• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 Abstract
 Introduction
 Review of research
 Procedures
 Results
 Discussion
 Appendix A: Training packet
 Appendix B: Sample introduction...
 Appendix C: Graphs
 Appendix D: Sample posttest
 Appendix E: Specific and generalization...
 Bibliography
 Biographical sketch














Group Title: effect of mathemagenic cues on retention of graphically presented data
Title: The effect of mathemagenic cues on retention of graphically presented data
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 Material Information
Title: The effect of mathemagenic cues on retention of graphically presented data
Physical Description: ix, 112 leaves : ; 28 cm.
Language: English
Creator: Kirk, Sandra K., 1943-
Publication Date: 1978
Copyright Date: 1978
 Subjects
Subject: Learning, Psychology of   ( lcsh )
Attention   ( lcsh )
Memory   ( lcsh )
Graphic methods   ( lcsh )
Curriculum and Instruction thesis Ph. D
Dissertations, Academic -- Curriculum and Instruction -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Sandra K. Kirk.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 109-111.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098083
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000079518
oclc - 04968017
notis - AAJ4826

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
    Abstract
        Page viii
        Page ix
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Review of research
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
    Procedures
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    Results
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
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        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Discussion
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
    Appendix A: Training packet
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
    Appendix B: Sample introduction page
        Page 98
        Page 99
    Appendix C: Graphs
        Page 100
        Page 101
        Page 102
        Page 103
    Appendix D: Sample posttest
        Page 104
        Page 105
        Page 106
    Appendix E: Specific and generalization cues
        Page 107
        Page 108
    Bibliography
        Page 109
        Page 110
        Page 111
    Biographical sketch
        Page 112
        Page 113
        Page 114
        Page 115
Full Text











THE EFFECT OF MATHEMAGENIC CUES
ON RETENTION OF GRAPHICALLY PRESENTED DATA













By

Sandra K. Kirk
















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA

1978














ACKNOWLEDGEMENTS


Among the many rewarding side-effects of pursuing this study were

the personal and professional relationships which developed and grew

because of mutual concerns. For me, at least, this was the greatest

value gained from this dissertation.

Dr. Roy Bolduc, chairman of my committee, put up with my strange

working schedule, while I put up with his terrible puns. Dr. Gordon

Lawrence, Dr. Mary Grace Kantowski, Dr. Don Bernard, and Dr. Charles

Nelson were helpful advisors and willing listeners.

Dr. Don Kauchak and Dr. Paul Eggen are my teachers, my colleagues,

and my friends in the very best sense of each of those words. They

are probably most responsible for getting me into whatever I am into.

Dr. Jim Cangelosi taught me most of what I know about things I am

supposed to know. Since these things include mathematics, measurement

and evaluation, and educational research, his contribution to this study

has been substantial.

Brenda Nichols edited, arranged, unscrambled, and typed, sometimes

into the late night hours. Her dedication and smiling attitude is

greatly appreciated.

My children, Bob, Neva, and Heidi, managed to stay healthy, happy

and well-adjusted, with only a part-time mother. Hopefully what they

gained will balance what they lost.

My husband served as sounding board, editor, typist, father and mother

to the children, and therapist to a sometimes frustrated and frustrating

wife.










Above all, I must thank my family and closest friends for under-

standing that love is qualitative not quantitative, and is not a

variable at all.
















TABLE OF CONTENTS


CHAPTER


ACKNOWLEDGEMENTS . . . . . . .


LIST OF TABLES . . . . .


ABSTRACT . . . . . . . . . . . . . .


I INTRODUCTION . . . . . .

Statement of Problem . .
Need for the Study . .
Overview of Research Design .
Scope of the Study . .
Definitions . . . . .
Organization of the Study .
Hypotheses . . . .
Limitations . . . . .


II REVIEW OF RESEARCH .... . . . . . . . .. .

Graphs . .. .......... .. . . . S
Mathemagenic Cues . . . . . . . . . . 10
Mathemagenic Cues Applied to Graphs .. . . . . . 16

III PROCEDURES .... . . . . . . . . . 19

Design. . . . . . . . . . . . 19
Hypotheses. ...... . . . . .. .. . 20
Materials and Instrumentation . . . .... .. . 21
Subjects and Sampling . . . ... . . . . 25
Data Collection . ... . .......... . 25
Data Analysis . ... . . . .. . . . . 27
Post Hoc Analysis . . . ..... ... .. . . 27

IV RESULTS. . . . . ... .. .. . . . 31


Statistical Analysis of
Statistical Analysis of
Statistical Analysis of
Post Hoc Analysis .


Hypotheses 1,
Hypothesis 4.
Hypotheses 5,


Page

. 11


. . . . vi


4
4
5
6

7
. . . . 1



. . . . 4
. . . . 4
. . . . 5

. . . . 6
. . . . 7


2, and 3.

6, and 7.














CHAPTER Page


V DISCUSSION. . . . . . .... .. .. .. . 71

Conclusions. .. . . . . .. . . 73
Implications . . . . . .... .. . 74
Future Research. .. . . . . . . 75
Summary. .. . . . .. . . ... 75

APPENDICES

A TRAINING PACKET ... . . . ......... . 78

B SAMPLE INTRODUCTION PAGE. . . . . ... ..... 99

C GRAPHS. . . . . . . . . . . . . 101

D SAMPLE POSTTEST .... . . . . ... . 105

E SPECIFIC AND GENERALIZATION CUES. . . . ... . 108

BIBLIOGRAPHY. . . ...... .. ...... 109

BIOGRAPHICAL SKETCH . .. .......... ... 112














LIST OF TABLES


Table Page

1 Sample Packets. . . ... .... . . . ..... 24

2 Time Schedule for Experimental Treatment. .... . .. ... 25

3 Means and Standard Deviation for Total Score for Each Level
of Treatment and Training . . ... ........ . 32

4 Analysis of Variance of Total Scores by Treatment Level and
Training Group. .. ... . . . . . . .33

5 Means and Standard Deviations on the Pretest and Posttest
Scores in the Training Booklet. .. ... . . . . 35

6 Means and Standard Deviations of Three Sections of Experimen-
tal Treatment ... .. ... . . . . . 37

7 Analysis of Variance of Total Scores by Treatment Level,
Training Group, and Grade Level . . . . . . . 39

8 Means and Standard Deviations for Total Scores by Training
Level and Treatment . . . . . . . .. . 40

9 Analysis of Total Score by Treatment for Each Level of
Training. .......... .... . . . . 41

10 Analysis of Total Score by Training Level for Each Treatment. 42

11 Interaction of Training and Treatment on Total Scores . .. 43

12 Means and Standard Deviations for Total Scores by Grade Level
and Treatment . . . . . . . ... .. . .. 45

13 Analysis of Total Score by Treatment for Each Grade Level .46

14 Analysis of Total Score by Grade Level for Each Treatment . 47

15 Interaction of Grade Level and Treatment on Total Scores. . 48

16 Means and Standard Deviations for Scores on Specific Cued
Questions ......... ..... . . . . 49

17 Analysis of Variance of Scores on Specific Cued Items by
Treatment . . .... . . . . . . . 50









LIST OF TABLES (Continued)


Table Page

18 Means and Standard Deviations for Total Scores on Specific
Questions . . . . . . . . . . 52

19 Analysis of Variance of Total Scores on Specific Cued
Questions. . . . . . . . . . .. 53

20 Means and Standard Deviations for Scores on Cued Generali-
zation Questions . . . . . . . . . 54

21 Analysis of Variance of Scores on Cued Generalization
Items by Treatment . . ... . ... . . ... 55

22 Means and Standard Devaitions for Uncued Generalization
Questions by Treatment . . . . . . .. .. 56

23 Analysis of Variance for Uncued Generalization Questions
by Treatment . . . . . . .... . . . 57

24 Means and Standard Deviations for All Groups of Internal
Comparison Items Not Related to Cues . . .. ... 59

25 Analysis of Variance for Internal Comparison Items Not
Related to Cues. . . . . . . . . . .60

26 Means and Standard Deviations for All Groups and Internal
Comparison Items Related to General Cue. . . . .. 61

27 Analysis of Variance for Internal Comparison Items Related
to General Cue ... . . . . . . .. 62

28 Means and Standard Deviations for Internal Comparison Scores
by Treatment and Training .. . . ....... . 63

29 Analysis of Internal Comparison Score by Treatment for
Each Training Level. . . . . . . . . ... 64

30 Interaction of Training and Treatment on Internal Comparison
Scores .... . . . . . . . . 66

31 Means and Standard Deviations for Internal Comparison Scores
by Treatment and Grade Level . .. ...... .... 67

32 Analysis of Internal Comparison Score by Each Grade Level
Treatment. . . . . . ... .. ... . . . 68

33 Interaction of Grade Level and Treatment on Internal Com-
parison Scores . . . . . . . . . . 69














Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy




THE EFFECT OF MATHEMAGENIC CUES
ON RETENTION OF GRAPHICALLY PRESENTED DATA

By

Sandra K. Kirk

August 1978

Chairman: J. Elroy Bolduc
Major Department: Curriculum and Instruction


This study was designed to measure the effects of two forms of mathe-

magenic cues and the effects of training in forming generalizations on

middle school age subjects' ability to learn from material presented in

the form of vertical bar graphs. A 2 x 3 completely randomized factorial

design was used with two levels of training (Trained and Untrained) and

three treatment conditions (Generalization Cue, Specific Cue, and Control).

Subjects were 140 students in grades 5 and 8. The experiment consisted of

the administration of a self-instructional training packet followed two

days later by the presentation of three bar graphs and a posttest on the

information contained in the graphs.

As has been found in studies of verbal prose, a given type of cue

did increase retention of the same type of information; that is, a specific

cue was related to better retention of specific factual information, and a

general cue resulted in better retention of the general relationships

shown in a graph. In this study, however, an interactive effect with

training was found, so that subjects who had been trained to generalize









from graphs were more likely to do this regardless of the type of cue

with which they were presented.

Examination of the data resulted in a post hoc analysis considering

grade level as a variable. Several interactive effects were found.

In general,trained subjects' scores were higher than untrained, and eighth

graders' scores were higher than fifth graders'. The treatment cues were

found to be significantly effective for untrained subjects and fifth grade

subjects. This led to the conclusion that the cues served to compensate

for lack of specific training in generalizing from graphically presented

data, or for lack of experience with graphs.

Implications for classroom application were discussed and areas for

further research were suggested.















CHAPTER I
INTRODUCTION


A citizen in today's society is confronted with an overwhelming

body of factual information. One technique designed to simplify or

clarify the often complex relationships which exist among these facts

is the use of graphs. Students are taught to read graphs in the

elementary school years; however, there are many questions which have

been raised concerning students' ability to learn from graphs, or even

attend to material presented in this form.


Statement of Problem

This study is designed to examine the effects of two different

types of attention directing cues and of specific training in generali-

zing from graphs on middle grade students' ability to learn from material

presented in graphic form. The study is an outgrowth of a body of

research dealing with various types of mathemagenic or attention direct-

ing cues which in this case have been applied to information contained

in bar graphs.


Need for the Study

The use of graphs to communicate meaningful information is a widely

accepted procedure. Graphs are used to organize, clarify, summarize,

and reinforce learning of prose material, or display it in visual form.

Although textbooks at all levels and in many content areas utilize

graphs of varioLs kinds, there is little research on the value of graphic









presentation. Reported research findings consistently show that students

do not effectively retain specific facts or form generalizations from

data presented in graphic form (Vernon, 1946). In one case, the use of

supplementary graphs actually had a deleterious effect on the retention

of material (Vernon, 1951). More recent results from the National

Assessment studies conducted in 1972-73 show that school age subjects

and adults from the general population do not perform well on graph-

related questions. For items requiring interpretation, percentage of

correct responses for thirteen year olds range from 7% to 45%. For

seventeen year olds, the range is 21% to 70%, and for adults it is 26%

to 70% (Carpenter et al., 1978).

Since 1950, the use of graphs to accompany prose material has in-

creased dramatically. A survey conducted in January, 1950, showed that

more than 176 graphs appeared in the New York Times during that month.

A similar survey conducted in January, 1978, found 294 graphs during

that period. The necessity for improvement of reading and interpreta-

tion of graphic material is emphasized in the report of the National

Advisory Committee on Mathematical Eduation (1975), which specifically

cells for greater development of statistical ideas at all levels of

instruction.

In another area of investigation, several researchers (Faw & Waller,

1977; Wilson & Koran, 1976) have found that certain types of questions

(mathemagenic cues) interspersed in prose material have significantly

increased retention of information. They have concluded that these

inserted questions served as cues which stimulate searching or attending

behaviors, focusing the learner's attention on important aspects of the

text. The type of question inserted has been directly related to the










kind of information retained by the learner (Richards, 1976; Kirk,

Kauchak & Eggen, 1978). Therefore, based on these studies, in order to

direct students' attention to a general trend or relationship repre-

sented in graphically displayed data, it is appropriate to use a cue

focusing on these types of relationships. Research in this area,

however, has shown inconsistent results (Kauchak, Eggen & Kirk, 1978).

In addition, although graph reading is most often initially taught

in the elementary school years, very few studies of mathemagenic cues

have utilized upper elementary, elementary, or middle school students.

Very little is known about the effectiveness of graphic presentation

for students of this age.

A study is needed which examines the effects of mathemagenic cues

of different types, particularly those involving generalization about

material presented in graphic form. In addition, the question of whether

these effects will be found with middle grade students should be studied.

Since Vernon's findings (1946) and the results of National Assess-

ment (Carpenter et al., 1978) indicate that readers are unlikely to be

skilled at generalizaing from graphs, specific training in this skill seems

desirable. The question of how this can be taught, particularly to mid-

dle grade students, has not, up to this time, been investigated.

The present study provides further information on the utilization

of mathemagenic cues to focus a reader's attention on information con-

tained in graphs. It differs from previous studies in the following

respects:

1. Cues are in the form of multiple choice questions requiring

identification of a correct generalization and in the form of specific

factual questions.









2. Subjects are students in grades five and eight.

3. An experimental group received training in generalizing from

graphic material. This group is compared with a control group who did

not receive such training.


Overview of Research Design

The study utilized a completely randomized 2 x 3 factorial design.

The variables were training which had two levels, Trained and Untrained,

and treatment, which had three conditions, General Cue, Specific Cue,

and Contrcl. Training was provided in the form of a self-instructional

packet for all subjects. For the Trained group, the packet also included

specific training in generalizing from graphs.

Two days after the training was completed, all subjects were admin-

istered one of the three experimental treatments containing information

to be learned. All subjects were then posttested on the data contained

in the: graphs.


Scope of the Study

The purpose of this study is to measure the effect of mathemagenic

cues in the form of questions requiring identification of a correct gen-

eralization and questions requiring specific factual answers on retention

of material presented in graphic form. In order to investigate whether

retention cf graphic material is enhanced by concurrent presentation of

cues, subjects were given graphs accompanied by these two types of

questions to be answered by inspection of the graphs.

Another question examined concerns the effect of training in form-

ing generalizations about graphs. Specifically, the study investigated

whether prior training in generalizing from graphs enabled the subjects









to better respond to questions concerning information presented in

graphic form.

The following questions were posed:

1. Will middle school age subjects given a graph accompanied by

four possible generalizations, and asked to inspect the graph to deter-

mine which generalization is supported by the data, achieve signifi-

cantly higher scores on a posttest than subjects receiving the graph

accompanied by a specific factual question, or subjects receiving only

the graph, accompanied by directions to study it carefully?

2. Will subjects receiving training in generalizing from graphic

data achieve higher scores on the posttest than students not receiving

such training?


Definitions

Graph. A graph is a diagram visually representing the successive

changes in the value of a variable. In this study, all graphs were

vertical bar graphs.

Mathemagenic. Attention directing.

Mathemagenic Cue. A question or statement designed to direct a

reader's attention to a particular piece of information.

General Cue. A cue question asking for a generalization about

given data.

Specific Cue. A cue question asking for specific factual informa-

tion.

Training. Specific instructions given in reading and generalizing

from graphs.









Organization of the Study

In order to gather information pertinent to the questions described

above, a sample of 140 fifth and eighth graders were selected. Approxi-

mately one half were randomly selected and given the training in general-

izing from graphic data. Subsequent to this, all subjects were randomly

assigned to one of three treatments which involved presentation of

material in graphic form and posttested to measure retention of various

kinds of information. Data from these procedures were then analyzed

to test the following hypotheses.


Hypotheses

The following major hypotheses were tested in this study.

Hypothesis 1: There is no interaction between treatment and
training.

Hypothesis 2: Scores of subjects in three treatment groups
do not differ.

Hypothesis 3: Scores of subjects receiving training in the
use of cues do not differ from scores of
untrained subjects.

The secondary hypotheses were as follows:

Hypothesis 4: Scores on the pretest administered prior to
completion of the training packet do not differ
from scores on the posttest following comple-
tion of the packet.

Hypothesis 5: Scores on the section following the water graph
do not differ from scores on the section follow-
ing the soil graph.

Hypothesis 6: Scores on the section following the water graph
do not differ from scores on the section follow-
ing the sunlight graph.

Hypothesis 7: Scores on the section following the soil graph
do not differ from scores on the section follow-
ing the sunlight graph.









Several additional hypotheses were tested in the post hoc

analysis (see Chapter 4).


Limitations

Due to the specific sample chosen and the restraints of time and

budget, certain limitations must be considered.

1. The subjects used in this study may not be representative of

typical fifth and eighth grade students. The two schools involved both

serve residents of Clay County with middle level socio-economic situations

and relatively high educational backgrounds. Although this was not a

variable in this study, it may prevent generalization to other popula-

tions.

2. The training booklets took approximately thirty minutes for

most students to complete. More long-term training in generalizing

from graphic data might produce quite different results.

3. The measures of retention of graphically presented information

were taken immediately after subjects studied the graphs. Measures of

longer range retention might show results which differ from these in some

respects.

4. All procedures in this study were dependent upon a student's

ability to read written prose material. These procedures are probably

not generalizable to non-reading students.















CHAPTER II
REVIEW OF RESEARCH


The question of whether mathemagenic cues can be utilized to in-

crease students' retention of material presented in graphs draws on two

different lines of research. This chapter summarizes the pertinent

research in three sections and a summary. The first presents the

research on learning from graphs. The second section describes research

on mathemagenic or attention directing cues. A third section discusses

previous research combining the use of mathemagenic cues with graphs.


Graphs

The ability of learners to comprehend, interpret, and retain

material presented in graphic form was studied by Vernon more than 25

years ago. In her earliest study (1946), she investigated the ability

of 231 students, airmen, and soldiers, to recall factual information

presented in graphs. She found that they generally were not able to

interpret the material, generalize from the facts, or separate the

data from preconceived ideas on the topic. She also found that in order

for facts to be retained, subjects had to be able to fit them into an

interrelated system of general information. This ability was related

to intelligence and level of education. Subjects with lower intelli-

gence or less education tended to forget, ignore, or transform the data

to fit preconceived ideas.









