home (i.e., 10:30-11:30),
she had no way of showing
that one person was never
home alone. She resolved
this dilemma first by using
0:00 to show that at no
time was there only one
person home. In her second
draft of the chart she
further resolved it by
omitting altogether the
category of one person
home. See Figure 4.
Figure 4
Many children left no place for more than the maximum number of people present or the maximum speed of the car in the particular situation they were illustrating. Many wrote times on their graphs only at exact times when people came and went, rather than at regular intervals. Furthermore, most children omitted evening and night hours on the graphs of population of their classroom when no one would be there, but included them on the graphs of population of their homes. When we asked some students why they had not listed the night hours, they inferred we meant there would be some people in the classroom then. One child asked her teacher how late she stayed at school; another asked whether the janitor would be present. See Figure 5.
Figure 5
Discussion
The development of any representation integrates:- The information to be conveyed
- The rules of the representational system
- The intended use of the representation
When faithfulness to the information to be communicated is the most critical determinant in the representation, we refer to it as data-driven. Most of the children’s examples we have described fall into this category. The children utilized elements of a system -- a well-defined set of symbols, the ordering of the symbols, and so forth -- but their systems were simple. They had only one layer of meaning (each symbol meant only one thing), and they were full of external elements such as figural depictions or illustrations that were not systematic. The children wanted to replicate the context they were representing. For them, their graph was complete if it illustrated all the data to be communicated. They were not concerned that the graph include all possible options.
On the other hand, when
the emphasis of the representation
is on using rules for coherence,
we refer to this as a system-driven
representation. In this
type of representation,
efforts are made to eliminate
elements that are not part
of the symbolic system
or that are not consistent
with the system’s
internal rules.
Each emphasis (data or system) has its own advantages. A data-driven representation:
- Facilitates communication when the system is unfamiliar to the reader. If we are not sure that the reader understands the representational system, it is helpful to resort to external elements such as keys and illustrations to provide additional cues.
- Preserves the information. A data-driven system keeps all information thought to be relevant, whereas one of the implications of a system-driven representation is that some pieces of information may be lost.
A system-driven representation:
- Helps us envision new possibilities and questions. A data-driven representation tends to reflect no more than what we already know. A system-driven representation allows us to perceive patterns and generalizations. A well-known example is the Periodic Table, where the system of listing chemical elements allowed scientists to recognize missing possibilities.
- Conveys more information with fewer symbols. A sophisticated system-driven representation is a network of tacit relationships that become part of the message.
In developing a representation we have to decide how to come up with a good combination of data- and system-driven elements. The bias toward a system- or data-driven representation is an issue of weighing the pros and cons of each. This is why we talk about the tension between them. Researchers report a developmental trend from data-driven to system-driven (Bamberger 1988; diSessa et al. in press; Ferreiro 1988, 1982; Karmiloff-Smith 1979). The research also stresses that this transition does not correspond simply to children’s failure to communicate using less systematic representations. Apparently children tend to adopt a more systematic approach even when their former practices were successful. Children eventually perceive that systematicity, internal consistency, "cleanness" or elimination of external elements, and avoidance of redundancies are advantageous to communication.
From a pedagogical point of view, teachers generally believe that conventional graphing methods have to be taught. For example, they tell students to mark time in regular hour or half-hour intervals. However, we have observed that students are typically introduced to a system-driven representation at a time when they still perceive it as a wrong or meaningless way of conveying information. They appropriate it in terms of a data-driven representation. We hope that a better understanding of spontaneous representations by children will help us to bridge this gap between children’s conceptions and what is required in a representational system.
Footnotes
1This study was conducted initially with eight students, and its observations were supported at a later date through work with an additional 45 fourth-graders.
References
Bamberger, J. (1988). Les structurations cognitives de l’appréhension et de la notation de rythmes simples. In H. Sinclair (Ed.), La Production de Notations Chez le Jeune Infant. Paris: Presses Universitaires de France.
diSessa, A. A., Hammer,
D., Sherin, B., and Kolpakowski,
T. (in press). Inventing
graphing: Meta-representational
expertise in children.
Journal of Mathematical
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Ferreiro, E. (1988). L’ecriture
avant la lettre. In H.
Sinclair (Ed.), La
Production de Notations
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Ferreiro, E. and Teberosky,
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Karmiloff-Smith, A. (1979). Micro- and macro-developmental changes in language acquisition and other representational systems. Cognitive Science, 3, 91-117.
Acknowledgements:
This article is an adaptation of a longer paper.
The research was supported by National Science Foundation grants Measuring and Modeling (MDR-8855644) and Mathematics: Investigations in Number, Data and Space (MDR-9050210).
Author Info:
Cornelia Tierney is a seventh-grade mathematics teacher and senior researcher at TERC on the project Mathematics: Investigations in Number, Data, and Space. Ricardo Nemirovsky is director of the TERC Measuring and Modeling project and is also developing elementary math curricula.
Other Articles by Cornelia C. Tierney:
Mathematics of Change: Ins and Outs or Ups and Downs
Patterns in the Multiplication Table
Other Articles by Ricardo Nemirovsky: