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Tbe development of
one student's graph
interpretation skills
By Vanessa
R.
Pitts Bannister, Idorenyin Jamar, and Jomo W Mutegi
T
eaching students how to read and interpret graphs
is a challenge we continually face as science teach
ers,
but
it
is
an important
one.
By
helping students
Co
develop graphing skills,
we not
only strengthen
students' understanding
of
mathematics concepts
but
also help them recognize
the
importance
of
mathematical
models
in
science
and
show them
how
they
can be
used
meaningfully
in
science inquiry.
In this article,
we
examine
the
learning progress
of
one
fifthgrade student—Jelani—with regard
to the
develop
ment
of
her graph interpretation skills
as she
participated
in the Junior Science Institute (JSI),
a
twoweek, science
intensive summer camp
in
which participants engaged
in
microbiology' research
and
application.
By showcasingjelani's development of graphinterpreta
tion skills,
we
hope
to
make apparent some ofthe cogni
tive processes students
may go
through
as
they attempt
to
master this important inquiry' skill
and
thus provide fellow
teachers with insight
as to how to
more effectively help
develop these skills
in
their own students.
Initial Understandings
An important understanding
for
students
to
develop
is
"the idea that
a
mathematical model
has
both descriptive
and predictive power" (NCTM2000,p. 162). For instance,
when data
is
displayed
in
graphical form,
the
graph
not
only represents the actual obsen'ations (descriptive power)
but might also provide insight into patterns underlying the
relationships (predictive power).
In
this way,
it
informs
the
leamer about the phenomenon under investigation (Lein
hardt, Zaslavsky,
and
Stein 1990).
For
example, reading
a
line graph showing bacterial growth over a number of
days,
a student could learn that data values
can be
determined
within
and
outside
a set of
known values obtained during
an investigation. Students could discover data values
by
reasoning about relationships between known data values.
For example,
if
the data values
for
Days One, Three,
and
Five
are
10, 20,
and
30,
a
student could infer that the values
on Days Two, Four,
and Six
would
be
15, 25,
and
35.
To move students toward deeper understanding
of
graphs,
on the
first
day
ofthe program, participants were
presented with
an
exercise asking them
to
interpret
a
line
graph
to
determine
how
many times
a
student (Tamika)
e
1.
Jelani's responses
on an
initiol exercise.
500
4S0
400
«
3SO
ex
a
300
'a
250
•Huw Many TInivs Tnnilkn Jumped Rope
30 Science
and
Children
jumped rope at various time intervals (Figure
1).
This preas
sessment was intended to
give
camp mstnactors insight into
students' initial level ofunderstanding of graphing
skills.
It
was also intended to serve as an introductory instructional
tool to get students thinking about the role and use of line
graphs.
Reading the line graph, Jelani determined accurate val
ues for points that were indicated with dots (1, 5, and 10
minutes). Howe\'er, in attempting to figure out how many
jumps occurred in 3 minutes (a figure not indicated on the
graph),
Jelani's responses indicated that she didn't know
how to use the graph effectively.
Moving Forward
After the preassessment, students began the introductory^
lesson, "A Growthy Wodd.'' During this lesson, teachers
guided students as they grew bacteria cultures, measured
Fiaure
2.
A representation of Jelani's first graph.
Day One
ThursdayDay Two
FridayDay Three
Saturday
Days
Day Four
Sunday
their results over
a
period of several days, and recorded and
discussed their results using tables and line graphs. Tliey
recorded the measurements in their journals and held group
discussions centered on the following questions: "Where
can bacteria be found? How do we know they can be found
there''^
What
evidence do we have?
How many
bacteria would
you
guess
are in the nutrient
broth?
What do you think will
happen tomorrow?" Students were then asked to interpret
their graphs to determine the approximate measurements
for the days in which they did not record data (i.e.. Days
Three and Four).
Jelani first plotted the points for Days One, Two, and
Five.
She did not know how to determine the \aIues for
Days Three and Four, so she asked for assistance. Based
on Jelani's graph (Figure 2), the teacher and Jelani had the
following conversation;
Teacher; What do you think the \'alues
for Days Three and Four
will
be?
Where do you think the points
may lie?
Jelani; Day Four..., (she paused)
Teacher: Why don't you draw the line
first? (The intent here was to help
Jelani observe that the line graph
serves as a tool to determine
values within the set of known
data values (i.e., Days Three and
Four) and (eventually) values
outside the known values such
as Days Six and Seven.)
.\fter Jelani connected the data values
for Days One, Two, and Five to draw
her line graph (Figure 3), the discussion
continued as follows:
Day Five
Manday
A representation of Jelani's second graph.