In a later study, Vernon (1950) compared interpretation of three

different types of graphs with and without a written text. Subjects were

39 students, ages 16 to 18. No consistent differences were found among

different types of graphs. Surprisingly, the graph and text combination

produced less learning and increased confusion. The students were able

to answer only 50% of the specific and general questions over the content

of the graphs. When discrepancies between textual and graphic materials

existed in one of the experimental conditions, only 3 out of 16 students

noted the discrepancy; however, the discrepant material was less well-

learned when measured on a retention test.

In her most recent study, Vernon (1951) found that students could

retain specific facts without understanding of the general meaning of

a graph; in fact, the number of specific facts recalled was inversely

related to general understanding. It seemed that if difficulty in com-

prehension of a graph was encountered, the subject resorted to memori-

zation of assorted details. In addition, she fouhd that presentation of

factual graphic material sometimes interfered with understanding of a

central textual argument.

The implication of these studies is that students are not able

to identify and focus on important aspects of graphic material. The

1972-73 National Assessment of Education Progress supports this conclu-

sion. On a question for which the answer could be directly read from

the graph, 91% of thirteen year olds responded correctly. However, on

a question requiring interpretation of a relationship presented in the

graph, only 45% of thirteen year olds, 69% of seventeen year olds, and

63% of adults responded correctly. On another question which required

use of a scale in which data were represented in thousands of units, the








percentage of each age group responding correctly dropped to 7%,

21% and 25%, respectively (Carpenter et al., 1978).

Much recent research dealing with graphic representation has

focused on comparisons of different types of graphs or different label-

ing styles (Culbertson & Powers, 1959) and is not directly relevant to

this study.

There are a series of studies in progress at the University of

Georgia, Department of Science Education,which deal with teaching tech-

niques for graph reading, but the results of these studies have not yet

been published.


Mathemagenic Cues

There is an extensive body of research with both theoretical and

practical implications investigating the utilization of mathemagenic

cues.

'Mathemagenic" is a word created by Ernst Z. Rothkopf to describe

the inspecting or attending behaviors which facilitate learning (Rothkopf,

1965). It is derived from two Greek roots, mathemain, meaning "that which

is learned," and gignesthai, meaning "to be born." Mathemagenic behaviors

are those which give birth to learning.

Rothkopf has investigated these behaviors in a series of studies

beginning in 1963. The majority of this research has involved utili-

zation of varying types of written cues to induce mathemagenic behavior.

Rothkopf has identified three classes of mathemagenic activity.

Class I. Orientation: Getting Ss into the vicinity of
instructional objects and keeping them there for suitable
time periods. In certain institutional settings, this also
involves control over activities that may distract or
disturb other students.

Class II. Object Acquisition: Selecting and procuring
appropriate instructional objects. Maintenance of selection and
procurement activities.









Class III. Translation and Processing: Scanning and sys-
tematic eye fixations on the instructional object; translation
into internal speech or internal representations; the mental
accompanyments of reading: discrimination, segmentation,
processing, etc. (p. 328)

Rothkopf further states the following:

The mathemagenic activities subsumed in Classes I and II
are of a gross, molar character, are directly observable,
and are relatively easy to measure. Class III activities
include only a few components that are directly observable
.they are inferred indirectly from observations. The
theoretical inventions in Class III are designated mathema-
genic simply because . they are theoretically designated
to affect substantive learning relative to objectives.
(Rothkopf, 1970, p. 328)

These behaviors are directly involved in the reading process and

it is within this class that would be considered the "reading" of graphs.

The effects of rehearsal in different forms (rereading, rereading

with deleted word, and non-rehearsal) and at different intervals was one

of the earliest variables investigated (Rothkopf & Coke, 1966). Results

of this study indicated that cues which provoked frequent rereading were

most effective.

The next reported study (Rothkopf, 1966) examined the effects of

testlike events. Using 159 college students assigned to one of five

experimental groups or a control group, the study produced results

which indicated that questions presented after reading the relevant

passage had both specific and general facilitative effects, while questions

before the passage facilitated only question-specific retention. It was

not necessary for the answers to be presented for the effects to be seen.

Rothkopf and Bisbicos (1967) supported these finds in a similar study.

More recently, Rothkopf has investigated other types of questions

for mathemagenic effects. In a study using 179 high school students

(Rothkopf, 1972) oral questions by teacher/monitors were found to be more

effective than written questions, and the test-relevance of the questions









was found to be the critical factor rather than the social interaction.

Questions used for indirect review and priming (Rothkopf & Billington,

1974) were investigated in a very complex three-part study involving

high school and college students. The results showed that indirect

review (mental review of material rather than rereading) could be induced

by adjunct questions, and that this review did result in improved per-

formance on related and unrelated questions on a subsequent posttest.

Goal-descriptive directions (specific directions to remember certain

types of information) were used as mathemagenic cues in a study by Gagne

and Rothkopf (1975) using 157 high school students. They found signi-

ficant increases in goal-relevant learning, and also noted decreases in

incidental learning not related to stated goals. A unique feature of

this study was that some goal-cues suggested that subjects recall certain

information which was not available to them, thereby creating a condition

where frustration may have reduced incidental learning (personal commun-

cation with Dr. Gagne, February, 1977).

Another series of studies conducted by Frase beginning in 1967

essentially supports the findings of Rothkopf. Frase (1967) examined

the effects of pre- and post-questions as cues and the effect of know-

ledge of results of response to cues using 72 college students and found

that both types of cues increased performance on cue-related posttest

questions. Post-questions were associated with better scores on non-

related questions, and knowledge of results interacted with questions

position so that if results were known, question position was no longer

a significant factor.

In another study Frase (1968a) found statistical difference between

the use of two different types of cue questions, but in a related follow-

up study found that general questions produced less retention than specific







questions on both total scores and specific questions. Both of these

studies employed college students as subjects. In another study,

Frase (1968b) supported the use of frequent post-questions, while pre-

questions were found to be more effective if they were less frequent.

Increased motivation was provided by monetary rewards of 10, 3f, or

Of per question to a group of 270 undergraduate students (Frase, Patrick

& Schumer, 1970). These rewards stimulated pre-question groups to out-

perform post-question groups; however, increased frequency of questions

inhibited this effect. Frase and Schwartz (1976) recently reported a

study involving 48 high school age subjects in which students were asked

to construct questions for or respond to questions constructed by partners,

both of which had a favorable effect on recall.

Bruning (1968) compared the use of statements to questions and

found questions superior for 69 college students although both types of

cues provided more retention than the control treatment.

A series of studies examined the effects of questions at varying

cognitive levels. An early study utilizing 300 high school seniors was

conducted by Watts and Anderson (1971). Students who had application

level questions were superior to those who had repeated answer or recall

of name questions on the posttest, even though they did not do as well

on those questions while studying the text. Apparently the more diffi-

cult questions required more processing of the data.

Building on the findings of Watts and Anderson, a study was con-

ducted by Felker and Dapra (1975), examining the effects of verbatim

recall or comprehension questions as cues, and also comparing different

question placements. This study utilized 95 college students involved

in a verbal problem solving task. They found that comprehension cues

following the verbal material resulted in the best performance on both

verbatim and comprehension posttests.









Slightly different results were found in a study by Richards (1976)

using 75 college students. In his study conceptual question cues were

compared with verbatim cues. He found an interaction between cognitive

level of cues and question position so that conceptual pre-questions were

better than conceptual post-questions as cues, but verbatim pre-questions

were less effective than verbatim post-questions. The conceptual pre-

question group and the verbatim post-question group both performed signi-

ficantly better than control on the immediate posttest; however, only

the conceptual group scored better than control on a retention test one

week later.

In a very recent study, Yost, Avila, and Vexler (1977) reported

that more complex cues were associated with better performance on both

specific and incidental questions. This is the only reported study

utilizing junior high school age subjects (193 seventh graders). The

authors defined a cue as more complex if it referred to more factors or

more levels of a factor. They hypothesized that improved performance

on the posttest might be related to increased inspection time.

The only reported study with elementary school children as subjects

used 109 fifth and sixth graders. It examined the effects of question

position and question-type, on an immediate posttest and a test one

week later (Swenson & Kulhavy, 1975). Results showed that post-questions

were the most effective cues for both the immediate and delayed posttests.

The type of question, either a verbatim or a semantically equivalent

question, produced no significantly different results. These findings

seem to indicate that the effects of mathemagenic cues are not limited

to adult populations.










Taken together, the body of research on mathemagenic cues is quite

consistent. Only one reported study (Markle & Capie, 1976) finds mathe-

magenic cues to be totally ineffective. This study was done using only

28 subjects and no control group, and therefore must be interpreted

cautiously at best.

In reviewing the early literature through 1974, Faw and Waller (1977)

and Wilson and Koran (1976) found that there was a consistent pattern of

findings:

1. Inserted questions which appear before the material
to which they relate (pre-questions) have a substantial
specific facilitative effect but no general facilitative
effect.

2. Inserted questions which appear after the material
to which they relate (post-questions) have a substantial
specific facilitative effect as well as a small general
facilitative effect on posttest scores.

That is, both pre-questions and post-questions facilitate
intentional learning, but only post-questions facilitate
incidental learning.

This pattern has also been found on delayed measures of retention

(Faw & Waller, 1977).

In summary, the research on mathemagenic cues clearly indicates that

insertion of certain types of questions in textual material improves re-

tention, apparently by modifying the attending behavior of the subject.

Frequent questions following the written material are apparently the

most effective.

Two theoretical explanations have been presented to account for the

effectiveness of mathemagenic cues. One suggests that they direct a

forward shaping of the information processing. This theory is supported

by studies citing increased retention of specific information associated

with mathemagenic cues placed prior to verbal material.








