Introducing addition and subtraction symbols to first graders

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Introducing addition and subtraction symbols to first graders
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Colvin, Suzanne McWhorter, 1960-
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Mathematical notation -- Study and teaching   ( lcsh )
Arithmetic -- Study and teaching   ( lcsh )
Curriculum and Instruction thesis Ph.D
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Thesis:
Thesis Ph.D.--University of Florida, 1987.
Bibliography:
Bibliography: leaves 144-148.
Statement of Responsibility:
by Suzanne McWhorter Colvin.
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Typescript.
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Vita.

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INTRODUCING ADDITION AND SUBTRACTION
SYMBOLS TO FIRST GRADERS







By

SUZANNE MCWHORTER COLVIN


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

UNIVERSITY OF FLORIDA


1987













ACKNOWLEDGEMENTS


I wish to express my sincere appreciation to Dr.

Suzanne Krogh, Dr. Doreen Ross, Dr. Roy Bolduc, Dr. Linda

Lamme, Dr. Alan Agresti, and Dr. David Knauft for their

support and advice during the course of this study and

preparation of this manuscript. Special thanks are extended

to Dr. Suzanne Krogh, chairperson of my committee, for her

time, expertise, patience, and sense of humor.

I would like to thank Jean Cunningham, Carol James, and

Debbie Smart for allowing me to use their classrooms as a

part of this study. I would also like to thank Debra Krank

for permitting me to conduct the study at Duval Elementary

School and for the support and encouragement she provided.

I will forever be indebted to my parents, William

Horace and Kathleen Story McWhorter, for their undying faith

in me and continued support and encouragement. They

instilled in me a love for education and learning, a desire

to succeed, and the faith in myself and my abilities needed

to undertake such an endeavor. They have been a source of

constant support and continued love and affection throughout

my education.

I wish to thank my father- and mother-in-law, Geral

Daniel and Claudette Colvin, for their love and

encouragement as well as their patience and kindness.









Finally, I owe most of my gratitude to my husband,

Danny, who helped to provide the incentive needed to

complete this degree. He displayed an enormous amount of

patience during the preparation of this manuscript and

always supplied me with the love, encouragement, and

confidence needed to sustain me during the course of my

education.


iii














TABLE OF CONTENTS


PAGE
ACKNOWLEDGEMENTS ................................... ii

ABSTRACT .............. ......... ....................... vi

CHAPTER 1. INTRODUCTION ........................... 1

Statement of the Problem ............. 5
Significance of the Study ............ 7
Definition of Terms .................. 11
Statement of Hypotheses .............. 14

CHAPTER 2. REVIEW OF THE LITERATURE 16

Solving Addition and Subtraction
Number Sentences .................... 16
Solving Addition and Subtraction Word
Problems ............ ..... ......... 22
Initial Instruction On Addition and
Subtraction Symbols .................. 27
Summary ............................. 33

CHAPTER 3. RESEARCH METHODOLOGY .................. 36

Subjects .............. ........ ...... .. 36
Data Collection Instruments and
Procedures ......................... 37
Procedures ........................... 44
Data Analysis ...................... 49

CHAPTER 4. ANALYSIS AND INTERPRETATION OF THE DATA 52

Preassessment Results ............... 53
Postassessment Results ............... 57
Clinical Interviews .................. 87

CHAPTER 5. SUMMARY, CONCLUSIONS, DISCUSSION,
LIMITATIONS, AND IMPLICATIONS ......... 109

Summary of the Study ................. 109
Conclusions and Discussion .......... 111
Observations ......................... 125
Limitations of the Study ............. 126
Implications of the Study ............ 128
Suggestions for Further Research ..... 129
Concluding Remarks .................. 130










APPENDICES ................................. .. 134

A SUMMARY TABLES OF STATISTICAL ANALYSES. 134

B SAMPLE RESEARCH INSTRUMENTS .......... 139

C FORMAT FOR CLINICAL INTERVIEWS ........ 142

REFERENCES .............. ........................... 144

BIOGRAPHICAL SKETCH ................................ 150













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



INTRODUCING ADDITION AND SUBTRACTION
SYMBOLS TO FIRST GRADERS

By

Suzanne McWhorter Colvin

May, 1987


Chairperson: Dr. S. 'L. Krogh
Major Department: Instruction and Curriculum


It was hypothesized that the time of introduction to

addition and subtraction symbols would have no effect on

first graders' ability to solve, produce, and interpret

number sentences, as well as on their ability to solve

story problems. A pretest-posttest comparison group design

was used to test this hypothesis. Subjects were 66 first

graders at Duval Elementary School, Gainesville, Florida.

Pretests were administered to three separate classrooms

measuring subjects' ability to solve, produce, and interpret

addition and subtraction number sentences and solve addition

and subtract n story problems. There were no statistically

significant differences in mean scores among the classrooms

( > .10).

Each classroom received a separate treatment. The

traditional group was introduced to number sentences before









solving related story problems, the immediate group

immediately after solving related story problems, and the

delayed group after solving story problems orally for 5

weeks.

Posttests identical to the protests were administered

following the treatment period. Mean scores for the

immediate and delayed groups were significantly higher than

for the traditional group on the measure of total symbol

understanding (p < .05). The immediate group had a

significantly higher mean score than the delayed and

traditional groups on measures of ability to solve and

produce number sentences. And the delayed group had a

significantly higher mean score on the measure of ability to

interpret number sentences.

Four weeks later, subjects were given a second set of

posttests identical to the protests and posttests

administered earlier. There were no significant differences

on the measure of ability to solve number sentences. The

immediate and delayed groups had significantly higher means

than the traditional group on measures of ability to produce

and interpret number sentences and solve story problems.

Before and after instruction, clinical interviews were

conducted to examine strategies employed when solving word

problems. Overall, subjects used more problem solving

strategies after instruction than before instruction as well

as more efficient problem solving strategies.


vii














CHAPTER 1
INTRODUCTION



It seems that even before formal instruction, young

children have intuitively formed ideas about addition and

subtraction (Hiebert, 1984). Children who are of the ages

of 3, 4, and 5 understand that addition increases numerosity

and subtraction decreases it (Brush, 1978). They also

realize that addition and subtraction are inverse operations

(Starkey & Gelman, 1982). Not only do children have an

understanding of many of the properties of addition and

subtraction prior to formal instruction, they are able to

use their understandings to solve addition and subtraction

story problems (Carpenter, Hiebert, & Moser, 1981; Ibarra

& Lindvall, 1982). In a study by Carpenter, Hiebert, and

Moser (1981), first graders were able to solve a variety of

story problems employing a variety of strategies before ever

receiving instruction in school.

As children proceed through school, they seem to lose

their intuitive knowledge of addition and subtraction and

their natural ability to solve problems (Wearne & Hiebert,

1984). Previous to formal instruction, first graders tend

to use a variety of problem solving strategies to solve

problems. After carefully analyzing the problem, they

decide upon or invent a strategy which makes sense to them









and use that strategy to solve the problem. After the

introduction to formal arithmetic, however, children begin

relying upon single strategies learned through formal

instruction rather than using their own abilities to analyze

and solve problems (Carpenter, Hiebert, & Moser, 1981).

