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# Students’ Interpretations of Misleading Graphs

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## Abstract and Figures

Since graphical displays of data are increasingly used in magazines, newspapers and on television to communicate relationships among numerical data, it is important to expose middle school students to various types of contexts where interpretation of graphical data is necessary. The selected problems focus on statistics and data analysis, primarily concerning the use and misuse of statistics in our everyday lives. Students were expected to use statistical concepts to analyze and communicate valid interpretations of the graphical and tabular forms of data from a variety of real-world contexts. Topics: Students, Mathematical problems, Mathematics education, Statistics, Crime rates, Mayors, Mathematics teachers, Pictographs, Numbers, Political campaigns
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SINCE GRAPHICAL DISPLAYS OF DATA ARE
increasingly used in magazines, in newspapers,
and on television to communicate relationships
among numerical data, it is important to expose
middle school students to various types of contexts where
interpretation of graphical data is necessary. The NCTM
(2000) concurs that students in grades 6–8 should “begin
to compare the effectiveness of various types of displays in
organizing the data for further analysis or in presenting the
data clearly to an audience” (p. 49).
Friel, Curcio, and Bright (2001) distinguish between
graph comprehension—“the graph readers’ abilities to de-
rive meaning from graphs created by others or by them-
selves”—and behaviors that seem to demonstrate a pres-
ence of graph sense (p. 132). One characteristic of students’
having graph sense is of particular interest: “To recognize
the components of graphs, the interrelationships among
these components, and the effect of these components on
the presentation of information in graphs” (p. 146). In How
to Lie with Statistics, Huff (1954) speaks freely about the
ease with which the creator of a graph can manipulate the
viewer’s interpretation of the data. By asking students to
play data, I hope to further understand students’ percep-
tions and interpretations of graphical displays of data (see
figs. 1, 2, and 3). I invite you to try these problems with
your students to see whether their performance reveals
some interesting results.
340 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Students’ Interpreta
S U Z A N N E R. H A R P E R
SUZANNE HARPER, harpersr@muohio.edu, is an assistant pro-
fessor of mathematics education at Miami University in Oxford,
Ohio. Her research interests include the appropriate use of tech-
nology to teach K–12 mathematics, the content knowledge of
prospective mathematics teachers, and the teaching and learn-
ing of geometry.
Edited by JANE KEISER, keiserjm@muohio.edu, a mathematics
educator in the Department of Mathematics and Statistics, Miami
University, Oxford, OH 45056
“Take Time for Action” encourages active involvement in research
by teachers as part of their classroom practice. Readers interested
in submitting manuscripts pertaining to this theme should send
them to “Take Time for Action,” MTMS, NCTM, 1906
Association Drive, Reston, VA 20191-1502.
Fig. 1 Problem 1: Average minimum temperature in San Francisco,
California
The average minimum
monthly temperatures for
San Francisco, California,
have been recorded in the
table, shown at right. The
data have been correctly
represented by both
graphs A and B.
Graph A
Graph B
a) Which graph would be best to help convince others
that the average minimum temperature in San Fran-
cisco is much colder in January than in August? Ex-
plain your reason for making this selection.
b) Why might people who thought that there was little
difference between the average minimum tempera-
ture in January and August consider the graph you
MONTH °F
January 45.6
February 48.0
March 48.9
April 49.8
May 51.1
June 52.9
July 53.6
August 54.5
20.0
30.0
40.0
50.0
60.0
70.0
80.0
Month
Average Min Temp
January
February
March
April
May
June
July
August
45.0
47.0
49.0
51.0
53.0
55.0
Month
Average Min Temp
January
February
March
April
May
June
July
August
SOURCE: ADAPTED FROM AN EIGHTH-GRADE NAEP TEST ITEM, BLOCK: 1996-8M12, NUMBER 9 (NAEP 2002)
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Discussion of the Problems
THE SELECTED PROBLEMS FOCUS ON STATISTICS AND
data analysis, primarily concerning the use and misuse of
statistics in our everyday lives. Students were expected to
use statistical concepts to analyze and communicate valid
interpretations of the graphical and tabular forms of data
from a variety of real-world contexts.
Although viewing data graphically allows the reader to see
the trend of the data easily, the choice of scaling along the x- and
y-axis of the graph could influence their interpretation of the
data. For example, a question similar to problem 1 was adminis-
tered on the 1996 National Assessment of Educational Progress
(NAEP) eighth-grade exam (see fig. 1). Only 2 percent of these
students answered the question correctly by identifying this in-
formation: (1) Graph B is the appropriate choice, since it ap-
pears to show a large increase between the smallest and largest
data point; and (2) graph B is misleading since the y-axis scale
exaggerates a relatively small increase. Of the students, 19 per-
cent partially answered the question correctly by identifying
graph B with an incomplete or partially correct explanation. The
remaining students either incorrectly or minimally answered
the question or omitted the test item altogether.
VOL. 9, NO. 6 .FEBRUARY 2004 341
Fig. 2 Problem 2: United States trash production
Study the pictograph below.
a) Write 1–2 sentences about the information displayed
in the graph above.
b) The pictograph shown above is misleading. Explain
why.