The other theory suggests that cues serve to direct a backward review

involving a mental review or reprocessing of information. The support

for this theory is primarily drawn from the studies citing improved

performance on incidental or general information associated with questions

placed after verbal material.

It seems likely that in actual practice, both of these theories are

involved in the explanation of mathemagenic effects.


Mathemagenic Cues Applied to Graphs

A study by Kauchak, Eggen, and Kirk (1975) applied the use of mathe-

magenic cues to material presented in graphic form. Subjects (143 stu-

dents in fourth, fifth, and sixth grade) read a passage which included

four graphs. One group responded to general questions placed after

each graph. The second group responded to specific questions. A third

group served as control, and was only directed to carefully look at the

graphs. Subjects were given a 20-item posttest containing cued-specific

questions, non-cued-specific questions, and generalizing questions. The

group receiving specific questions was significantly superior to the

other two groups.

Another study by the same investigators (Eggen, Kauchak & Kirk, 1978),

compared subjects given graphs accompanied by two possible generalizations

to a control group having only the graph. Subjects were juniors in the

College of Education, University of North Florida, enrolled in a required

class in general methods and curriculum. The control group scores were

significantly higher than the group receiving the generalization cues

in every comparison. The reason for these unexpected results is not

clear, but it was hypothesized that perhaps the complexity of the cues

was so great that the subjects were not able to utilize the cues as

intended.








A third study (Kirk, Kauchak, & Eggen, 1978) intended to provide

additional information about the results stated above was conducted.

In this study upper level college students were randomly assigned to one

of four conditions, each of which utilized a different form of cue. Cues

were either in the form of Overall Cue which consisted of a generaliza-

tion referring to both weeks of a two-week experiment; General Cue which

consisted of a generalization about the final results of an experiment;

Specific Cue, which asked Ss to respond to a factual question, and Control

which received only the graph and directions to study it carefully.

This study yielded a large number of interesting findings. In

general, the General Cue group and Specific Cue group scored higher than

the Overall Cue group and Control. This tended to support the hypothesis

that relatively complex cues were less effective. Performance on non-

cued questions indicated that the General Cue was most effective in

producing scanning of the graph. When comparisons were made using only

those groups which correctly responded to the cue, differences were

even more dramatic. Only in this case were differences found favoring the

Overall Cue group, again indicating that cue complexity was a problem,

since the cue was effective for those who were able to respond correctly

to it.

In planning the present study, it was decided to use only General

and Specific Cues, since these two kinds are apparently most effective,

and since cue complexity might be expected to be an even greater problem

with younger subjects.

The present study extends the research in three important ways.

First, the subjects of this study are elementary and middle school stu-

dents. Very little of the research either on mathemagenic cues or on

graphic interpretation has utilized young students as subjects.








Second, this study includes a training component, so that some

subjects are first trained in forming generalizations about material pre-

sented in graphic form.

The third difference is that the cues followed the graphs rather

than preceding them, a difference which has shown to be very significant

in research on mathemagenic cues used with verbal material. The pre-

vious studies with graphic materials have, in every case, placed the

cues before rather than after the graph.















CHAPTER III
PROCEDURES


The problems in this study were to determine whether mathemagenic cues

can be utilized to increase students' retention of material presented in

graphic form and whether specific training in generalizing from graphs

enhances the effectiveness of the cues.

In order to determine whether presentation of different types of

cues to accompany graphic material results in better retention of infor-

mation from these graphs, the following question was examined:

1. Will subjects given a graph accompanied by four possible

generalizations, and asked to inspect the graph to determine

which generalization is supported by the data achieve signifi-

cantly higher scores on a posttest than subjects receiving the

graph accompanied by a specific factual question, or subjects

receiving only the graph, accompanied by directions to study

it carefully?

To determine whether training in generalizing from graphs results

in better retention of appropriate generalizations, the following ques-

tion was examined:

2. Will subjects receiving training in generalizing from graphic

data achieve higher scores on the posttest?


Design

This study was conducted as a 2 x 3 factorial experiment, using

two levels of training (trained and untrained), and three treatment

groups (Generalization Cue, Specific Cue and Control).












-0
c)
'c
Ca)
s-
+1


C
ra
c=n
-i


TREATMENT

General Cue Specific Cue Control

General Cue Specific Cue Control

Trained Trained Trained


General Cue Specific Cue Control

Untrained Untrained Untrained


The dependent variables were the scores on a combined set of three post-

tests which are described in detail later in this chapter. A total of

140 subjects were randomly assigned to each condition as described below.

Based on this design the following model was assumed:

Xijk = / + i( + Bj + ~d ij + k(ij)


where Xijk = posttest score of a subject k in treatment level i and

training level j

I/ = the grand mean posttest score

S= the effect of treatment level i

S= the effect of training level j

9R = the interaction of treatment and training

= the random error

: N (0, ) and independent


Hypotheses

The following null hypotheses were tested:

Hypothesis 1: There is no interaction between treatment and
training.

(Ho : (4 ij = 0).










Hypothesis 2: Scores of subjects in three treatment groups do
not differ.

(H : i = 0).


Hypothesis 3: Scores of subjects receiving training in the
use of cues do not differ from scores of untrained
subjects.

(Ho : j 0).


In order to assess the direct effect of the training packet, the

following null hypothesis was tested:

Hypothesis 4: Scores on the pretest administered prior to
completion of the training packet do not
differ from scores on the posttest following
completion of the packet.

In order to determine whether the three randomly ordered sections of

the test were equivalent, the following hypotheses were tested:

Hypothesis 5: Scores on the section following the water graph
do not differ from scores on tie section follow-
ing the soil graph.

Hypothesis 6: Scores on the section following the water graph
do not differ from scores on the section follow-
ing the sunlight graph.

Hypothesis 7: Scores on the section following the soil graph
do not differ from scores on the section follow-
ing the sunlight graph.


Materials and Instrumentation

Experimental Materials

Three graphs were prepared, each showing the results of a hypothe-

tical experiment involving bean plants grown for two weeks under

varying conditions. Variables used were amount of water (W),

organic material in soil (S), and sunlight (L). Each graph









showed the results of manipulating one of these variables, in terms of

height of bean plants at the end of one and two weeks. Bar graphs were

used and were designed so that all three are of equal difficulty based

on the amount of information conveyed. The data, though reasonable, were

counter-intuitive, so that test results reflected information gained from

the graph rather than previous knowledge (Appendix C ). In one treat-

ment, General Cue, the graphs were followed by four generalizations, one

of which was correct based on the data. For subjects in the Specific Cue

treatment, the graph was followed by a specific factual question (Appen-

dix E ).

A fifteen item multiple choice test was prepared for each graph,

consisting of three types of questions: (a) two questions requiring iden-

tification of a generalization about the data in the graph, (b) three

questions requiring identification of specific facts, and (c) ten ques-

tions involving comparison of internal relationships within the data

(Appendix D). Reliability of the tests was assessed using the formula

Kuder-Richardson 20, resulting in r = .80. The validity of the tests

was assessed by a panel of three professors of science education to

verify that the items actually measured the content indicated.

An introductory page preceded each graph to explain the particular

bean plant experiment, and to direct the subject to carefully examine the

graph which appeared on the next page.

These materials were assembled into four page sets, each consisting

of an introductory page, a graph page, and two test pages, in that order.

Packets were formed, using one set for each of the three bean plant

variables, arranged in random order, resulting in a 12-page packet.

Although within each set the order of the four pages was constant

(introduction, graph, test), the sequence of the three sets was varied.









Therefore, it was possible to control for any difference in difficulty

by randomizing the sequence for each packet. In addition, the equi-

valence of the three sections of the test was tested (see Hypothesis 4).

The contents of three hypothetical packets are shown in Table 1.


Training

Two versions of a training booklet were written. These were designed

to present two kinds of training.

One version of the training booklet was made up of a pretest, two

types of training material, and a posttest. The pretest consisted of

seven questions measuring the subjects' knowledge of parts of a graph,

ability to read factual information from the graph, and ability to iden-

tify an appropriate generalization based on the data in the graph. The

posttest was identical to the pretest.

The training material was interactive in nature. It presented a

short discussion of a topic, then questions to measure subjects' under-

standing of the discussion. Immediate feedback consisting of the correct

response to the question was provided in all cases. The first type of

training material described the parts of a graph and discussed the direct

reading of factual material. Within this section subjects were required

to read data from a graph and were provided with feedback on their

responses. The second type of material was designed to help subjects form

generalizations about data in graphs. Various possible generalizations

were discussed for several different graphs, and again an interactive

format with immediate feedback was utilized.

The second version of the training booklet was identical to the

first except that the second type of training material (the section

which provided training in forming generalizations) was replaced by a
















Table 1. Sample Packets.


Control Specific Cue General Cue

Page Bean Type of Bean Type of Bean Type of
Variable Page Variable Page Variable Page


1 Water Intro Soil Intro Light Intro

2 W Graph S Graph & L Graph &
Spec Cue Gen Cue

3-4 W Test S Test L Test

5 L Intro L Intro W Intro

6 L Graph L Graph & W Graph &
Spec Cue Gen Cue

7-8 L Test L Test W Test

9 S Intro W Intro S Intro

10 S Graph W Graph & S Graph &
Spec Cue Gen Cue

11-12 S Test W Test S Test









placebo section designed to equalize the time needed to complete the

two packets without providing additional training to the control

group.


Subjects and Sampling

The subjects for this study were two classes of fifth and three

classes of eighth grade students in Clay County schools, Orange Park,

Florida. At the elementary school, two fifth grade classes were chosen

at random. At the middle school, all three eighth grade classes were

used. From these, half of the students in each class were selected at

random to receive the training in generalizing from graphs. Two days

prior to the administration of the experiment, subjects were randomly

assigned to one of the two training conditions, and all subjects completed

one of the two versions of the booklets. Subjects worked at their own

pace, and no time limitations were imposed. All training and testing

periods were under the direction of the experimentor.