Many students blindly apply memorized rules and algorithms

which have little or no apparent meaning and hope their

answer is correct. Before long, mathematics is viewed as

meaningless and incomprehensible with little or no

relationship to the outside world (Kamii, 1982; Wearne &

Hiebert, 1984). The National Assessment of Educational

Progress (1983) reported that by the time students are

17, they perceive mathematics as an application of rules,

not necessarily understood, used to get a "right" answer.

It is when children are introduced to the formal

symbolism of addition and subtraction that they experience

the most difficulty (Ginsburg, 1977; Hiebert, 1981; Hiebert,

1984). While children can be taught to use numerical and

operational symbols, there is no assurance that they have an

understanding of those symbols. Many children can be

proficient in computation without understanding the

computational process and the associated symbols (Ginsburg,

1977). For example, Ibarra and Lindvall (1980) found that

many primary children able to successfully solve simple

addition and subtraction number sentences have difficulty in

explaining the meaning of number sentences.









There are many who believe young children are incapable

of giving meaning to the arbitrary symbolism of mathematics

(Hiebert, 1984; Piaget, 1952) and that introduction to

mathematical symbolism should be delayed (Driscoll, 1979;

Lovell, 1971). Others have suggested that it is not that

children are incapable of understanding the symbolism

associated with addition and subtraction, rather that

teachers rely upon traditional teaching methods which fail

to help form a link between what children already understand

and are using and the symbols they are being introduced to

in school (Hiebert, 1981).

Typically, children are introduced to addition and

subtraction symbolism through instruction on solving number

sentences or equations (Campbell, 1984; Hiebert, 1984;

Kamii, 1985). Students are presented with a number sentence

and are shown a technique, sometimes involving manipulatives

or pictures, for solving the number sentence. Instruction

on the meaning of individual symbols is usually provided

along with the rules for using those symbols (Hiebert,

1984). Quite often, teachers will provide a story problem

to go with the number sentence after it has been introduced

(Campbell, 1984; Kamii, 1982). In this way, teachers feel

that they are helping children to see the relationship

between number sentences and the actions represented by

those number sentences. Research indicates that this is not

the case. Students fail to connect the word problem with

the number sentence representing it, treating number









sentences and story problems as if they were completely

unrelated (Carpenter, Hiebert, & Moser, 1983; Groen &

Resnick, 1977; Hiebert, 1984; Kamii, 1985; Kantowski, 1983;

Lindvall & Ibarra, 1981; Wearne & Hiebert, 1984).

Many instructional approaches introduce addition and

subtraction symbols before direct experience with objects,

actions, or verbal problem solving. Researchers are

suggesting that introducing symbols before verbal problems

may not be appropriate. They suggest that to be successful

in providing a link between the written form and the

mathematical concept, teachers must do more than simply

relate a verbal problem to a number sentence (Grouws, 1972)

or use verbal problems as a means of seeing if students can

apply previously learned computational skills (Kantowski,

1981). Instead, they suggest that story problems serve as

the context in which addition and subtraction symbols are

introduced with equations serving as a representation of the

actions associated with the story problems (Campbell, 1984;

Hiebert, 1981; Kamii, 1985; Wearne & Hiebert, 1984). Only

after a great deal of experience with verbal problems and

real objects and events should students be introduced to

operational equations used to represent those experiences

(Baratta-Lorton, 1976; Hiebert, 1981; Kamii, 1985; Lovell,

1971).

Others agree that the introduction of symbols should

occur after direct experiences with objects or verbal

problems, but contend that the presentation of the symbols









should immediately follow the experience. Once a student

solves a problem verbally or through the manipulation of

objects, representation of that problem solving experience

should take place by writing a number sentence. That way,

frequent links can be made between actions and symbolic

representations (Campbell, 1984; Hiebert, 1981; Wearne &

Hiebert, 1984).

In order to determine at what point in the

instructional process the introduction of symbols should

occur, it is necessary to investigate the effects of a

variety of instructional approaches within a classroom

setting (Weaver, 1982). This study investigated three

instructional approaches designed to introduce first grade

children to addition and subtraction symbolism. The time at

which symbols were introduced during instruction varied with

each instructional approach.



Statement of the Problem



Past research indicates that children experience a

great deal of difficulty when introduced to the formal

symbolism of addition and subtraction. Children seem to

lose their intuitive ability to solve problems and are

unable to understand and interpret addition and subtraction

symbolism (Hiebert, 1981; Hiebert, 1984; Ibarra & Lindvall,

1982; Wearne & Hiebert, 1984). The point at which symbols

are introduced during instruction seems to be crucial in









children's understanding of addition and subtraction

symbolism. Some researchers contend that the introduction

of symbols should be postponed until students have had a

variety of experiences with objects, events, and verbal

problems (Driscoll, 1979; Lovell, 1971; Piaget, 1952).

Others contend that symbols should be introduced immediately

following each activity with objects and verbal problems

(Campbell, 1984; Hiebert, 1981; Wearne & Hiebert, 1984).

And still others endorse introducing addition and subtrac-

tion symbols in the traditional manner in which children

are first presented with a number sentence, instructed as to

how to use the symbols in the number sentence, and offered

techniques or procedures for successfully solving the number

sentence (Hiebert, 1984). It was the intent of this study

to examine the effectiveness of each of these approaches in

helping children to successfully solve and interpret

addition and subtraction problems and word problems.

More specifically, the purpose of this study was to

examine three instructional approaches designed to introduce

first graders to addition and subtraction symbolism and the

effect of each of these instructional approaches immediately

and over time on first graders' ability to do the following:



1. solve written addition and subtraction number
sentences;

2. give meaning to addition and subtraction
symbols as indicated by the ability to solve,
produce, and interpret addition and subtraction
number sentences;









3. solve addition and subtraction story problems
presented orally.

In addition, the effect of instruction on the processes and

strategies children used to solve addition and subtraction

story problems was investigated.



Significance of the Study



One of the greatest weaknesses of school mathematics is

the failure to provide children with an understanding of the

written symbolism of mathematics (Ginsburg, 1977),

especially since the majority of school mathematics is

comprised of learning to manipulate symbols in computation

(Ginsburg, 1977; Greenes, 1981). School arithmetic is based

on written numbers and symbols such as addition and

subtraction symbols. In order to be successful in

arithmetic, children must learn to use symbols properly in

computation and must understand symbols (Ginsburg, 1977).

Otherwise, mathematics becomes a process of applying

meaningless rules and procedures to get the "right answer"

with little thinking and reflection on the part of students

(Kamii, 1982; Wearne & Hiebert, 1984).

While children can learn to use addition and

subtraction symbols very early they apparently have little

or no understanding of those symbols. This is evident in

their inability to interpret symbols (Ginsburg, 1977;

Hendrickson, 1979; Lindvall & Ibarra, 1980) as well as in

their inability to assess the reasonableness of their









answers. Students apply memorized rules and procedures and

assume their answer is correct regardless of what the

symbols say (DeCorte & Verschaffel, 1981). For example,

students are often taught the rule that a plus sign

indicates that two numbers should be combined. When

students are faced with problems such as 3 + = 7, they

are very likely to blindly apply the rule they have learned

and combine the 3 and the 7 to arrive at an answer of 10.

