ArticlePDF Available

Line Graph Learning

Authors:

Abstract

In this article, the learning progress of one fifth-grade student is examined with regard to the development of her graph interpretation skills as she participated in the Junior Science Institute (JSI), a two-week, science intensive summer camp in which participants engaged in microbiology research and application. By showcasing the student's development of graph interpretation skills, the authors hope to make apparent some of the cognitive 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 develop these skills in their own students.
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
fifth-grade student—Jelani—with regard
to the
develop-
ment
of
her graph interpretation skills
as she
participated
in the Junior Science Institute (JSI),
a
two-week, 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 11-19.)
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): 1-64.
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
... Veri temsil şekilleri arasında grafik, tablo, sözlü ifade ve sembolik gösterim bulunmaktadır. Bunlar arasında grafik, hem günlük yaşamda sıklıkla karşılaşılan (Zacks, Levy, Tversky, & Schiano, 2002) hem de matematik, fen bilimleri ve sosyal bilgiler gibi farklı disiplinlerin anlamlandırılmasında önemli görülen bir temsil türüdür (Bannister, Jamar, & Mutegi, 2007). Bu çalışma, grafik türlerinden çizgi grafiği üzerine odaklanmıştır. ...
... Grafiklerin matematikte önemli bir temsil türü olmasına karşın, öğrenciler grafikleri anlamlandırmakta ve diğer temsil türleri ile ilişkilendirmekte zorluklar yaşamaktadırlar (Bannister, Jamar, & Mutegi, 2007;Glazer, 2011;Leinhardt, Zaslavsky & Stein, 1990;Li & Shen, 1992). Öğrencilerin kavramakta zorluklar yaşadığı grafik türlerinden birisi çizgi grafiğidir (Bell, Brekke, & Swan, 1987;Wainer, 1980). ...
... "Yağmur aynı hızla yağmaya devam ederse, 25. dakikada kovada ne kadar su birikir?" sorusu verilerin ötesini okuma düzeyindedir. Öğrencilerin grafik yorumlama becerilerini inceleyen araştırmalar, öğrencilerin grafikte açıkça işaretlenen noktaları kolay okuyabildiklerini, ancak ara değer veya tahmin gerektiren değerlere daha zor karar verdiklerini ortaya koymaktadır (Bannister, Jamar, & Mutegi, 2007;Curcio, 1987;Glazer, 2011). ...
Full-text available
Article
ÖZ Bu çalışmada, ortaokul öğrencilerinin çizgi grafiğini yorumlama ve oluşturma becerilerini değerlendirmede kullanılabilecek bir ölçme aracı geliştirilmiştir. Geliştirilen bu ölçme aracı 7 şubede bulunan toplam 166 yedinci sınıf öğrencisine uygulanmıştır. Ölçme aracında bulunan soruların madde analizi için, düzeltilmiş madde toplam korelasyonu, Klasik Test Kuramı ile hesaplanan ayırtedicilik ve Madde Tepki Kuramı ile hesaplanan ayırtedicilik değerleri kullanılmıştır. Doğrulayıcı Faktör Analizi kullanılarak ölçme aracının yapı geçerliği test edilmiştir. Bu analizler sonucunda son hali verilen ölçme aracı, 14 sorudan oluşmaktadır. Bulgular, ölçme aracının geçerli ve güvenilir olduğunu ve maddelerinin ayırtedicilik değerlerinin yüksek olduğunu göstermiştir. Ayrıca, ölçme aracının, grafik bilgi ve becerilerini dört boyutta (verileri okuma, veriler arasını okuma, verilerin ötesini okuma ve grafik oluşturma) ölçtüğü doğrulayıcı faktör analizi ile gösterilmiştir. Geliştirilen ölçme aracı, ortaokul öğrencilerinin çizgi grafiğini yorumlama ve oluşturma ile ilgili başarı düzeylerini genel olarak ve her alt boyut için ayrı ayrı belirlemekte ve öğrencilere bu boyutlar için geri dönüt vermekte kullanılabilir. Anahtar kelimeler: Grafik yorumlama, grafik oluşturma, test geliştirme, klasik test kuramı, madde tepki kuramı, doğrulayıcı faktör analizi. Assessing Line Graph Comprehension and Construction Skills ABSTRACT In this study, a line graph comprehension and construction test for middle grade students has been developed. The test was administered to 166 seventh grade students in 7 classrooms. Item analysis of the test items was conducted through corrected item total correlation, discrimination index calculated by Classical Test Theory and by Item Response Theory. Confirmatory factor analysis was used to evaluate the construct validity of the test. In the final form, the test had 14 items. Findings showed that the test was reliable and valid, and the discrimination indexes of the items were high. Also, confirmatory factor analysis showed that the test measured graphing skills under 4 dimensions: read the data, read between the data, read beyond the data, and graph construction. The test can be used to measure middle grade students’ line graph comprehension and construction skills and to provide feedback related to subdomains of graphing skills. Keywords: Graph comprehension, graph construction, test development, classical test theory, item response theory, confirmatory factor analysis.
Article
Learning and analyzing graph data is one of the most fundamental research areas in machine learning and data mining. Among numerous graph‐based data structures, this paper focuses on a graph bag (simply, bag), which corresponds to a training object containing one or more graphs, and a label is available only for a bag. This type of a bag can represent various real‐world objects such as drugs, web pages, XML documents, and images, among many others, and there have been many researches on models for learning this type of bag data. Within this research context, we define a novel problem of dynamic graph bag classification, and propose an algorithm to solve this problem. Dynamic bag classification aims to build a classification model for bags, which are presented in a streaming fashion, ie, frequent emerging of new bags or graphs over time. Given such changes made to the bag dataset, our proposed algorithm aims to update incrementally the top‐m most discriminative features instead of searching for them from scratch. Incremental gSpan and incremental gScore are proposed as core parts of our algorithm to deal with a stream of bags efficiently. We evaluate our algorithm on two real‐world datasets in terms of both feature selection time and classification accuracy. The experimental results demonstrate that our algorithm derives an informative feature set much faster than the existing one originally designed for targeting static bag data, with little accuracy loss.
Article
This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.