Data Collection

The experimental data were collected on the third day following the

completion of the training packets. All subjects in each class were

randomly assigned to one of the three treatment conditions. General

directions were given to the group. Subjects were encouraged to do their

best and assured that this study was intended only to gather informa-

tion, and that their individual performances would not be reported.

Subjects worked through the experimental packet and were permitted to

turn pages only when directed to do so by the experimentor. Subjects

were given 30 seconds to read the introductory page, 120 seconds to study

the graph, and 210 seconds to complete the posttest. A summary of this

schedule is given in Table 2. These time allowances were adequate for all
riiin+r -f-i rnmnlnot o.rk cnri4nm
















Table 2. Time Schedule for Experimental Treatment.




Elapsed Time Type of Page to Begin Reading
in
Minutes and Seconds


0 0 First Introductory Page

0 30 First Graph

2 30 First Test

6 0 Second Introductory Page

6 30 Second Graph

8 30 Second Test

12 0 Third Introductory Page

12 30 Third Graph

14 30 Third Test

18 0 Experiment Completed








Data Analysis

Hypotheses 1, 2, and 3 were tested using two-way analysis of vari-

ance in a 2 x 3 completely randomized factorial design. Where inter-

actions were found to be significant, simple main effects were tested

at each level of each variable. Otherwise, main effects were tested

across the other variable. In cases where treatment was found to have

significant effect, Tukey's HSD procedure was used to determine differ-

ences among treatments. Hypotheses 4, 5, 6 and 7 were tested using a

matched T-test. All analyses were conducted at the .05 level of signifi-

cance.


Post Hoc Analysis

Upon examination of the raw data which was collected in this study,

it became apparent that there might be differential performances on

the experimental posttest by fifth grade subjects as compared to eighth

grade subjects.

Another question raised by examination of the data was whether the

treatment groups performed differently on the different types of ques-

tions in the posttest.

In order to answer these two questions, a post hoc analysis of the

data was completed. The procedures followed are discussed in the follow-

ing two sections.


Analysis Incorporating the Effect of Grade Level

In order to examine the possible differential effect of grade level

on total scores the following model was utilized:


X =: I + di + j Y k + ij + tik +3 Yijk +1l(ijk)




i0i


where Xikl = the posttest score of a subject 1 in treatment level i,

training level j, and grade level k, and where,

S= the grand mean posttest score

o{ = the effect of treatment level i

= the effect of training level j

> = the effect of grade level k

a = the interaction of treatment and training

C y = the interaction of treatment and grade level

P = the interaction of training and grade level

O\P = the interaction of treatment, training and grade level

= random error

S: N(O,r- ) and independent.

The following null hypotheses were tested:

Hypothesis 8: There was no interaction between treatment,
training and grade level.

(Ho :; jk = 0).

Hypothesis 9: There was no interaction between treatment and
training.

(Ho : 0 ij = 0).


Hypothesis 10: There was no interaction between treatment and
grade level.

(Ho :oYik 0).

Hypothesis 11: There was no interaction between training and
grade level.

(Ho : OYij 0).









Hypothesis 12: Scores of subjects in the three treatment groups
do not differ.

(Ho : 0 i = 0).


Hypothesis 13: Scores of subjects receiving training do not
differ from scores of untrained subjects.

(Ho : = 0).


Hypothesis 14: Scores of subjects in grade 5 do not differ
from scores of subjects in grade 8.

(Ho : k = 0).


The data were analyzed using three-way analysis of variance. Signifi

cant interactions were further analyzed to determine simple main effects,

and Tukey's HSD procedure was used to determine the specific differences

between groups. Results of these analyses are reported in Chapter IV.


Analysis to Identify Differential Performance on Types of Test Items

Examination of the data indicated that there might be differential

performance on different types of test items. This was a reasonable

finding since previous research has indicated that posttest performance

on questions of a given type is best facilitated by cues of the same type

(Watts & Anderson, 1971; Kauchak, Eggen & Kirk, 1975).

In order to investigate this question, the experimental posttest

scores were separated into subsections as follows:

1. Specific Cued Question (posttest question identical to specific
cue)

2. Total on Specific Questions

3. Cued Generalization Question (posttest question identical to
general cue)

4. Uncued Generalization Question

5. Internal Comparison of Data Questions not Related to General Cue

6. Internal Comparison of Data Questions Related to General Cue









One-way analysis of variance was used to determine any differences

which existed in performance on these subtests at different levels of

the variables treatment, training, and grade level. Where necessary

Tukey,'s HSD procedure was used to determine which treatment groups differed

significantly. Results of these analyses are reported in Chapter IV.















CHAPTER IV
RESULTS


The purpose of this study was to assess the effects of mathemagenic

cues in two forms and of training in forming generalizations from graphs

on subjects' retention of material presented in graphic form. The

results of the statistical analysis of the data are presented in this

chapter, organized by sets of hypotheses.


Statistical Analysis of Hypotheses 1, 2, and 3

These three hypotheses were tested using two-way analysis of variance

in a 2 x 3 factorial analysis. The means and standard deviations for

each group are presented in Table 3 and the results of the analysis are

presented in Table 4.

Hypothesis 1: There is no interaction between treatment
training.

The results shown in the table indicate that there was no statis-

tically significant interaction between training and treatment. Therefore,

hypothesis 1 could not be rejected.

Hypothesis 2: Scores of subjects in three treatment groups
do not differ.

The results in Table 4 show that the effect of treatment was not

associated with statistically significant differences in total scores.

Based on these findings, it was not possible to reject the second null

hypothesis.

Hypothesis 3: Scores of subjects receiving training in the
use of cues do not differ from scores of untrained
subjects.















Table 3. Means and Standard Deviation for
of Treatment and Training.


Total Score for Each Level


Treatment 1 Treatment 2 Treatment 3 Totals




Trained M 26.39 26.84 25.32 26.16

SD 7.79 7.15 4.84 6.53

N 18 25 25 68



Untrained M 21.95 25.30 25.76 24.35

SD 8.54 7.33 6.65 7.63

N 24 23 25 72


TOTALS M 23.86 26.11 25.54 25.32

SD 8.43 7.21 5.76 7.15

N 42 48 50 140
















Table 4. Analysis of Variance of
Training Group.


Total Scores by Treatment Level and


Source SS df MS F p



Treatment 103.83 2 51.91 1.03 .36


Training 98.33 1 98.33 1.95 .16


Treatment x Training 134.25 2 67.12 1.33 .26


Within Groups 6749.32 134 50.37



Total 7102.55 139









The results in Table 4 show that the effect of training was not

associated with statistically significant differences in total scores.

Based on these findings, it was not possible to reject the third null

hypothesis.

Although it was not possible to reject any of the first three hypo-

theses, examination of the raw data raised questions about whether sub-

jects in the fifth grade groups had performed differently from subjects

in the eighth grade groups, and the possibility that these differences

may have masked differences caused by treatment and/or training. This

led to the rather extensive post hoc analysis which is described in

Chapter III, the results of which are presented later in this chapter.


Statistical Analysis of Hypothesis 4

Hypothesis 4: Scores on the pretest administered prior to
completion of the training packet do not differ
from scores on the posttest following comple-
tion of the packet.

This hypothesis was tested using a matched T-test, resulting in

t(139) = 2.77, which was significant with p <.01. Means and standard
deviations for the Pretest Scores and Posttest Scores are reported in

Table 5. Based on this the null Hypothesis 4 was rejected, and it was

concluded that the training packet did result in slightly improved perfor-

mance on the training posttest.


Statistical Analysis of Hypotheses 5, 6, and 7

Hypothesis 5: Scores on the sections following the water graph
do not differ from scores on the section follow-
ing the soil graph.

Hypothesis 6: Scores on the section following the water graph
do not differ from scores or the section follow-
ing the sun light graph.

Hypothesis 7: Scores on the section following the soil graph do
not differ from scores on the section following
the sun light graph.















Table 5. Means and Standard Deviations on the Pretest and Posttest
Scores in the Training Booklet.



M SD T P



Pretest 5.33 1.59
2.77 .01

Posttest 5.52 1.65


(n 140, possible score = 7)









These three hypotheses were tested to determine whether the three

sections of the experimental treatment were equivalent. Results of

these tests are reported in Table 6. There were no significant differ-

ences among any of the three sections, so the null hypotheses stating

that the sections were equivalent could not be rejected. This indicated

that the sections were essentially equivalent, so that the order in

vhich they were encountered was not a concern in this study. This problem

was also controlled for by the random ordering procedure described in

Chapter III.


Post Hoc Analysis

Examination of the results of the data raised several questions, and

in order to investigate these, a fairly extensive post hoc analysis was

conducted.


Statistical Analysis of

The first question

the results of posttest

was constructed and the

Hypothesis 8:


Hypothesis 9.


Hypothesis 10:


Hypothesis 11:


Hypothesis 12:


Hypothesis 13:


Hypotheses 8 through 14

examined concerned the effect of grade level on

performance. To assess this, a revised model

following hypotheses were tested.

There was no interaction between treatment,
training and grade level.

There was no interaction between treatment and
training.

There was no interaction between treatment and
grade level.

There was no interaction between training and
grade level.

Scores of subjects in the three treatment groups
do not differ.

Scores of subjects receiving training do not
differ from scores of untrained subjects.
















Table 6. Means and Standard
mental Treatment.