Even as unreasonable as this answer is, many students would

not recognize it as such since they had, in their minds,

correctly applied the rule.

Before formal instruction, children solve problems by

analyzing and interpreting a problem, deciding upon a

strategy based upon their interpretation of the problem, and

applying the strategy to solve the problem (Carpenter,

Hiebert, & Moser, 1981; Hiebert, 1984; Wearne & Hiebert,

1984). If an answer is unreasonable, children will usually

notice it, unlike many children who have received formal

instruction, because they have carefully analyzed the

problem first. As a result, they are able to successfully

solve a variety of problems using a variety of strategies

and be reasonably sure that their answers are correct

(Carpenter, Moser, & Hiebert, 1981).

Because children who have received formal instruction

tend to apply memorized rules and procedures without

considering the problem structure, they are likely to

incorrectly solve problems unfamiliar to them which have no









memorized algorithms that can be directly applied

(Kantowski, 1981). The National Assessment of Educational

Progress (1981) showed that the majority of students at all

age levels could not solve nonroutine problems (problems

which have no algorithm that can be applied to guarantee a

solution) which required some analysis. This is an

indication that while students are learning to use

mathematical symbols and prescribed procedures, they have

limited understanding of those symbols and procedures.

It is crucial that students be introduced to formal

arithmetic in such a way that they are able to use and

understand the symbolism while retaining their abilities to

analyze and solve problems including unfamiliar or

nonroutine problems. Apparently, many of the instructional

methods used today are not sufficient in accomplishing this

and new instructional techniques must be investigated.

Not only is it crucial to examine how symbols are

introduced, it is also important to examine when they are

introduced. It has been suggested that verbal problems

should serve as a context in which to introduce addition and

subtraction symbols (Campbell, 1984; Hiebert, 1984; Wearne &

Hiebert, 1984). Questions pertaining to whether it is

better to introduce symbols before or after experience with

verbal problems should be addressed as well as how soon

before or after experiences with verbal problems. Some

researchers suggest postponing the introduction of symbols

until students have had a great deal of experience with









verbal problems and manipulatives (Driscoll, 1979; Kamii;

1985; Lovell, 1971; Piaget, 1952). Others feel symbolism

should be introduced immediately following experiences with

verbal problems and concrete manipulatives to establish

frequent and clear links between verbal and concrete

representations and the written symbolic form (Campbell,

1984; Hiebert, 1984; Lindvall & Cica, 1982; Wearne &

Hiebert, 1984). It is necessary to find out at what point

in instruction it is better to introduce symbols. This

this issue was investigated as a part of this study.

In addition to finding out how methods of instruction

affect children's ability to use and understand addition and

subtraction symbols, how children's problem solving

processes were affected by instruction was examined as well.

Educators are increasingly concerned with the way in which

children process information and how instruction may affect

that processing. Much of the current mathematics research

has been more concerned with children's processing of

addition and subtraction rather than whether or not children

can be taught to correctly solve addition and subtraction

problems (Romberg, 1982). Research has focused on how

children process information before, during, and after

formal instruction (Carpenter, Hiebert, & Moser, 1981).

Formal instruction has often been found to discourage

processes of analysis, thinking, and interpretation, and

encourage processes based upon memory abilities (Hiebert,

1981; Kamii, 1982; Wearne & Hiebert, 1984). It is









important, therefore, to investigate whether certain

instructional procedures enhance or interfere with thinking.



Definition of Terms



Addition and Subtraction Symbols

Addition and subtraction symbols refer to written

numbers and operational signs of addition and subtraction.

The plus sign (+), for example, is the symbol for addition

while the minus sign (-) is the symbol for subtraction.

Addition and subtraction symbols are usually presented in

the form or a number sentence or equation in the form of

"a + b = c" or "a b = c".

Meaning of the Symbol "a + b = c" or "a b = c"

Meaning of the symbol "a + b = c" or "a b = c", as

defined by Fordham (1974/1975) and Hamrick (1976), refers to

the pairing in an individual's mind of the symbol with an

action that is appropriate for that symbol. Appropriate

actions are those which lead from a representation of the

problem to an answer to the question, "How many are

represented by the symbol?". An appropriate action for the

symbol "5 3 = would be to construct a set of five

objects and remove a set of three objects to count or

identify the remaining objects.

Producing Number Sentences

Producing number sentences involves writing number

sentences which accurately represent story problems or









actions with objects. For example, after listening to a

story about a child who had 3 marbles and obtained 2 more, a

student might produce the number sentence, "3 + 2 = 5".

This number sentence would appropriately represent the story

problem.

Interpreting Number Sentences

Interpreting number sentences involves telling a story

or demonstrating an action which accurately represents a

written number sentence. For example, when shown the number

sentence, "7 4 = 3", a child might tell a story about a

friend who had 7 cookies and ate 4 of them and only 3 were

left.

Story Problems

Story problems are computational problems presented in

story format. In this study, story problems consisted of

addition and subtraction story problems. Story problems are

also referred to as verbal problems. An example of a simple

addition story problem is, "There are three children drawing

on the blackboard. If two more children draw on the

blackboard, how many children will be drawing on the

blackboard altogether?"

Addition Story Problems

Addition story problems are those which require

students to use the operation of addition to solve the

problem. Carpenter, Hiebert, and Moser (1981) divided

addition story problems into the following classes:









1. join problems -- a set increases in quantity as in

the following problem: "Danny had 3 pennies. His

father gave him 2 more. How many pennies did Danny

have altogether?"

2. combine problems -- two or more sets are combined as

in the following problem: "Adam has 6 red blocks and

3 blue blocks. How many blocks does he have

altogether?"

3. compare problems -- two quantities are compared as

in the following problem: "Katie has 3 pieces of

candy. Tim has 4 more pieces of candy than Katie.

How many pieces of candy does Tim have?"

Subtraction Story Problems

Subtraction story problems are those which require that

students use the operation of subtraction to solve the

problem. Carpenter, Hiebert, and Moser (1981) divided

subtraction story problems into the following classes:

1. separate problems -- a smaller quantity is removed

from a larger quantity as in the following problem:

"Bill had 7 marbles. He lost 4 of them. How many

does he have left?".

2. combine problems -- there is no action direct or

implied; two quantities may be considered as parts

of a whole as in the following problem: "There are 6

people on the playground. Four are girls. How many

are boys?"









3. compare problems -- two quantities are compared as

in the following problem: "Martha has 4 kittens.

Robin only has 2 kittens. How many more kittens

does Martha have than Robin?"

4. equalize problems -- students must decide how to

make two quantities equal as in the following

problem: "Kathy picked 8 flowers. Lil only had 5

flowers. How many flowers does Lil have to pick to

have as many flowers as Kathy has?"

Addition and Subtraction Number Sentences or Equations

Addition and subtraction problems presented in the form

of "a + b = c" or "a b = c" are referred to as number

sentences or equations. The letters "a", "b", and "c"

represent numerals.

Problem Solving Processes or Strategies

Problem solving processes or strategies are techniques

used to solve a problem such as modeling with fingers or

objects, using counting sequences, or recalling basic number

facts.