Mayor McCann is running for a second term as mayor against the challenger, Representative Royce. One of the hot topics
of the campaign is crime prevention. Each graph displays the number of crimes per 1000 citizens for the four years of
Mayor McCann’s current term.
a) What are the differences between the two graphs?
b) If you were Representative Royce’s campaign manager, which graph would you choose to use in the campaign? Why?
c) If you were Mayor McCann’s campaign manager, which graph would you choose to use in the campaign? Why?
Crime Rate Graph A
23
24
25
26
27
28
29
1999 2000 2001 2002
Year
Crime Rate
Crime Rate Graph B
051015 20 25 30
1999
2000
2001
2002
Crime Rate
Year
Fig. 3 Problem 3: Crime rate of Metropolis
THE UNITED STATES
IS PRODUCING MORE TRASH
Millions of Tons of Trash
80 Million Tons
160 Million Tons
200
100
0
1960 1980
SOURCE: EIGHTH-GRADE NAEP TEST ITEM, BLOCK: 1992-8M15, NUMBER 17 (NAEP 2002)
SOURCE: ADAPTED FROM MUSSER, BURGER, AND PETERSON 2003, P. 457
Problem 2 was also used on a NAEP eighth-grade exam
in 1992 (see fig. 2). From the posted numerical values, we
see that in 1960 the United States was producing half as
much trash as it did in 1980. However, the volume of the
trash-can icons does not appear to be in a proportion of 1:2
as the numerical data indicates. It appears that both the ra-
dius and the height of the 1960’s trash can have doubled,
causing the ratio of the volume of the cylinders to be 1:8.
for their solution if they indicate that (1) both width and
height have been doubled; (2) the trash can in 1980 holds
much more than twice the amount of the 1960 trash can; or
(3) the ratio of the amount in the 1960’s can to the amount
in the 1980’s can is less than 1:2. On the NAEP exam, only
8 percent of student responses were correct, 86 percent
were incorrect, and 6 percent omitted the problem.
Although the first two problems were chosen to illus-
trate specific characteristics of misleading graphs, the
third problem was selected to explore various reasons why
certain groups of people would display data differently
(see fig. 3). As Kosslyn (1994) describes, “A good graph
forces the reader to see the information the designer
wanted to convey” (p. 271). By having Mayor McCann’s
campaign manager use graph A, which shows a drastic
drop in the crime rate, the desired effect to which Kosslyn
refers is achieved.
Results of Students’ Responses
THREE CLASSES WITH A TOTAL OF 59 SEVENTH-GRADE
prealgebra students were asked to respond to the prob-
lems. (Summary tables characterizing the “correctness” of
the students’ answers are displayed in tables 1, 2, and
3.) All the students were taught by the same teacher who
recently completed a three-week data analysis unit. The
unit’s objectives were (1) to analyze a set of data using and
comparing combinations of measures of central tendency
(mean, mode, median) and measures of spread (range,
quartile, interquartile range) and describe how the inclu-
sion or exclusion of outliers affects those measures; and
(2) to read, create, and interpret box-and-whisker plots,
stem-and-leaf plots, and other types of graphs.
In problem 1, all but a single student (who omitted the
question) correctly identified graph B as being an appropri-
ate display to convince others that the average minimum
temperature in San Francisco is much colder in January
than in August. However, the vast majority of students did
not provide valid reasoning for their selection or explain
why graph B could be interpreted as misleading, which
classified their responses as partially correct (see table 1).
A typical response from this group of students reads, “On
graph A, I really can’t see any changes in temp so I
wouldn’t be able to tell the difference in temp between Jan-
uary and August. On graph B, you can really tell because it
is spaced out and more clear.” Even though the student de-
scribed the physical characteristics of the graph, he failed
to mention how the scale or range of numbers along the y-
axis changed the appearance of the graph. Just over one-
fourth of students (27.1%) responded that some aspect of
the scale along the y-axis had changed. One student cor-
rectly reasoned why graph B might be considered mislead-
ing by responding, “There was little difference [between
the temperatures]. They might think that graph B is mis-
leading because it shows a big change.” This student’s ex-
planation indicates his understanding of the numerical dif-
ference in temperature from January to August and the
effects the different scales have along the y-axis.
In response to problem 2, 83 percent of students incor-
rectly justified why the graph was misleading (see table 2).