Deviations of Three Sections of Experi-


Section Mean SD




Water 8.92 3.28


Light 8.15 3.89


Soil 8.43 3.49



Note: T value for Water:Light = 1.85, p = .11
T value for Light:Soil = 1.62, p = .15
T value for Water:Soil = 1.44, p = .07
(n = 140)









Hypothesis 14: Scores of subjects in grade 5 do hot differ
from scores of subjects in grade 8.

These hypotheses were tested using a three-way analysis of variance,

the results of which are reported in Table 7.

The three-way interaction was not significant; therefore, the two-

way interactions were examined.

Since the interaction between Treatment and Training and the inter-

action between Treatment and Grade Level were both significant, it was

necessary to reject null hypotheses 9 and 10, but it was not possible

to reject hypothesis 11. Because of the two significant interactions,

it was not appropriate to consider main effects of these variables, so

the F's obtained in testing hypotheses 12, 13, and 14 were not meaning-

ful, although in some cases they were statistically significant.


Further Analysis of Hypothesis 9

Since the interaction between Treatment and Training was significant,

the simple main effects of each of these variables at each level of the

other variable were examined. Means and standard deviations for each of

these subgroups are reported in Table 8, and the analyses by level are

reported in Tables 9 and 10. It is apparent that treatment differences

are greatest in the Untrained group, and that training had its greatest

effect on the Control group. Although none of these simple main effects

were significant, the effect of training on the Control group was associ-

ated with an F value of 2.98, and a p value of .09.

The combined effect of these interactions is shown in Table 11.


Further Analysis of Hypothesis 10

Because the interaction between Treatment and Grade Level was signifi-

cant, the simple main effects of each of these variables at each level of
















Table 7. Analysis of Variance of Total Scores by Treatment Level,
Training Group, and Grade Level.



Source SS df MS F p




Treatment 70.78 2 35.39 .82 .44

Training 136.05 1 136.05 3.16 .07

Grade 963.28 1 963.28 22.40 .01

Treatment x Training 265.73 2 132.86 3.09 .04

Treatment x Grade 357.05 2 178.53 4.15 .02

Training x Grade 9.10 1 9.10 .21 .64

Treatment x Training x
Grade 84.58 2 42.29 .98 .37

Within Groups 5244.09 128 40.97


Total 7102.55 139















Table 8. Means and Standard Deviations for Total Scores by Training
Level and Treatment.



Specific General
Treatment Control Cue Cue



Trained M 26.39 26.84 25.32

SD 7.79 7.15 4.84

N 18 25 25


Untrained M 21.95 25.30 25.76

SD 8.54 7.33 6.55

N 24 23 25















Table 9. Analysis of Total Score by Treatment for Each Level of Training.




Source SS df MS F p



Trained

Treatment 30.14 2 15.07 .35 .70

Within 2823.07 65 43.43

Total 2853.22 67



Untrained

Treatment 207.93 2 103.97 1.83 .16

Within 3926.38 69 56.90

Total 4134.31 71
















Table 10. Analysis of Total Score by Training Level for Each Treatment.



Source SS df MS F p


Treatment 1


Training

Within

Total


201.91

2711.23

2913.14


201.91

67.78


Treatment 2


Training

Within

Total


28.25

2414.22

2442.47


28.25

52.48


Treatment 3


Training

Within

Total


2.42

1624.00

1626.42


2.42

33.83


2.98 .09








.54 .46








.07 .79


















Table 11. Interaction of Training and Treatment on Total Scores.


~~6


A


Control


Specific Cue

TREATMENT


General Cue


Trained

Untrained ---------









the other variable were examined. Means and standard deviations for

each of these subgroups are reported in Table 12, and analyses by level

are reported in Tables 13 and 14.

When the effect of Treatment was assessed for Grade 5, it was found

to be statistically significant. Tukey's HSD procedure was used to deter-

mine which Treatment group scores were significantly different from the

others, with the following results: HSD = 3.40, indicating that an

absolute difference between means of 3.40 would represent a statistical

difference. Based on this, the General Cue group was found to have scored

significantly higher than the Control group.

When the effect of Grade Level was determined for each Treatment

group, Grade level was found to have a significant simple main effect

for both the Control group and the Specific Cue group, but not for the

General Cue group. This seems to indicate that the effect of the General

Cue was powerful enough to overcome the expected superior performance

of the eighth graders over the fifth graders. This will be discussed

further in Chapter V.


Statistical Analysis of Performance on Specific and General Posttest

Questions. Six different types of posttest questions were examined.

Results of these analyses are reported in sections identified below.


Specific Cues Questions

These are questions on the posttest which are identical to the cues

given to the Specific Cue group. It was predicted that the scores of this

Treatment group would be higher than the other two treatment groups on

these items of the posttest. This prediction was supported by the data,

which is reported in Tables 16 and 17. Using Tukey's HSD = 3.35, it
















Table 12. Means and Standard Deviations for Total Scores by Grade
Level and Treatment.



Treatment Control Specific Cue General Cue

(n = 42) (n = 48) (n = 50)



Grade 5 M 19.26 22.63 24.35

SD 7.06 5.66 4.51

N 19 19 20


Grade 8 M 27.65 28.39 26.33

SD 7.64 7.29 6.41

N 23 29 30
















Table 13. Analysis of Total Score by Treatment for Each Grade Level.




Source SS df MS F p


Grade 5


Treatment

Within

Total


259.50

1850.65

2120.15


129.05

33.83


Grade 8


Treatment

Within

Total


63.35

3962.71

4026.06


31.68

50.16


3.84 .02








.63 .53
















Table 14. Analysis of Total Score by Grade Level for Each Treatment.




Source SS df MS F p


Treatment 1


Grade

Within

Total


732.24

2180.90

2 913.14


732.24

54.52


Treatment 2


Grade

Within

Total


379.23

2063.24

2442.48


379.23

44.85


Treatment 3


Grade

Within

Total


47.20

1579.21

1626.41


47.20

32.90


13.43 .01








8.46 .01








1.43 .23
















Table 15. Interaction of Grade Level and Treatment on Total Scores.


29

28 -

27
-4
26

25

2 24

v 23

22

21

19

18



Control Specific Cue General Cue

TREATMENT

Grade 5

Grade 8 ----------















Table 16. Means and Standard Deviations for Scores on Specific Cued
Questions.




Treatment Mean SD ,



Control (n = 42) 1.66 .90


Specific Cue (n = 48) 2.31 .90


General Cue (n = 50) 1 .82 1.00















Table 17. Analysis of Variance of Scores on Specific Cued Items by
Treatment.




Source SS df MS F p



Treatment 10.52 2 5.26 5.95 .01


Within 121.02 137 .88


Total 131.54









was found that, as predicted, the Specific Cue group scores were

significantly higher on these questions than both the General Cue and

Control groups.


Total on Specific Questions

This score consists of the total score on the first three items of

each of the three sections of the experimental posttest. These questions

involve retention of specific factual information. Data for this analysis

arereported in Tables 18 and 19. Although the Specific Cue group appears

to have done better than the other group on this subscore, these differ-

ences were not statistically significant, since the F value of 2.15,

is associated with a p value of .12.


Cued Generalization questions

This score consists of the total on the generalization questions

which are identical to the cues given to the General Cue group. It

was predicted that the scores of this group would be higher than the

other two Treatment groups on these items of the posttest. This pre-

diction was supported by the data which are reported in Tables 20 and 21.

Using Tukey's HSD = 3.35, it was found that the General Cue group

scores were significantly higher than the Control group on these items.


Uncued Generalization Questions

This score consists of the total score on the three uncued general-

ization questions. These questions referred to different data than the

cued generalizations; therefore, it was predicted that there would be

no differences in scores among the three treatment groups. This was

supported by the data which appear in Tables in 22 and 23.















Table 18. Means and Standard Deviations for Total Scores on Specific
Questions.




Treatment Mean SD



Control (n = 42) 5.42 2.07


Specific Cue (n = 48) 6.25 1.84


General Cue (n = 50) 5.81 1.57
















Table 19. Analysis of Variance of Total Scores on Specific Cued
Questions.




Source SS df MS F p


Treatment


Within


Total


14.32


429.34


443.66


7.16


3.33
















Table 20. Means and Standard Deviations for Scores on Cued Generalization
Questions.




Treatment Mean SD



Control (n = 42) 1.21 .95


Specific Cue (n = 48) 1.44 .99


General Cue (n = 50) 1.74 1.03















Table 21. Analysis of Variance of Score on Cued Generalization Items
by Treatment.




Source SS df MS F p



Treatment 6.43 2 3.22 3.28 .04


Within 134.50 137 .98


Total 140.93 139















Table 22. Means and Standard Deviations for Uncued Generalization
Questions by Treatment.




Treatment Mean SD



Control (n = 42) .90 .76


Specific Cue (n = 48) 1.12 .84


General Cue (n = 50) 1.00 .88
















Table 23. Analysis of Variance for Uncued Generalization Questions by
Treatment.




Source SS df MS F p



Treatment 1.10 2 .55 .80 .45


Within 94.87 137 .69


Total 95.97 139









Internal Comparison Questions Not Related to General Cue

This score consists of a total of 15 items, 5 on each section, each

of which required subjects to determine which of two conditions pro-

duced better results in bean plant growth. These questions were included

to measure subjects' retention of relationships within the graph which

were not specifically stated or referred to in any of the cues.

The scores on these items were analyzed using three-way analysis of

variance. The means and standard deviations for each level of each

variable are reported in Table 24. The results of the analysis are re-

ported in Table 25. Since none of the two-way or three-way interactions

were significant, only the main effects need be considered.

Main effects for both Training and Grade Level were found to be

significant, with trained subjects scoring higher than untrained, and

eighth graders scoring higher than fifth graders on this set of questions.