Statement of Hypotheses



The following null hypotheses were tested in this

study immediately following the treatment period as well as

4 weeks later:



1. The time of introduction of addition and

subtraction symbols will have no effect on first










graders' ability to write answers for written

addition and subtraction number sentences.

2. The time of introduction of addition and

subtraction symbols will have no effect on first

graders' ability to produce addition and subtraction

number sentences.

3. The time of introduction of addition and

subtraction symbols will have no effect on first

graders' ability to interpret addition and

subtraction number sentences.

4. The time of introduction of addition and

subtraction symbols will have no effect on first

graders' ability to demonstrate the meaning of

addition and subtraction symbols as measured by

their ability to solve, produce, and interpret

number sentences.

5. The time of introduction of addition an subtraction

symbols will have no effect on first graders'

ability to solve addition and subtraction story

problems presented orally.

Once data were collected, the results were analyzed in

order to determine whether the investigation produced

evidence that supported the above hypotheses. Any

hypothesis tested which was found to be significant at the

.05 level of significance or below would be rejected.















CHAPTER 2
REVIEW OF THE LITERATURE



For years, an overriding concern in mathematics

education has been the children's learning of arithmetic.

Teachers, educational administrators, and parents all expect

children to learn to add, subtract, multiply, and divide

with efficiency and accuracy. As a result, a great deal of

research investigating how children acquire operational

concepts and skills has emerged. This is especially true in

the area of addition and subtraction, the first two of the

four basic operations to be taught in school. The following

is a review of the research which examines how children are

introduced to and learn addition and subtraction.

The review of the literature is divided into the

following sections: (a) Solving Addition and Subtraction

Number Sentences, (b) Solving Addition and Subtraction Word

Problems, and (c) Initial Instruction on Addition and

Subtraction Symbols.



Solving Addition and Subtraction Number Sen*ences



The majority of the early research on addition and

subtraction examined children's ability to compute and was









concerned with problem difficulty (Carpenter & Moser, 1982).

There were attempts to rank number facts from most difficult

to least difficult so that teachers could better know in

what order basic facts should be introduced (Brownwell,

1941). Findings among various studies were inconsistent,

however, and as a result no real rankings of the most to the

least difficult basic addition and subtraction facts emerged

(Carpenter & Moser, 1983). Researchers do agree, however,

that in general, subtraction problems are more difficult to

solve than addition (Baroody, 1984), problems with larger

numbers are more difficult than problems with smaller

numbers (Carpenter & Moser, 1983), and problems with a

missing addend tend to be among the most difficult to solve

(Grouws, 1972; Weaver, 1971).

Current researchers investigating children's

understanding of basic number concepts attempt to describe

not only what problems can be solved but how those problems

are solved as well. Research related to addition and

subtraction has focused on what strategies are used to solve

problems and how children process addition and subtraction

information (Carpenter & Moser, 1982). Methods used to

identify these processes usually involve some kind of

interviewing technique (Carpenter & Moser, 1982; Steffe,

Thompson, & Richards, 1982). Children are individually

presented problems, and by observing and recording responses

to probing questions, researchers infer how particular

problems are solved. There are many limitations to using









interviews, such as the inaccuracy of children's

explanations, the effect an interviewer may have on

children's problem solving strategies, and the subjective

judgment of the experimenter. Regardless of these

limitations, the clinical interview is recommended as the

most direct and accurate measure of the processes that

children use in solving problems (Carpenter & Moser, 1983).

From using interviews, several basic addition and

subtraction problem solving strategies have been identified.

Addition Strategies

Carpenter and Moser (1982) have identified three levels

of addition strategies: strategies based on direct modeling

with fingers or physical objects, strategies based on the

use of counting sequences, and strategies based on recalled

number facts. In the most basic strategy, children use

physical objects or fingers to represent each addend and the

union of the two addends is counted starting with one. This

strategy is called counting all with models and is used

frequently by first grade children (Carpenter & Moser,

1982).

Three strategies based on the use of counting sequences

have been identified. The first identified by Suppes and

Groen (1967) and Groen and Parkman (1972) is called the

counting all without models whicn is similar to the counting

all with models except that no physical objects or fingers

are used. This requires that children keep track of the

number of counting steps in order to know when to stop









counting. Often children will use a rhythmic or cadence

counting. The other two strategies are the counting on from

first strategy and the counting on from larger strategy.

The first of these strategies involves counting forward from

the first addend in the problem. The counting sequence

begins with the first number given in the problem and

continues the number of units represented by the second

number. The answer is the final number in the sequence.

The counting on from larger strategy is identical except

that the child begins counting from the larger addend and

continues the number of units represented by the smaller

addend. Again, the answer is the final number in the

sequence. Sometimes fingers or objects are used with

counting strategies to help children keep track of how many

units have been counted (Baroody, 1984).

The final addition strategy involves the application of

a known or derived addition fact. A derived fact is

generated from a small set of known basic facts usually

based on doubles or numbers whose sum is 10 (Carpenter &

Moser, 1982).

Subtraction Strategies

Several subtraction strategies have also been described

by Carpenter and Moser (1983). Among the strategies

identified are those involving direct modeling with objects

or fingers, those involving counting sequences, and those

involving the use of number facts. The following describes

each of those strategies.









Direct modeling strategies include the separating

from, the adding on strategy, and the matching strategy.

The separating from strategy involves a subtractive action

where a larger quantity is represented and the smaller

quantity is removed from it. With the separating from,

children use concrete objects or fingers to construct the

larger set, then remove the smaller set counting the objects

that remain. For example, 6 4 would involve counting out

six objects or fingers, counting and removing two of the

items, and counting the remaining items to arrive at an

answer.

Another strategy using objects or fingers is a strategy

involving an additive action called the adding on strategy.

With the adding on strategy, a child starts with the smaller

quantity and constructs the larger. A number of objects is

set out equal to the smaller addend then objects are added

to that set until the new collection is equal to the larger

addend. Counting the number of objects added gives the

solution.

The final strategy involving the use of objects or

fingers is the matching strategy. A child puts out two sets

of objects and sets are matched one-to-one. Counting the

unmatched objects gives the answer. The choice strategy

involves a combination of counting down from and counting up

from given strategies. In this case, a child decides which

strategy requires the fewest number of counts and solves the

problem accordingly. This can only be observed when a child

is asked to solve several different problems.









Among the strategies involving the use of counting

sequences are the counting down from strategy and the

counting up from given strategy. With the counting down

from strategy, children initiate a backward counting

sequence beginning with the larger number and the last

number uttered of the counting sequence is the answer. The

child must count backwards a certain number of steps. This

procedure usually entails a forward count to keep track of

the subtrahend or some other device to keep track such as

matching the backward count to fingers. In effect, this

procedure involves a forward count in order to keep up with

how many fingers were counted down. Thus counting down

requires two simultaneous processes that in effect go in

opposite directions (Baroody, 1984). As a result, the

counting down from strategy was found to be very difficult

and rarely used by first graders (Carpenter & Moser, 1982).

In the counting up from given strategy, a child

initiates a forward counting process beginning with the

smaller given number and ends with the larger given number.

By keeping track of the number of counting words uttered,

the child arrives at an answer. Counting up strategies

model a missing addend approach to subtraction (Carpenter &

Moser, 1982).