90
80
70
60
50
40
30
20
10

Day One
ThursdayDay Two
FridayDay Three
Saturdoy
Dayi
Day Four
Sunday
<V
Day Five
Monday
Teacher; Looking at your line graph, can
you answer now, where v\'ould
the points for Day Three and
Day Four be on the line?
Jelam; (says softly) Day Four would be
64.
(The teacher assumed that
this answer
was
a guess, because
if reading the graph correctly, an
appropriate range for Sunday—
Day Four would be 1119.)
Teacher; Day Four will be 64?,. .But 64 is
not on the line.
Jelani: How come is it more? (At this
point,
it's
clear that Jelani doesn't
realize the line is a continuous
display and that she has to guess
the answer.)
October 2007 31
Line Graph Learning
Teacher: Let's look at the known values for your bacteria
(the teacher points to the values for Days One,
Two.
and Five on the line graph). It went from
63 to 75 and then to
81,
right? And Days Three
and Four
are
between days Two and
Five,
so let's
draw a point at Day Three.
Jelani: Right here? (Jelani pointed to
a
point that was
not on the line.}
Teacher; But that's not on the line. This line
we
just drew
is called the curve. (For emphasis, the teacher
traced Jelani's line graph with a pencil.)
It was clear that Jelani still did not understand that the
line she drew
was
the curve and that the curve could be used
to estimate \'alues for Days Three and Four. However,
as
the conversation continued, Jelani began to consider the
predictive nature ofthe line graph.
Teacher: What is the percent going to be on Day Three?
Just give me a guess—between what numbers?
(The intent here was to determine whether she
could provide an appropriate range—thus read
the graph,)
Jelani: Between 75 and 80.
Teacher; Okay, how about for Day Four?
Jelani; Between 78 and 79.
It appeared from Jelani stating fas she pointed to the ap
propriate point on the line graph) an appropriate range of
values for Day Three that she had begun to realize that a
line could be used to determine an appropriate
value.
Aft:er
plotting appropriate points for Days Three and Four on the
line graph. Jeiani abruptly said, "I thought they just made
lines,
I didn't know you could read them."
Deepening Understanding
Toward the end ofthe two weeks, students conducted an
exercise that required them to apply their
skills
reading line
graphs to detennine how much bacteria "Sam the Genn Ex
pert"
grew duringa week.
"Sam"
collected data on Monday,
Wednesday, and Friday, thus the line graph (curve) served
as
a
means to approximate values on days when he did not
collect data (Figure 4). Based on Jelani's response of
"20"
for Tuesday, she continued to exhibit an understanding of
the following:
Since the
data
is
displayed using
a
line graph,
although data was not collected for Tuesday,
a value
can be
determined by using
a
line graph.
Through these learning experiences, students both
learned scientific methods of data coUection and analysis
and explored relationships between quantities. Such ex
plorations encouraged students to come to grips with the
descriptive and predictive natures of graphical displays.
Jelani's burgeoning understanding of a line graph high
lights the potential of bringing mathematical ideas alive
in the context of science activities while emphasizing the
importance of considering students' initial and developing
conceptions.
Vanessa R. Pitts Bannister {pittsbannister@vt.edu)
is an assistant professor of mathematics education
at Virginia Tech in Blacksburg, Virginia, ldorenyin
Jamar (jamar@pitt.edu) is
a
fellow ofthe Institute
for Learning at the University of Pittsburgh Learning
Research and Development Center (LRDC) in Pitts
burgh,
Pennsylvania. Jomo W. Mutegi (jmutegi@
sankoreinstitute.org) is the executive director ofthe
Sankore Institute in
Cleveland,
Ohio.
References
Leinhardt, G., O. Zaslavsky, and M.K. Stein. 1990. Func
tions,
graphs, and graphing:
Tasks,
learning, and teaching.
Review of Educational Research 60(11): 164.
National Council
of
Teachers
of
Mathematics (NCTM),
2000.
Principles
and
standards
for
school mathematics.
Reston, VA: Author,
National Research Council (NRC).
1QQ6.
National
science
educa
tion standards. Washington, DC: National Academy Press.
Fiaure 4.
Jelani's responses on a final exercise.
eo
7O
BO
5O
IO
30
Sam's £. CoU Farm
Mon Tues
Wed
Thurs
Fri Sal Sur
Day
Connecting to the Standards
This article relates to the following National
Science
Education Standards
(NRC 1996).
Science Education Progrann Standards
Standard
C
The science program should
be
coordinated with
the mathematics program
to
enhance student use
and understanding
of
mathematics in the study
of
science and
to
improve student understanding
af
mathematics.
32 Science and Children