Internal Comparison Questions Related to General Cues

This score also consists of a total of 15 items, which again required

subjects to recall internal relationships within each graph. This set of

questions referred to the same set of data as the General Cue, although

the cue did not refer to any specific comparisons such as those required

in these test questions.

The scores on these items were analyzed using three-way analysis of

variance. The means and standard deviations are reported in Table 26

and the results of the analysis are reported in Table 27.

Since both the two-way interactions for Treatment by Training and for

Treatment by Grade were significant, analysis was completed for each level

of each variable at each level of the other variable.

The results of this analysis for Treatment and Training are reported

in Tables 28 and 29. The combined effect of this interaction is shown















Table 24. Means and Standard Deviations for All Groups of Internal
Comparison Items Not Related to Cues.




Groups Mean SD



Treatment

Control (n = 42) 9.26 3.24

Specific Cue (n = 48) 9.84 2.63

General Cue (n = 50) 9.02 2.49


Training

Trained (n = 68) 10.10 2.37

Untrained (n = 72) 8.63 2.96


Grade

Fifth (n = 58) 8.53 2.59

Eighth (n = 82) 9.96 2.77















Table 25. Analysis of Variance for Internal Comparison Items Not
Related to Cues.




Source SS df MS F p



Treatment 12.24 2 6.12 .86 .42

Training 56.46 1 56.46 7.92 .01

Grade 75.19 1 75.19 10.54 .01

Treatment x Training 19.93 2 9.97 1.40 .25

Treatment x Grade 12.66 2 6.33 .89 .41

Training x Grade 1.45 1 1.45 .20 .65

Treatment x Training x Grade 4.17 1 2.09 .29 .74

Within 912.64 128 7.13


Total 1088.91 139
















Table 26. Means and Standard Deviations for All Groups
Comparison Items Related to General Cue.


and Internal


Groups Mean SD



Treatment

Control (n = 42) 6.77 4.58

Specific Cue (n = 48) 7.25 4.06

General Cue (n = 50) 7.62 3.96


Training

Trained (n = 68) 7.64 4.16

Untrained (n = 72) 6.83 4.19


Grade

Fifth (n = 58) 6.08 3.59

Eighth (n = 82) 7.99 4.39















Table 27. Analysis of Variance
to General Cue.


for Internal Comparison Items Related


Source SS df MS F p



Treatment 30.19 2 15.09 .99 .37

Training 19.63 1 19.63 1.29 .26

Grade 97.40 1 97.40 6.38 .01

Treatment x Training 92.99 2 46.50 3.05 .05

Treatment x Grade 103.82 2 51.91 3.40 .03

Training x Grade 16.17 1 16.17 1.06 .30

Treatment x Training x Grade 83.33 2 41.67 2.73 .07

Within 1953.28 128 15.26


Total 2373.16 139















Table 28. Means and Standard Deviations
by Treatment and Training.


for Internal Comparison Scores


Treatment
Treatment Control Specific Cue General Cue
(n = 42) (n = 48) (n = 50)


Trained

(n = 68) M 8.29 7.77 7.00

SD 4.85 3.87 3.95

N 18 25 25


Untrained

(n = 72) M 5.59 6.73 8.30

SD 4.08 4.27 3.96

N 24 23 25
















Table 29. Analysis of Internal Comparison Score by Treatment
for Each Training Level.



Source SS df MS F p



Trained

Treatment 16.67 2 8.33 .47 .62

Within 1019.39 65 17.58

Total 1036.06 67


Untrained

Treatment 97.23 2 48.61 2.88 .05

Within 1029.88 69 16.88

Total 1127.11 71









in Table 30. Since the effect of treatment was found to be significant

for the untrained subjects, Tukey's HSD procedure was used to determine

which Treatment group differed significantly. Using HSD = 3.40, it was

found that the General Cue group scores were significantly higher than

the Control group scores.

The results of a similar analysis for Treatment and Grade Level are

reported in Tables 31 and 32, and the combined effect of the interaction

is reported in Table 33.

Since the effect of treatment was found to be significant for the

fifth grade subjects, Tukey's HSD procedure was used to determine which

Treatment group differed significantly. Using HSD = 3.40 it was found

that the General Cue group scores were significantly higher than the

Control group.

In summary, the significant differences relevant to the major

questions in this study are as follows:

1. For fifth graders, the General Cue treatment was better than

Control for total scores.

2. The Specific Cue treatment resulted in higher scores on specific

questions for both grades and training conditions.

3. The General Cue group scored higher than Control on cued generali-

zation questions.

4. Trained subjects scored higher than Untrained subjects on internal

comparison questions unrelated to cues.

5. Eighth graders scored higher than fifth graders on internal

comparison questions unrelated to cues.

6. For Untrained subjects, the General Cue group scores were higher

than Control on internal comparison questions.

















Table 30. Interaction of Training and Treatment on Internal Comparison
Scores.


Control


Specific Cue

TREATMENT


General Cue


Trained

Untrained -----------


r
















Table 31. Means and Standard Deviations for Internal Comparison
Scores by Treatment and Grade Level.



Treatment Control Specific Cue General Cue

(n = 42) (n = 48) (n = 50)




Grade 5 M 4.50 6.29 7.35

(n = 58) SD 3.74 2.76 3.79

N 19 19 20


Grade 8 M 8.34 7.85 7.80

(n = 82) SD 4.51 4.65 4.14

N 23 29 30
















Table 32. Analysis of
Treatment.


Internal Comparison Score by Each Grade Level


Source SS df MS F p


Grade 5


Treatment

Within

Total


68.27

561.41

629.67


34.13

11.94


Grade 8


Treatment

Within

Total


4.36

1420.62

1424.98


2.18

19.73


.11 .89

















Table 33. Interaction of Grade Level and Treatment on Internal
Comparison Scores.


- -


Control


Specific Cue

TREATMENT


General Cue


Grade 5

Grade 8









7. For fifth graders, the General Cue group scores were higher

than Control on internal comparison questions.

Further discussion of these differences is presented in Chapter V.















CHAPTER V
DISCUSSION


One purpose of this study was to determine whether mathemagenic

cues in the form of generalization questions or specific questions would

enhance middle school age students' retention of material in graphic form.

A second purpose was to determine whether training in forming generaliza-

tions about data presented in graphic form would result in greater reten-

tion of either factual information or general information from graphs.

The results reported in Chapter IV indicate that cues do have an

effect on retention of information. As has been found in previous studies

(Kauchak, Eggen, & Kirk, 1975), the type of cue influenced the type of

information retained; that is, specific cues resulted in greater reten-

tion of specific facts, and gereralization cues resulted in greater reten-

tion of generalizations about the information in the graphs.

The effect of the two types of cues on total scores is more complex,

and, as a result, more difficult to interpret. There were several inter-

active effects found which led to conclusions which must be interpreted

cautiously.

Since an interaction was found between grade level and treatment,

the effect of treatment at each grade level was considered. For fifth

graders, the General Cue treatment was found to be superior, but for

eighth graders there were no significant differences between treatments.

A possible explanation for this is that eighth graders were better able










to attend to the appropriate data in the graphs regardless of the type

of cue they received, and thus were less dependent upon the cues to direct

their attention. Fifth graders, however, possibly because of fewer years'

experience with graphs, were helped by the cues which directed their

attention to the important data in each of the graphs.

The effect of the training variable was most apparent on questions

requiring internal comparisons within the graph. An example of this type

of question might be to ask whether two ounces of water per day or four

ounces of water per day resulted in taller bean plants. These questions

were designed to measure the subjects' retention of relationships within

the graph, rather than retention of specific facts. Again, interpreta-

tion of the findings is complex because of interaction between training

and treatment and between training and grade level.

For untrained subjects, the General Cue treatment was found to be

significantly superior to the Control treatment, but this effect was not

found for all trained subjects. One interpretation of this might be that

for untrained subjects, the cue directed the students' attention to the

general relationships in the graph, thus enabling them to better respond

to the internal comparison questions. Trained subjects, however, were

more likely to look for the general relationships whether or not they were

provided with the cue; that is, the training compensated for the lack of

cues.

Further support for this hypothesis is found in the interaction between

grade level and treatment. Overall, eighth graders performed better than

fifth graders on internal comparison questions. For eighth graders no

significant treatment differences were found. For fifth graders, however,

the General Cue group scored higher than the Control group. Again it

annonc that thp pinhth nradprs wPrP ahlp to rp~Dnnd to the questions










whether or not they were provided with a cue, while the presence of a

cue made a significant difference for the fifth grade students.

In general, the data indicate that all three of the variables, treat-

ment, training, and grade level, have an effect on students' ability to

learn from material presented in graphic form. There appears to be a

balancing relationship among these variables; that is, the cues are

most helpful to students who have not had the benefit of specific training,

and to younger students. Training is most helpful to students who have

not been provided with attention directing cues, and again to younger

students.

These findings do not necessarily indicate that training in general-

izing from graphs and provision of mathemagenic cues are only valuable

for younger students. The lack of significant differences for eighth grade

subjects might be attributed to the design of this particular study. In

order to make the graphs and reading material appropriate for both fifth

and eighth grade students, a compromise in reading level probably produced

material that was slightly difficult for the fifth graders and perhaps less

difficult for the eighth graders. Therefore, the fifth graders may have

been more dependent on whatever assistance was provided by the training

of cues, and the eighth graders may have needed these aids less than might

be the case with reading material or information of higher difficulty level.


Conclusions

The results of this study led to the following conclusions.

1. Mathemagenic cues do increase the retention of graphically

presented information.

2. Students in upper elementary school and middle school can be

trained to generalize from graphs.
*i TL^ ..^^ ^A ^C / i~ r -f ^*wr t'lr^^ u rti-n --l--.l +-r. CfIi"! e-irnnr rt- i ~n-Fi iwm ';nn









retained; that is, specific cues result in better retention of specific

factual information, and general cues result in better retention of

general relationships shown in the graphs.