The final subtraction strategy as with addition

strategies, is the recalling number facts or using derived

facts. These strategies are more often employed by older

students as a result of instruction in school. Some first









graders use basic recall of facts if they have had

experience with addition and subtraction facts at home or in

preschools.



Solving Addition and Subtraction Word Problems



Helping children to become efficient problem solvers is

a widely recognized goal of mathematics education today.

The National Council of Teachers of Mathematic's (1980)

states that problem solving should be the primary focus of

school mathematics in the 1980s. Despite the accepted

importance of problem solving, many students are not capable

of solving relatively straightforward mathematics problems

(Carpenter, Mathews, Lindquist, & Silver, 1984; Silver &

Thompson, 1984; Zweng, 1979). The problems which seem to

present the most difficulty are nonroutine problems

(Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980) or

problems that can not be solved by a routine application of

a single arithmetic operation or algorithm (Kantowski,

1981). For example, most children would consider the

following a nonroutine problem: "Mary had 9 pieces of candy.

She gave her sister 2 pieces and her brother 4 pieces. Her

brother gave her 1 piece back. How many pieces of candy

does Mary have?" Difficulties in solving these problems

seem related to a lack of ability to first analyze and

understand a problem before applying a strategy to solve the

problem. In addition, memorized algorithms used to solve

problems seem to have little or no real meaning.









Most children in elementary school are introduced to

problem solving through verbal problems often referred to as

word or story problems (Kantowski, 1981). Generally, most

problem solving done at the primary level consists of

solving one-step word problems (Carpenter, Mathews,

Lindquist, & Silver, 1984). A majority of students can

solve these problems by mechanically applying a

computational algorithm they have learned. Research has

shown, however, that even children who have received no

instruction can successfully solve problems. This behavior

suggests that neither computational skills nor the learning

of algorithms are prerequisites for problem solving

(Ginsburg, 1977; Groen & Resnick, 1977; Hiebert, 1981;

Wearne & Hiebert, 1984). In fact, children may be better

problem solvers before instruction than after.

To solve a problem, students should first recognize the

structure of the verbal problem, select or invent an

appropriate strategy, and correctly apply that strategy

(Kantowski, 1981). It seems that prior to instruction,

children follow these three steps. Carpenter, Hiebert, and

Moser (1983) found that children apply a variety of

strategies in solving a variety of problems. In an 8-week

study examining the processes that first graders used to

solve selected addition and subtraction word problems, they

found not only that first graders prior to instruction could

solve a variety of story problems using a variety of

strategies, but that the strategies chosen were related to









the structure of the problem itself. This indicates that

instead of applying a memorized rule or algorithm, children

were carefully analyzing the problem first, unlike many

children who have received instruction.

Many teachers try to help children in understanding

problems by having them look for key words or cues which can

suggest what operation to perform. Teachers believe that by

providing student with these hints, they are helping them to

think about how the variables in a problem are related.

Unfortunately, this technique tends to prevent children from

thinking about the problem. Instead, they blindly apply the

"key-word" approach rather than analyzing the problem first,

and choosing or inventing a way to solve the problem

(Sowder, 1981).

A number of researchers have examined how the

mathematical structure of a task influences problem solving

performance. For example, multi-step problems have been

found more difficult than one-step problems as were problems

requiring the application of more than one operation (Silver

& Thompson, 1984). Researchers studying young children's

performance on addition and subtraction problems have found

that problems indicating an explicit action are easier to

solve than problems indicating an implied action (LeBlanc,

1968; Shore & Underhill, 1976). For example, a problem such

as "Katie had 4 cookies. Katie's father gave her 3 more

cookies. How many cookies does Katie have now?" was found

to be easier than a problem such as, "Katie has 4 cookies.









Her father has 3 cookies. How many do they have

altogether?". Shores and Underhill (1976) also found that

subtraction problems in which the action was "take-away"

were much easier to solve than "comparison" or "additive

subtraction" problems.

Carpenter and Moser (1983) identified four broad

categories of addition and subtraction problems: change,

combine, compare, and equalize. Change problems involve

actions that cause increases or decreases in some quantity

(Riley, Greeno, & Heller, 1983). Two types of change

problems are joining problems and separating problems.

Joining problems consist of putting together two quantities

such as in the problem, "Mary had 4 marbles. Jack gave her 5

more problems. How many marbles does Mary have now?"

Separating problems involve decreasing a quantity by the

removal of sets such as in the problem, "Mary had 9 marbles

and gave 5 marbles to Jack. How many marbles does Mary have

left?"

Combine and compare problems involve static relation-

ships for which there is no direct action. Combine problems

involve the relationship existing among a set and its

subsets. An example of a combine addition problem might be,

" Jimmy has 4 blue cars and 2 red cars. How many cars does

he have altogether?" An example of a combine subtraction

problem might be, "There are 5 people in a car. Two are in

the front seat and the rest are in the back seat. How many

people are in the back seat?".









Comparison problems involve directly comparing two

disjoint sets such as in the subtraction problem, "Bob has 6

cookies and Sue has 8 cookies. How many more cookies does

Sue have than Bob?" An example of a comparison addition

problem would be, "Bob has 6 cookies. Sue has 2 more

cookies than Bob. How many cookies does Sue have?"

Equalizing involves changing one of two quantities so that

both quantities are equal such as in the problem, "There are

4 girls and 6 boys at a birthday party. How many more girls

should come to the party so there are the same number of

boys as girls?" Equalize problems have been found to be

"awkward" for young children to solve due to the irregular

wording of the problems (Carpenter, Hiebert, & Moser, 1983).

Often equalize problems are not introduced until children

are older.

When examining how first graders solve various word

problems before instruction, Carpenter, Hiebert, & Moser

(1983) found that children basically use the same pattern

for solving addition. This was true after instruction as

well. When solving subtraction word problems before

instruction, however, first graders were found to use

several different strategies for different problem types.

Separate problems were primarily solved using a separate

strategy, combine problems were solved using separate and

add on strategies, and compare problems were usually solved

by using a matching strategy as were equalize problems.

After instruction, however, all problem types were primarily









solved by applying a single strategy -- the separate

strategy.

Somehow, as a result of the instructional process,

children move from inventing a variety of modeling and

counting strategies to solve a variety of problems, to using

one memorized strategy or algorithm. Before instruction

children are able to analyze and represent the structure of

different problems in order to solve them, and are able to

invent sophisticated strategies for solving problems with no

aid or dependence upon learned algorithms. Gradually,

children become less dependent upon their own abilities to

analyze and solve problems, and more dependent upon

memorized and mechanical procedures for adding, subtracting,

multiplying, and dividing with little regard for the content

of the problem (Carpenter, Corbitt, Kepner, Lindquist, &

Reys, 1980). It is crucial, therefore, that the influence of

instruction be carefully examined. At this point, the

effects of instruction are still unclear (Carpenter & Moser,

1983).



Initial Instruction on Addition and Subtraction Symbols



Prior to formal instruction, young children have

already developed sound, intuitive ideas about arithmetic

operations (Hiebert, 1983). In fact, young children are

"surprisingly competent" at simple arithmetic operations and

come to first grade with a substantial fund of knowledge.









