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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

answer questions where misleading graphs are used to dis-

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

chose to be misleading?

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)

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

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

tions of Misleading Graphs

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.

Using the NAEP grading rubric, students receive full credit

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

trash can had been doubled. The sample of seventh-grade

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

analyze. When the students were asked about which graph

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

were asked about Mayor McCann’s graph selection, fifteen

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

your students construct various “misleading” graphs to

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-

scribed in this article, will not only help them become

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.

Stop!

Don’t

Turn That

Page!

Take a moment to reflect on

what you’ve been reading in

the journal. What activities

or insights will you use in

your classroom, and when?

How can you share these ideas

with others? Pass this issue along

to a colleague or parent. Send a

letter to “Readers Write.” Tell oth-

ers how you use the journal.

Thanks for the moment! Go

ahead to the next page. . . .