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Comprehension of Mathematical Relationships Expressed in Graphs Author(s): Frances R. Curcio Source: Journal for Research in Mathematics Education, Vol. 18, No. 5 (Nov., 1987), pp. 382-393 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/749086 Accessed: 19-04-2018 13:51 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/749086?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms

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Journal for Research in Mathematics Education 1987, Vol. 18, No. 5, 382-393

COMPREHENSION OF MATHEMATICAL RELATIONSHIPS EXPRESSED IN GRAPHS FRANCES R. CURCIO, Queens College of the City University of New York In this study, the schema-theoretic perspective of understanding general discourse was

extended to include graph comprehension. Fourth graders (n = 204) and seventh

graders (n = 185) were given a prior-knowledge inventory, a graph test, and the SRA Reading and Mathematics Achievement Tests during four testing sessions. The unique predictors of graph comprehension for Grade 4 included reading achievement, mathematics achievement, and prior knowledge of the topic, mathematical content, and form of the graph. The unique predictors for Grade 7 were the same except that prior knowledge of topic and graphical form were not included. The results suggest that children should be involved in graphing activities to build and expand relevant schemata needed for comprehension.

Processing information in our highly technological society is becoming more and more dependent upon a reader's ability to comprehend graphs. Although a literal reading of data presented in graphical form is an impor-

tant component of graph-reading ability, the maximum potential of the graph is actualized when the reader is capable of interpreting and generalizing from the data presented (Kirk, Eggen, & Kauchak, 1980). The results of the Third National Assessment of Educational Progress indicated that 9and 13-year-olds have difficulty with high-level thinking skills such as interpreting graphs and drawing conclusions (Lindquist, Carpenter, Silver, & Matthews, 1983). The advent of a new cognitive perspective for explaining reading com-

prehension-schema theory (Adams & Collins, 1977; Smith-Burke,

1979)-may reveal some of the reasons students experience difficulty with reading graphs. Although an extensive amount of research analyzing comprehension of story-structured material has been done, how other types of

text are processed has not received much attention and should be examined (Kintsch, 1977; Sticht, 1977). An exploration of how schema theory is related to reading mathematics (Silver, 1979) and mathematical understanding (Greeno, 1978) is warranted.

This study was supported by grants from the National Institute of Education (Grant

No. NIE-G-80-0093) and St. Francis College. It is based on the author's doctoral

dissertation, completed at New York University in 1981 under the direction of Edward M. Carroll. Thanks are due to M. Trika Smith-Burke and Robert G. Malgady, New York University, and Stephen M. Kosslyn, Harvard University, for their com-

ments on an earlier version. Also, thanks to the anonymous reviewers for their

suggestions.

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383

SCHEMA-THEORETIC FRAMEWORK

A knowledge of the topic, content, and form of general discourse depen

upon the amount of previous meaningful exposure to the topic, cont and form of the discourse the reader has had. This exposure contribut the development, revision, modification, and editing of related schema The topic of general discourse is usually identified by its title and as

the reader in retrieving from memory relevant familiar information per

ing to the passage to aid in comprehension (Bransford & Johnson, 1 Sjogren & Timpson, 1979). The topic of a graph, which is identified by title, labels on axes, and key vocabulary words used in the title and la

may be one of the factors that requires prior knowledge for comprehend

the mathematical relationships expressed in the graph (Culbertson & P ers, 1959; Harper & Otto, 1934; Washburne, 1927).

The content of text material is the relationship between words and idea

the familiarity of which allows the reader to recognize, for example, a ca

effect relationship (Pearson & Johnson, 1978). The mathematical cont of a graph, which is the number concepts, relationships, and fundam operations contained in it, is a second factor about which prior know seems to be necessary for comprehension (Goetsch, 1936; Thomas, 1 Vernon, 1952). The form of a reading passage is its structure or framework, which ploys certain conventions. Knowledge about these conventions allows reader to make predictions and impose certain expectations about the (Royer & Cunningham, 1978). The form or type of graph, such as a graph, bar graph, pictograph, or circle graph, appears to be a third f about which prior knowledge is necessary (Janvier, 1978; MacDonald-R 1977). (For an extensive discussion of topic, content, and form, see Cu 1981a, 1981b.)