Even children as young as 4 years old can understand that

addition increases numerosity and subtraction decreases

numerosity (Brush, 1978) as well as recognize that addition

and subtraction are inverse operations in that the effect of

one cancels the effect of the other (Gelman & Gallistel,

1978).

As children enter school, they receive instruction in

formal arithmetic. It is at this point that many children

begin to experience difficulty and replace their sound

analytical problem-solving strategies with shallow,

meaningless procedures (Hiebert, 1984). Many attribute the

difficulty to the abstract formal nature of school

mathematics which is so different from the informal

mathematics children acquire naturally (Donaldson, 1978;

Ginsburg, 1977; Hiebert, 1984). Much of school mathematics

involves manipulating symbols according to prescribed rules

(Hiebert, 1984) to arrive at a correct answer. Teachers are

so concerned with the writing of correct answers, that there

is little time left for the development of extension of

thinking strategies (Kamii, 1982). As a result, many

children do not connect what they learn in school with the

mathematical knowledge they already possess (Wearne and

Hiebert, 1984). Mathematics becomes a process of applying

meaningless rules and procedures with little understanding

of what they represent (Kamii, 1982).

Understanding Symbols

Many children experience a great deal of difficulty

dealing with symbolic expressions and establishing meaning









for the symbols (Hiebert, 1981). Formal mathematics using

abstract notation uses structures very different from those

utilized in the informal, natural mathematics of a child

(Ginsburg, 1983). As a result, children often do not

establish a link between the formal (school mathematics) and

the informal (mathematical knowledge acquired naturally)

(Ginsburg, 1983; Wearne & Hiebert, 1984). The absence of

this link causes a shift from intuitive and meaningful

problem solving to mechanical, meaningless problem solving

(Hiebert, 1984).

Hiebert (1984) believes the key to success with

mathematical symbols is establishing a connection between

form (the symbolic phrases) and understanding (intuition and

ideas about how mathematics works). He has identified three

points in the problem-solving process where form and

understanding might be linked. Initially, the symbolic

representations in the problem can be linked with referents

that give them meaning by association with story problems,

real experiences, or concrete objects. Secondly, form and

understanding are linked when children connect a procedure

or algorithm with the underlying concept. For example,

regrouping can be tied to understandings of the base-10

numeration system and place value. And finally, the

solution to a problem represented symbolically could be

assessed as to its reasonableness or whether it is

consistent with other knowledge children have about

mathematics.









Concrete materials can be used to represent arithmetic

concepts and symbols physically. It is important, however,

that links are made between the physical representation and

the symbolic representation (Hiebert, 1981). When those

links between the physical and the concrete should be made

is debated by researchers. Some believe connections should

be made immediately while others recommend that children use

and manipulate objects for an extended period of time before

trying to connect symbols with their actions. (Kamii, 1982;

Lovell, 1971; Piaget, 1952).

Simply supplying children with concrete objects is not

enough. Children need to use objects and have the

opportunity to reflect upon their actions with them in order

for an activity to be meaningful (Evans, 1983). It is from

their actions with objects and their reflection upon those

actions that mathematical knowledge is developed (Piaget,

1952). It is important as well that teachers avoid telling

students how to use manipulatives to solve problems;

otherwise, manipulatives become no more than "physical

algorithms" with little meaning as symbolic algorithms.

Often children are introduced to addition and

subtraction through learning how to solve number sentences

or equations. They are instructed as to the meaning of

individual symbols and are shown a written sentence

comprised of those individual symbols (Hiebert, 1984). Some

recommend that when first introduced to number sentences,

children should use an elementary form of mapping rather









than the traditional form. For example, an arrow can be

used instead of the traditional "=". It was found that

children understood the implications of an arrow more so

than the "=" (Nuffield and Schools Council Project, 1969).

Several researchers have found that children have a great

deal of difficulty in interpreting "=" and believe it to

indicate an action rather than the equality between two

quantities (Ginsburg, 1983; Behr, Erlwanger, & Nichols,

1980).

Regardless of the form used in writing equations,

children have difficulty recognizing the meaning of those

equations unless they are tied to real world events and

experiences. Many teachers relate number sentences to story

problems in an attempt to make this tie (Ibarra & Lindvall,

1982). Grouws (1972) found that to be successful, teachers

need to do more than simply provide a story problem to go

with an equation after it is introduced. Hiebert (1981)

suggested that a variety of experiences need to be provided

in writing number sentences that represent verbal problems

and writing story problems to represent number sentences.

Problem Solving as a Means of Introducing Addition and

Subtraction

Many teachers wait to introduce verbal problems until

students can compute and have knowledge of several

algorithms. The thinking is that verbal problems provide a

means for seeing if students can correctly apply algorithms

and computational skills (Kantowski, 1981). It appears that









quite the opposite is true. Students do not necessarily

need to know how to compute in order to solve verbal

problems. Many students are quite competent at inventing

their own strategies for solving problems before learning to

"compute" (Carpenter, Moser, & Hiebert, 1979; Ginsburg,

1977; Groen & Resnick, 1977). It seems, in fact, that

children need to first be able to comprehend and solve story

problems if they are to master simple number sentences

(Ibarra & Lindvall, 1982). Therefore, verbal problems may

be a more appropriate context in which to introduce

operations and thus provide for a variety of interpretations

of addition and subtraction as well as a better

understanding of the basic operations (Hiebert, 1981).

Campbell (1984) proposed a means of using a problem

solving approach to introduce children to addition and

subtraction concepts. She suggested that teachers first

read a story problem to the students, have students

paraphrase the story taking from the story the essentials of

the problem, solve for the answer, and finally symbolize the

general form that had been described by writing a number

sentence and solving it. In this way, symbols are

introduced as recording devices that help children in

attributing meaning to the symbols. According to Campbell,

not only can children acquire computational skills and

concepts, they can improve their problem solving abilities

as well. While Campbell has proposed introducing symbols

immediately after interpreting and solving a story problem,









others have recommended that children have extended

experiences with verbal problems first, possibly for several

weeks, before introducing the symbols which represent them.

Hamrick (1976) investigated the effects of delaying the

introduction of symbols to students. Thirty-eight first

graders were classified as "ready" or "not ready" based upon

their ability to solve simple addition and subtraction

problems verbally. Half of those students classified as

"ready" were introduced to symbols immediately and half were

introduced to symbols 5 weeks later. The same was done for

those students classified as "not-ready". Half were

introduced to symbols immediately and half 5 weeks later.

Using a measure to assess their understanding of symbols and

their ability to solve simple addition and subtraction

problems, Hamrick found that students in the "ready" group

performed significantly better on the measure of symbol

interpretation and understanding than did those in the

"not-ready" group who were introduced to symbols

immediately. There were no significant differences between

those in the immediate group and delayed group among

students classified as "ready". There were also no

significant differences among groups in their ability to

solve addition and subtraction number sentences.