Implications

The results of this study indicate that upper elementary and middle

school students' learning from graphs can be increased by the use of

mathemagenic or attention directing cues. This has direct implications

for text book writers and classroom teachers. In texts where graphs are

utilized to present data, cues in the text can be used to facilitate stu-

dents' inspection of these graphs.

Similarly, classroom teachers can use cues to supplement textbook

Fresertations where no such cues are provided by the authors. These cues

might take the form of reading guides or study questions to accompany a

reading assignment.

The results which show that students can be trained to generalize

from graphs are extremely important. The primary value of graphic presen-

tation of data is to show relationships which exist within this data.

Therefore, if students can le trained to look for these relationships

rather than specific facts, the purpose of the graphic representation

will be better achieved.

The results which shove that specific factual cues increase the reten-

tion of specific factual information and general cues increase the reten-

tion of general relationships within the graphic further supports the

notion that the level of questioning is directly related to the level of

learning of any material. The implications of this are clear. If it is

desirable for students to generalize from graphic data, they must be asked

questions which require them to do this.








Future Research


Much further research is needed to clarify the answers to questions

raised by this study. It is not clear whether differences in performance

at the two grade levels were a result of the level of the experimental

material or whether there are developmental differences involved. A

study which presented similar material on different grade levels might be

designed to answer this question.

The training component of this study was rather limited in nature.

The effect of long term training in generalizing from graphs would.

almost surely be stronger, but studies should be designed to measure this

effect.

The question of whether typical classroom teachers can be taught to

utilize mathemagenic cues has not been investigated at any level or with

any kind of material. This is an extremely important area for research,

since it is of little value to find that mathemagenic cues are helpful

to learners urless it is also found that teachers can be trained to use

these cues.


Summary

The purpose of this study was to assess the effects of mathemagenic

cues in two forms, identification of a correct generaliztion about the

graphic data or response to a specific factual question about the data,

and the effects of training in forming generalizations about graphs on

middle school age subjects' learning of material presented in graphic

form.

Previous research has shown that mathemagenic cues facilitate

learning of prose material, and limited research has demonstrated that

similar results are found with graphic material. The present study altered








the forms of these cues anc' involved elementary and middle school age

children, since it is during these years that graph reading and interpre-

tation is first taught.

The results of the study show that these cues can result in increased

retention and that training does effect these results.

As has been found in studies of verbal prose, a given type of cue did

increase retention of the same type of information; that is, a specific

cue was related to better retention of specific factual information, and

a general cue resulted in better retention of the general relationships

shown in a graph. In this study, however, an interactive effect with

training was found, so that subjects who had been trained to generalize

from graphs were more likely to do this regardless of the type of cue

with which they were presented.

As might be expected, eighth graders consistently scored higher

than fifth graders on all types of questions and with all experimental

conditions.

General inspection was found to be best facilitated by the general

cues. This was most clearly shown by higher scores on a set of questions

requiring retention of internal relationships within the data shown in

the graphs.
































APPENDIX A
TRAINING PACKET










A graph is a picture or diagram which is used to give information

in a way that is easy to read. Graphs are often used to show how things

are related to each other, or how one thing changes when the other

changes.

Before we begin to learn about graphs let's take a pretest to see

how much you already know.



















(GRAPH DISPLAYED HERE)


The graph above has each of these sets of words:

A. AGE OF CHILD IN YEARS
B. AVERAGE TIME SPENT WATCHING TELEVISION EACH DAY
AT DIFFERENT AGES.
C. TIME IN MINUTES

Choose the set which answers each question below. Please circle
A, B, C for your answer. More than one answer might be needed.

1. Which set or sets of words give the title of
this graph? A B C

2. Which set or sets of words give labels for this
graph? A B C

3. In what units is the time spent watching TV measured?

a. Seconds
b. Minutes
c. Hours
d. Days
e. Years


















(GRAPH DISPLAYED HERE)


4. How much time did two-year olds spend watching TV each day?


10 minutes
10 hours
50 minutes
50 hours
100 minutes
100 hours


5. How much time did three-year olds spend watching TV each day?


50 minutes
60 minutes
70 minutes
80 minutes
90 minutes
100 minutes


6. Based on the information in the graph, which is the best
statement to make about the amount of time spent by twelve
year olds?

a. Twelve year olds probably spend less than 50 minutes per
day watching TV.
b. Twelve year olds probably spend more than 50 minutes per
day watching TV.
c. This graph does not give enough information to answer this
question about twelve year olds.


















(GRAPH DISPLAYED HERE)


7. Which of these generalizations
the graph?


best describes the data in


As children get older they spend more time watching TV.
As children get older they spend less time watching TV.
As children get older they spend more time watching TV up
to a point, then they watch TV less and less.
Age does not seem to be related to the amount of time
children spend watching TV.










Today we are going to learn about BAR GRAPHS.

This is a bar graph:








(GRAPH DISPLAYED HERE)













A graph is used to give information in a way that is clear and easy

to understand. It gives you a picture which would often take many, many

words to explain.

There are some important parts that every graph must have so that

everyone will know exactly what the graph means.

First of all, every graph must have a TITLE. The title of this graph

is "HEIGHT OF BEAN PLANTS GROWN IN DIFFERENT AMOUNTS OF WATER."

The title of a graph tells us what it is about, so it is very

important.

Another important part which every graph must have is a LABEL for

each side. The labels tell us what things we are comparing and what kind

of units are used to measure each thing.




















(GRAPH DISPLAYED HERE)


In this graph, one label "Ounces of Water Per Day" tells us that

we are comparing different amounts of water. It also tells us that the

water was measured in ounces.

The other label "Heights in Centimeters" tells us that we are com-

paring the heights of bean plants and the unit that was used to measure

the plants.

Without these labels the graph would not be clear . we would not

understand it. We need to know if the bean plants were measured in inches,

feet, centimeters or some other units.

According to the label on this graph, what kind of unit was used to

measure the bean plants?

a. inches
b. ounces
c. centimeters
d. meters
e. feet



















(GRAPH DISPLAYED HERE)















You should have chosen centimeters because the label says "Height in

Centimeters."

Instead of many words, this graph uses bars to give us information

about how tall each plant grew when it was given each different amount

of water.

Look at the bar which shows the height of the bean plant which got

2 ounces of water each day. You can see that this bean plant was 14 cen-

timeters tall. How tall was the bean plant which got 4 ounces of water

per day?

The correct answer is 16 centimeters. Look at the graph and be sure

you understand why this answer is correct.









Now let's look at another graph.


(GRAPH DISPLAYED HERE)


What is the title of this graph?



















(GRAPH DISPLAYED HERE)


The title of this graph is Amount of Paper Collected in Paper Drive.

To understand this graph you need to know how the amount of paper

was measured. Which one of these things was done to measure the paper?

a. They counted the number of papers.
b. They weighed the paper.
c. They stacked up the paper and measured the height of the stack.

You can tell by looking at the label which says "Weight in Pounds"

that they weighed the paper.

Now let's look at the data in the graph.

Notice that the graph shows how much paper students collected in two

months. The amount they had collected the first month is shown by the

shaded part of the bar, and the total amount at the end of two months is

shown by the height of the bar.

How much paper was collected by students in Grade 2 during the first


month?




















(GRAPH DISPLAYED HERE)


Look directly across from the shaded part of the bar for Grade 2

and you will see that the answer is 25 pounds.

How much paper was collected by these same students all together?

You should have looked at the top of the bar which is even with 45 pounds.

How much paper was collected by students in Grade 1 by the end of the

first month? ____ Ten pounds is the correct answer. How much was

collected all together by Grade 1?

That question is a little harder -- you can see that the bar for Grade

1 comes between two marks. The correct answer is 17 pounds.




















(GRAPH DISPLAYED HERE)


What grade collected the greatest amount of paper?

To answer that question you should look for the highest bar. On

this graph Grade 2 has the highest bar, so we know that Grade 2 collected

the greatest amount of paper.

Which grade collected the smallest amount of paper?

This time you have to look for the shortest bar. You can see that

Grade 5 collected the smallest amount of paper.










Now that you have become an expert at graph reading, you are ready

for the next step. One of the main reasons for using graphs is to show

a general picture or pattern so to give people a general idea about how

one thing affects another.

For example, let's look at another graph.







(GRAPH DISPLAYED HERE)













Suppose you were trying to give someone a general idea about what

this graph says. You would want to say something about how the amount of

time spent studying spelling words affects the students' grade on their

spelling test. Judging by the graph you might say:

"The more time spent studying the better the grade on the spelling

test."

A statement like this is called a generalization. It gives you a

general idea about how two things are related -- in this case, about how

study time is related to spelling grades.



















(GRAPH DISPLAYED HERE)














You have to be careful about making generalizations. For example,

we decide "More study time means higher spelling grades." But, if a

student spent 8 hours studying his spelling words he might be so tired

he would fall asleep in the middle of his test. We can only make our

generalization about study times between 5 minutes and 30 minutes.

In order to make a generalization about the information in a graph

you have to look at the pattern made by the bars. In this graph the bars

get taller as the study time gets greater.

If the bars made a different pattern we would made a different gener-

alization about it. Let's change this graph a little so that the bars

are different.



















(GRAPH DISPLAYED HERE)















Now our generalization about this graph would be quite different.

We might say "The more time spent studying, the lower the spelling grades."

The two patterns we have looked at have been very simple. Either

the graphs have gotten taller each time or shorter each time. But some-

times patterns are a little more difficult.

Let's change the graph again.




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