The purpose of this study was to extend the schema-theoretic perspecti

of understanding general discourse to graph comprehension by exami

the effect of prior knowledge on the ability to comprehend the mathema

relationships expressed in graphs. Because reading achievement and m ematics achievement are general predictors of success in school-rela tasks, these were also examined to determine whether prior knowle

contributes significantly to the ability to comprehend graphs over and ab

the contribution of reading and mathematics achievement. The study focused attention on the performance of fourth and sev graders. By the fourth grade, most of the elementary work with gr should have been accomplished, and children should have achieved a su cient command of reading and arithmetic skills, the tools of learning essary for reading graphs (Strickland, 1938/1972). By the seventh gra expected that growth and achievement in graph-reading ability would occurred (Bamberger, 1942), and any sex-related differences would be

ifested (Armstrong, 1975; Callahan & Glennon, 1975; Fennema, 1

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384

Graph

Comprehension

1977; Maccoby, 1966; Suydam & were also examined to determine when

confronting

The

research

1.

prior

Is

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questions

knowledge

mater

were

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m

related to comprehending the m graphs independent of mathemat 2. What is the optimal linear c matics

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kno

and of graphical form in predic relationships expressed in graphs 3. Is sex related to comprehendi in

graphs? METHOD

Subjects

The sample, restricted to native speakers of English, consisted of 204 fourth graders (101 boys and 103 girls) and 185 seventh graders (102 boys and 83 girls) from four elementary schools, two junior high schools, and one K-8 school located in a stable, middle-class community in 1 of the 32 New York City School Districts. Native English-speaking children were selected so that inability with language would not be a confounding factor. The superintendent and principals expressed interest and granted permission for the study to be conducted in their schools. As required by the public schools, parents gave written consent for their children to participate in the study. Variables

Graph comprehension was measured by a researcher-designed Graph Test composed of twelve graphs: three bar graphs, three circle graphs, three line

graphs, and three pictographs. Six multiple-choice items were constructed for each graph. The six items reflected three tasks of comprehension: two questions were literal items (requiring a literal reading of the data, title, or

axis label); two questions were comparison items (requiring comparisons and the use of mathematical concepts and skills to "read between the data"); and two questions were extension items (requiring an extension, prediction,

or inference to "read beyond the data"). One of the graphs with its six comprehension questions is presented in Figure 1. (For a detailed description

of test construction and the instruments used in the study, see Curcio, 1981a, 1981b.) Prior to the study, the reliability of the Graph Test was estimated. Seventyfive fourth graders and 67 seventh graders from a K-8 school in the school district of the main study were given the test in the spring of 1980. All the

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386

Graph

children

were obtained.

Comprehension

were

native

speakers

of

Prior knowledge was measured by a researcher-designed Prior Knowledge

Inventory, consisting of three subtests (Topic, Mathematical Content, and Graphical Form). The inventory was designed to match the topic, mathematical content, and graphical form of each of the twelve graphs. Items that

match the graph in Figure 1 can be found in Table 1. Prior to the study, the reliability of the Prior Knowledge Inventory was

estimated. Sixty-seven fourth graders from the K-8 school in the district were given the inventory during the spring of 1980. A KR-20 reliability coefficient of .97 was obtained. The subtests (Topic, Mathematical Content, and Graphical Form) had reliabilities of .93, .97, and .93. Seventy-three seventh graders were given the same test and a KR-20 reliability coefficient

of .96 was obtained. The subtests had KR-20 reliabilities of .90, .95, and .86.

Reading and mathematics achievement were measured by Level D and Level F (for fourth and seventh grades, respectively) of the Reading and Mathematics tests of the SRA Achievement Series (Naslund, Thorpe, & Lefever, 1978). Sex was measured by a dichotomous variable (0 for boys; 1 for girls). Procedure

Nineteen seniors in a college teacher-training program were recruited and

trained as test proctors. Each had completed a course in tests and measurements and attended one orientation session to insure that uniform testing conditions and procedures were followed. During the fall of 1980, each of the four tests was administered by the proctors during one of four testing sessions. Data on sex and native-English-

speaking status were collected on the cover sheet of the Prior Knowledge Inventory, the first test given.

Data Analysis Correlational and multiple regression analyses were computed by grade. To avoid having the results confounded with other cognitive components (S. M. Kosslyn, personal communication, 21 January 1981), second-order partial correlations were computed to determine the unique contribution of

prior knowledge of topic, of mathematical content, and of graphical form to graph comprehension, partialing out reading and mathematics achievement. First-order partial correlations of graph comprehension with reading

and mathematics achievement, controlling for mathematics and reading achievement, respectively, were also computed. These coefficients can be found in Curcio (1981a, 1981b).