Summary



As is evident in the literature, the way in which

children are introduced to addition and subtraction









influences their ability to understand the concepts and

processes associated with addition and subtraction, as well

as the symbols representing those operations. It also

influences the way in which they approach and solve

problems. It seems that as a result of the instructional

process, children move from inventing and using a variety of

problem solving strategies to using a single memorized

algorithm. What aspects of the instructional process create

this effect, however, is still unclear (Carpenter & Moser,

1983). It is important, therefore, to carefully examine the

instructional process and determine what approaches are most

appropriate when introducing children to formal arithmetic

and how those approaches affect their ability to solve both

verbal and written problems and understand the symbols

associated with those problems.

When children are introduced to addition and

subtraction symbols (number sentences), they seem to

experience a great deal of difficulty. Research has

indicated that children have a better understanding of

symbols if they are introduced after experiences in solving

verbal problems with the aid of concrete manipulatives. How

soon after these experiences symbolS should be introduced is

still unclear, however. Should symbols be introduced after

several weeks of concrete problem solving experiences, or

should they be introduced immediately following concrete

experiences as a means of representing or recording those

experiences? In addition, it is not clear how the









introduction of symbols affects children' ability to solve

varying kinds of word or story problems or how the processes

used to solve such problems are affected.

In this study, such questions were investigated by

examining the effects of three instructional approaches

designed to introduce addition and subtraction symbols to

first graders. Although all three instructional approaches

used word problems as part of the instructional process, the

timing of the introduction of symbols is varied. One

approach introduced symbols first followed by an associated

word problem. The second approach introduced word problems

then introduced symbols immediately after as a means of

representing the problem. And the third approach introduced

symbols after several weeks of solving problems verbally.

Chapter III describes the procedures that were followed when

conducting the investigation.















CHAPTER 3
RESEARCH METHODOLOGY



Subjects



The subjects for this study were 66 first grade

students from Duval Elementary School in Gainesville,

Florida. The school is predominantly Anglo-American

(approximately 90%) with the majority of students coming

from low to middle socioeconomic backgrounds. All three of

the first grade classes participated in the study. At the

beginning of the school year, students were assigned to

classrooms on the basis of sex, race, and reading

achievement as measured by the Ginn Basal Reading series

to assure an even distribution of boys, girls, Blacks,

whites, and students in low, average, and high achieving

reading groups in each classroom. Approximately 26 students

were assigned to each first grade class. Each classroom

was randomly assigned to one of three treatment groups: (a)

traditional group, (b) immediate group, and (c) delayed

group.

Prior to the treatment period, students were given a

variety of numeration tasks to assess their understanding of

basic number concepts. Tasks included (a) numeral

recognition through nine, (b) rational counting through









nine, (c) recognition of sets through nine, (d) corre-

sponding numerals to sets through nine, (d) number

constancy, and (e) understanding of "more" and "less". Only

students who mastered all of the above numeration tasks were

assigned to treatment groups. All other students received

individualized instruction and tutoring by their regular

classroom teachers.

Additionally, students who mastered all pretest measures

were excluded from treatment groups. These students were

assumed to have mastered all of the concepts and skills

being presented through each treatment and did not need to

receive any of the treatments designed for this study. They

too were provided with individualized tutoring by their

classroom teachers.



Data Collection Instruments and Procedures



The following is a description of each of the data

collection instruments and procedures used in this study.

Each instrument and procedure was used in preassessments as

well as postassessments. The researcher administered all

instruments and conducted all data collection procedures.

The Addition and Subtraction Test

Task One of the measure of symbol understanding was

a measure of ability to provide answers for written addition

and subtraction number sentences. This task was analyzed

separately and used to determine if any differences existed









among treatment groups in ability to solve number sentences

(Hypothesis 1).

Measure of Symbol Understanding

As noted in Chapter I, subjects demonstrated that they

knew the meaning of a symbol in one of the following two

ways: (a) given a symbol, they interpreted that symbol by

describing or carrying out an action that is appropriate for

the symbol; (b) given an action or description of an action

that is appropriate for a symbol, they produced the symbol.

When the symbol was an addition or subtraction number

sentence, it was also necessary that students stated the

correct sum or difference. It was assumed that subjects

able to do all of the three tasks described, demonstrated

that symbols were more meaningful to them than students who

could do only one or two of the above tasks (Hamrick,

1976/1977).

The measure used to assess a subject's understanding of

the meaning of symbols (Hypothesis 4) was directly derived

from the instrument used by Hamrick (1976/1977) in a study

sponsored by the National Science Foundation assessing first

graders' understanding of addition and subtraction symbols.

The instrument was reviewed by a peer review panel of

mathematicians, university professors, and representatives

of the National Science Foundation from various locations

throughout the U.S. The panel agreed that the instrument

was a valid measure of symbol understanding based upon the

content of the instrument and the construct being measured.









Test-retest reliability coefficients were computed by

the researcher prior to the implementations of the study

using a sample of 48 first graders in Gainesville. Each

task was administered on one day, then once again on the

following day. Pearson product-moment correlation

coefficients were calculated for each of the three tasks as

well as for the entire instrument showing the relationship

between scores on the first administration of the test to

scores on the second administration of the test. The

reliabilities of task one (solving number sentences), task

two (producing number sentences), and task three

(interpreting number sentences) were calculated to be .97,

.92, and .92, respectively. The reliability of the entire

instrument was calculated to be .97.

The instrument was comprised of three tasks: (a)

solving written addition and subtraction number sentences,

(b) producing written addition and subtraction number

sentences, and (c) interpreting addition and subtraction

number sentences. Number sentences used in each task were

randomly selected from a pool of addition number sentences

and a pool of subtraction number sentences whose sums were

less than 10 and differences were greater than 1 with no

addend greater than 9 or less than 1.

Task one: solving addition and subtraction number

sentences. The first task contained 10 written addition

number sentences and 10 written subtraction number

sentences. Subjects were asked to provide a written answer









for each number sentence. This portion of the test served

as the measure of ability to solve addition and subtraction

number sentences (Hypothesis 1) as well as one part of the

measure of symbol meaning understanding (Hypothesis 4). The

assumption was that part of understanding of the meaning of

a symbol was knowing how to correctly solve number sentences

which use that symbol.

Task two: producing addition and subtraction number

sentences. In this task, students observed an action while

listening to a story or description and decided whether the

action or description represented the operation of addition

or subtraction. They then had to decide which symbols

represented that operation and write them in the form of a

number sentence. For example while manipulating blocks, the

examiner might have said, "I have 2 blocks and you have 3

blocks. If you take my 2 blocks and put them with your 3

blocks, you will have 5 blocks." The subject was then asked

to write a number sentence appropriate for the action such

as, 2 + 3 = 5. Subjects were asked to produce 10

subtraction number sentences and 10 addition number

sentences. The results from this task were used to test

Hypothesis 2.

Task three: interpreting addition and subtraction

number sentences. Subjects were shown 10 addition number

sentences and 10 subtraction number sentences each presented

separately and in random order. They were then asked to

manipulate objects, tell a story, or draw a picture which









represented each number sentence. For example, when

presented with the number sentence, 3 + 4 = 7, a child might

have told a story like, "John had 3 cookies. His mother

gave him 4 more cookies. Then he had 7 cookies." The

results from this task were used to test Hypothesis 3.

In order to determine which treatment groups seemed to

have had a better understanding of symbols overall, scores

of each of the three tasks described were combined and

statistically analyzed. These scores were used to test

Hypothesis 4.