Of these incorrect responses, only four mentioned the rela-
tive size of the trash cans. Sixteen students addressed either
the scaling of the horizontal or vertical axes as being the
graph’s misleading attribute. One student concentrated on
the numerical values along the y-axis, “It is misleading be-
cause there are no numbers in between the 0–100 and
100–200 to tell exact measurements.” Another answer fo-
cused on the x-axis scaling, “This pictograph is misleading
because it should have went by years like 1960, 1970, 1980
besides 1960 then 1980.” A couple of students thought that
the type of trash can was misleading readers to think that 80
342 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Table 1
NUMBER OF PERCENTAGE
PROBLEM 1STUDENTS OF STUDENTS
Correct 7 11.9%
Partially Correct 34 57.6%
Minimally Correct 17 28.8%
Omitted 1 1.7%
Table 2
NUMBER OF PERCENTAGE
PROBLEM 2S
TUDENTS OF STUDENTS
Correct 4 6.8%
Incorrect 49 83.0%
Omitted 6 10.2%
Table 3
N
UMBER AND
N
UMBER AND
N
UMBER AND
P
ERCENTAGE
P
ERCENTAGE
P
ERCENTAGE
OF
S
TUDENTS OF
S
TUDENTS OF
S
TUDENTS
W
HO
C
HOSE
W
HO
C
HOSE
W
HO
O
MITTED
P
ROBLEM
3G
RAPH
AG
RAPH
BQ
UESTION
Royce’s 24 25 10
Manager (40.7%) (42.4%) (16.9%)
McCann’s 33 11 15
Manager (55.9%) (18.7%) (25.4%)
VOL. 9, NO. 6 .FEBRUARY 2004 343
million tons of trash would physically fit in that type of can.
Only four students produced correct responses, such as
“The picture of the trash can in 1960 is supposed to be half
of the trash can in 1980, but it’s much more than half.” It was
interesting to find that none of the students’ responses indi-
cated they noticed that both width and height of the 1960
student responses to both problems 1 and 2 remarkably par-
alleled those of the reported 1992 and 1996 NAEP results.
The responses to problem 3 were the most interesting to
Representative Royce should use in his campaign, they re-
sponded with no clear favorite (see table 3). Approximately
41 percent favored graph A, 42 percent favored graph B, and
17 percent did not respond to the question. Of the students
who selected graph A, the majority of explanations were
based on the look of the graph: “It’s more professional and
accurate,” “. . . easier to read,” “It has less amount of crime
rate,” and “. . . shows the difference in the number.” In com-
parison, the responses of students who chose graph B were
more likely to concentrate on the candidate’s campaign to
win the election. Their reasons were primarily politically
motivated: “We don’t want him [McCann] to win. And that
[graph] doesn’t show much information”; “It shows the
crime rate going up and needs to be fixed”; and “It doesn’t
really look like the rate has gone down.” When the students
students who preferred the line graph for Royce also chose
to use the line graph for McCann, almost all citing the exact
same physical reasons as before. However, of those stu-
dents who responded to the question, 75 percent over-
whelmingly selected graph A. Most students agreed that
the crime rate “looks like it drops a lot more” in graph A,
helping Mayor McCann secure his bid for reelection.
Implications for Teaching
ALTHOUGH MOST MIDDLE SCHOOL STUDENTS HAVE
learned how to read and extract information from various
types of graphs, it might not be obvious to them how par-
ticular graphs can be misleading or inaccurate. In this
study, many students noticed the graphs’ axis scaling as a
misleading feature; however, they were not as keen to ob-
serve pictorial embellishments or consider the source of
the graph. A possible classroom idea would be to have
communicate information they wish to convey on a partic-
ular issue. Students could discuss the different messages
that each graph provides. Having students study graphs in
current magazines and newspapers, like the problems de-
more “statistically literate” but also aware of the possible
political agenda in some media sources. Finding informa-
tion about the individuals who have created a graph and
the target audience of the graph can lead to some insight-
ful classroom discussions and conclusions.
References
Friel, Susan N., Frances R. Curcio, and George W. Bright. “Mak-
ing Sense of Graphs: Critical Factors Influencing Comprehen-
sion and Instructional Implications.” Journal for Research in
Mathematics Education 32 (March 2001): 124–58.
Huff, Darrell. How to Lie with Statistics. New York: W. W. Norton,
1954.
Kosslyn, Stephen M. Elements of Graph Design. New York: W. H.
Freeman & Co., 1994.
Musser, Gary L., William F. Burger, and Blake E. Peterson.
Mathematics for Elementary Teachers: A Contemporary Ap-
proach. New York: John Wiley & Sons, 2003.
National Council of Teachers of Mathematics (NCTM). Princi-
ples and Standards for School Mathematics. Reston, Va.:
NCTM, 2000.
National Assessment of Educational Progress (NAEP). The Na-
tion’s Report Card Sample Questions Tool Version 2.0. Wash-
ington, D.C.: National Center for Educational Statistics, 2002.
April 2, 2003. nces.ed.gov/nationsreportcard/itmrls/.
The author would like to recognize the seventh-grade students in
Cincinnati, Ohio, who participated in this study and to express grati-
tude to their teacher for taking time out of her busy schedule to
accommodate her.
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ers how you use the journal.
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Mathematics for Elementary Teachers: A Contemporary Approach National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics National Assessment of Educational Progress (NAEP). The Nation's Report Card Sample Questions Tool Version 2.0
• Gary L Musser
• F William
• Blake E Burger
• Peterson
Musser, Gary L., William F. Burger, and Blake E. Peterson. Mathematics for Elementary Teachers: A Contemporary Approach. New York: John Wiley & Sons, 2003. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. National Assessment of Educational Progress (NAEP). The Nation's Report Card Sample Questions Tool Version 2.0. Washington, D.C.: National Center for Educational Statistics, 2002. April 2, 2003. nces.ed.gov/nationsreportcard/itmrls/.