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Frances

R.

Curcio

387

Table 1

Sample Items From the Prior Knowledge Inventory

Prior Knowledge of Topic 1. What does "height" mean in the following sentence? In school today, the teacher measured Tommy's height. a. How much Tommy weighs b. How old Tommy is c. How smart Tommy is d. How tall Tommy is 2. How can we determine who is the tallest in the class?

a. b. c. d.

By By By By

eating a lot of food sleeping the most standing next to one another dressing properly

Prior Knowledge of Mathematical Content 3. Which of the following is a correct statement? a. 1 centimeter is greater than 1 inch b. 1 inch is less than 1 centimeter

c. 1 inch equals 1 centimeter

d. 1 centimeter is less than 1 inch 4. 105 - 85= a. 20

b. 25 c. 80 d. 190

Prior Knowledge of Graphical Form Use the following picture to answer questions 5 and 6:

100-

25-T

A

5.

B

C

D

Which

repre

a. A b. B c. C

d. D

6. What amount does C represent? a. 0

b. 75 c. 100

d. 125

Sex, reading achievement, and mathematics achievement were controlled in the regression analysis to determine whether prior knowledge contributed to the prediction of graph comprehension independently of any of the pre-

viously entered measures. After sex had been entered, reading and mathe-

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3

8

8

Graph

matics

Comprehension

achievement

were

entered

se

bution of each. Then, because the support entering one of the varia together.

as

a

set,

Before

each

the

three

component

compone

was

ente

contribution. RESULTS

The means and standard deviations for raw-score data of the variables

by grade are reported in Table 2. As expected, the seventh graders, being older and, in general, having had more experiences to build and expand a knowledge base, outperformed the fourth graders on the researcher-designed tests. Because different levels of the SRA Achievement Series (i.e., Levels D and F) were administered and raw scores used in the analysis, no comments can be made to compare reading and mathematics achievement across the grades. Table 2

Means and Standard Deviations for All Variables by Grade Grade 4 Grade 7

(n = 204) (n = 185) Variable

Topic

27.80

M

SD

10.33

M

39.41

SD

5.54

Content 34.68 14.00 51.42 6.31 Form 16.14 5.97 21.91 3.48

Reading achievement 40.61 9.45 53.58 14.66

Mathematics achievement 47.35 13.18 34.09 11.74

Graph comprehension 30.73 11.07 41.38 11.28

Relation of Prior Knowledge to Comprehension

The second-order partial correlations controlling for read ematics achievement are given in Table 3. For Grade 4, even and mathematics achievement were partialed out, the corre

the remaining independent variables and graph comprehension

icant (p < .01). For Grade 7, the correlations were significa topic and content but not form. Table 3

Second-Order Partial Correlations for Grade 4 (n = 204) and Grade 7 (n = 185) Controlling for Reading and Mathematics Achievement

Graph

Variables Topic Content Form comprehension Topic

-

.03

Content .31** Form .18*

.15*

.23**

- .21** .38** .14 -.24**

Graph comprehension .15* .34** .11 Note. Entries above the diagonal are for Grade 4. *p < .05. **p < .01.

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Frances

R.

Curcio

Prediction

The

4

of

results

and

Table

389

Compreh

of

5

the

for

reg

Gr

mathematics achievemen tasks, account for the g sion. As Equations 2 and

accounted

mathematics achievement.

for

by

read

Table 4

Regression Analysis With Graph Comprehension as the Dependent Variable,

Grade 4

Variables

Equation in equation R2 AR 2 df F 1 1 .000 (1,202) 0.098 2 1,2 .491 .491 (2,201) 96.856* 3 1,3 .454 .454 (2,201) 83.472* 4 1,2,3 .598 .598 (3,200) 99.153* 5 1,2,3,4 .618 .020 (4,199) 80.569* 6 1,2,3,5 .656 .058 (4,199) 95.039* 7 1,2,3,6 .620 .022 (4,199) 81.172* 8 1,2,3,4,5,6 .681 .083 (6,197) 70.129*

Note. Code for variables: 1 = sex; 2 = reading achievement; 3 = m 4 = topic; 5 = content; 6 = form.

*p < .01.

Table 5

Regression Analysis With Graph Comprehension as the Dependent Variable,

Grade 7

Variables

Equation in equation R2 AR 2 df F 1

1

.022

(1,183)

4.173*

2 1,2 .474 .452 (2,182) 81.951** 3 1,3 .478 .456 (2,182) 83.198** 4 1,2,3 .600 .578 (3,181) 90.311** 5 1,2,3,4 .608 .008 (4,180) 69.905** 6 1,2,3,5 .644 .044 (4,180) 81.564** 7 1,2,3,6 .604 .004 (4,180) 68.758** 8 1,2,3,4,5,6 .647 .047 (6,178) 54.359**

Note. Code for variables: 1 = sex; 2 = reading achievement; 3 = math 4 = topic; 5 = content; 6 = form. * p < .05. **p < .01.

When each of the three aspects of prior knowledge (topic, content, and form) was entered (see Equations 5, 6, and 7 in Tables 4 and 5), although the contribution to the variance was small, it was significant (p < .01) in

each case.

To determine the optimal linear combination of sex, reading and mathematics achievement, and prior knowledge of topic, of mathematical content, and of graphical form in predicting graph comprehension, the beta weights for each grade were calculated (see Table 6). Predictors of graph

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390

Graph

Comprehension

comprehension

graph

for

Grade

comprehension

achievement

and

prior

4

for

inclu

Gra

knowledg

Table 6

Variables in the Graph Comprehension Regression Equations at

Grades 4 and 7

Grade 4 Grade 7

Variable Beta F(1,197) Beta F(1,178) Sexa -0.05 1.49 0.01 0.03 Mathematics achievement 0.21 14.13** 0.30 21.67**

Reading

achievement 0.29 23.76** 0.33 23.32**

Topic 0.17 9.25** 0.04 0.30

Content 0.26 28.53** 0.26 17.95** Form 0.11 3.98** 0.05 0.78

aBoys

=

0;

girls

=

1.

*p < .05.

**p < .01.

Relation between Sex and Comprehension

Table 7 contains the zero-order partial correlations for both grades. Fo Grade 4, correlations between sex and the other variables were low and n significant. Although the correlations between sex and the achievement and comprehension variables for Grade 7 were significant (p < .05), they wer very low. Table 7

Zero-Order Partial Correlations for Grade 4 (n = 204) and Grade 7 (n = 185)

Reading Mathematics Graph Variables Sexa Topic Content Form achievement achievement comprehension Sexa

-

.00

.09

.00

.10

.09

.02

Topic .11 - .37** .53** .65** .55** .63**

Content .06 .54** - .48** .43** .52** .61* Form

.01

.45**

.39**

-

.60**

.58**

.62**

Reading

achievement .16* .70** .50** .47** - .58** .70*

Mathematics achievement .23* .46** .62** .40** .58** - .67*

Graph

comprehension .15* .57** .65** .44** .69** .69** -

Note. Entries above the diagonal are for Grade 4. aBoys = 0; girls = 1. *p < .05. **p < .001.

DISCUSSION

The results of this study support previous research indicating tha are no significant sex-related differences with respect to graph com

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Frances

R.

Curcio

391

sion (Peterson & Schram the lack of sex-related d to

the

construction

of

th

types. The possibility th school-related tasks that gested by some research The

salience

of

the

comprehension pected,

seventh

three

seems

graders

to

ap

topics and graph forms failure of topic and form hension at Grade 7. It is had a greater need for aspects of a graph, whic knowledge of mathemat stract and embedded wi three aspects of prior kn Elementary school child world" data to construct aged to verbalize the re lected data (e.g., larger t this way, the application students' concept develo matics schemata they ne tionships expressed in g REFERENCES

Adams, M. J., & Collins, A. (1977). A schema-theoretic view of reading (Tech. R Urbana, IL: Center for the Study of Reading. (ERIC Document Reproduction Serv ED 142 971)

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Suydam, M. N., & Riedesel elementary school mathem (ERIC Document Reprodu Thomas, K. C. (1933). The ab teaching of geography (32n pp. 492-494). Bloomington Vernon,

M.

D.

Occupational

Washburne, methods

361-376.

of

(1952).

The

u

Psychology,

J.

N.

(1927).

presenting

qua

AUTHOR

FRANCES R. CURCIO, Assistant Professor, School of Education, Queens College of University of New York, Flushing, NY 11367

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