Assessment of First Graders' Ability to Solve Addition and

Subtraction Story Problems.

Two types of addition story problems and three types of

subtraction story problems were included in the assessment

of ability to solve addition and subtraction story problems

presented orally (Hypothesis 5). Addition problem types

included combine and compare problems. Join problems were

not included because research has shown that children use

the same processes to solve join and combine problems

(Carpenter, Hiebert, & Moser, 1983). Subtraction problems

included separate, combine, and compare problems.

The problems conformed to the following rules: (a) all

addends were greater than 1 and less than 10, (b) all sums

were less than 10, and (c) the difference of the addends was

greater than 1. Four separate problems, 4 combine problems,

and 4 compare problems were read to students in random

order. After each problem was read, students were asked to









record their answers on a numbered answer sheet. Students

were allowed to use manipulatives or to draw pictures to aid

them in solving the problems. Test-retest reliability

coefficients were obtained by the researcher prior to the

implementation of the study using a sample of 48 first

graders in Gainesville. The reliability was found to be

.99.

Identification of Children's Processes Used to Solve Story

Problems.

Identification of the processes and strategies used by

students in solving story problems was accomplished through

revised clinical interviews. The revised clinical interview

was first developed by Piaget when he concluded that the

verbal interview method was sometimes inadequate, especially

with younger children who often have difficulty expressing

themselves verbally. In the revised clinical method,

concrete materials can be used. Data of interest in the

revised method are both verbalizations and aspects of

nonverbal behavior (Ginsburg, Kossan, Schwartz, & Swanson,

1983).

The aim of the interviews was to get an idea of the

kind of cognitive processes children employed to solve

problems. In other words, what problem solving strategies

did children employ to solve verbal story problems and did

these strategies differ with the type of problem? While

there was a set of procedures to be followed for conducting

the revised clinical interview, it was important that there










also be flexibility. In particular, the examiner needed to

be able to invent critical tests or questions as the subject

responded and to continue to be a neutral influence upon the

subject (Ginsburg, Kossan, Schwartz, & Swanson, 1983).

Interviews were conducted by the investigator and were

audio-taped for future analysis.

Interviews were administered individually to each

subject by the researcher. The procedures that were

followed when conducting the revised clinical interviews

were adapted from Ginsburg, et al. (1983) and were as

follows:

1. Initial Presentation of the Task the interviewer

told a story problem randomly chosen from one

of the five problem types and asked the subject to

show or tell how he/she arrived at a solution

to the problem.

2. Concreteness the interviewer repeated the

story emphasizing crucial elements of the story to

ensure that the subject had heard and understood

the story.

3. The Demand for Reflection the subject was

asked to verbalize or demonstrate his/her thoughts

with objects or by acting out or drawing.

Questions were asked such as, "How did you get

the answer?", "Can you show me how you got that

answer?", or "Can you tell me how you got that

answer?"









4. Contingency sometimes the interviewer's

questions were contingent upon the subject's

response.

5. Standardized Questions a set of standardized

questions was developed to use with each

problem type. The order or wording of the

questions changed depending upon the response

or actions of the subjects.

6. Use of Naturalistic Observation as the child

reacted to questions, observations were made of

his/her actions, facial expressions, etc. as

possible indicators of his/her thought processes.

For example, some children bounced their heads as

they solved problems counting their bounces to

arrive at an answer.



Procedures



The explanation of the procedures to be followed in

implementing this study is divided into three sections: (a)

preassessment, (b) treatment conditions, and (c) post-

assessment.

Preassessment

During the first week of the study, the following

protests were administered to all subjects:

1. numeration tasks,

2. measure of ability to solve addition and









subtraction number sentences (task one of the

measure of symbol understanding),

3. measure of symbol understanding,

4. measure of ability to solve addition and

subtraction story problems, and

5. revised clinical interviews identifying first

graders' problem solving processes.

Results of the numeration tasks determined what

subjects were "ready" to participate in the study. Only

students mastering all tasks were deemed as ready to be

introduced to addition and subtraction and were included in

the study. Students not included in the study were provided

with instruction in numeration by their classroom teachers.

Results of all other protests were used to decide which

students were too advanced to participate in the study.

Students who already knew how to add and subtract, solve

word problems, and identify the symbols associated with

addition and subtraction did not need to be reintroduced to

those concepts and thus did not need to participate in this

study. Students who mastered all protests were considered

too advanced and were excluded from participation in

treatment groups. They too were provided with instruction

by their regular classroom teachers.

Pretest scores also served to determine if any

statistically significant differences existed among the

three treatment groups in their ability to solve addition

and subtraction number sentences and story problems and in









their ability to understand addition and subtraction

symbols. Audio-tapes of revised clinical interviews were

examined to determine the problem-solving strategies used by

first graders before instruction.

Treatment Conditions

Prior to the treatment period, students had received

instruction in numeration skills and concepts by their

regular classroom teachers. The first grade curriculum for

mathematics at Duval Elementary consisted primarily of

skills and concepts taught in the Heath Mathematics Series.

The following is a list of those skills and concepts and the

order in which they were normally presented:

1. numbers 0 through 10

2. addition with sums up to 10

3. subtracting from 10 or less

4. place value

5. time and money

6. sums through 12

7. subtracting from 12 or less

8. geometry and measurement

9. addition and subtraction through two place digits

10. fractions

11. sums through 18 and subtracting from 18

At a meeting with the first grade teachers in August,

1986, it was agreed that no addition and subtraction would

be taught by them prior to and during the treatment period.

Instead, teachers provided instruction in numeration,









telling time, coin identification, and geometry. The

treatment period began once the majority of students had

mastered the skills and concepts of numeration.

Each first grade class was randomly assigned to one of

three treatment groups. Whole classrooms comprised each

treatment group with the exception of those students who did

not qualify to be in the study. The following is a

description of each treatment group.

The traditional group. In the traditional group,

students were instructed on how to solve addition and

subtraction number sentences first and were then presented

with a story problem to go with each number sentence.

Dramatizations and concrete objects were often used when

presenting stories. Students first solved number sentences

orally and were later asked to solve number sentences by

writing answers and number sentences. Problems included all

of the five problem types described earlier although the

majority of time was spent teaching combine addition and

separate subtraction problems. This basic procedure was

repeated each day for 5 weeks.

The immediate group. In the immediate group, students

were presented with a story problem with the aid of

dramatization and concrete objects and were asked to solve

the problem orally. Immediately following, a number

sentence was introduced which represented the story problem

just solved. Immediately following experiences with

objects, dramatization, and word problems, students solved









number sentenced orally and were later asked to solve number

sentences by writing answers and number sentences. Problems

included all of the five problem types described earlier

although the majority of time was spent solving combine

addition and separate subtraction problems. This basic

procedure was repeated for 5 weeks.

The delayed group. Students in the delayed group were

presented with story problems with the aid of dramatization

and concrete objects and were asked to solve the

problems orally. Problems included all of the five problem

types described earlier although the majority of time was

spent on combine addition and separate subtraction problems.

This basic procedure was repeated for 4 weeks. On the fifth

week, students were introduced to number sentences as a

means of representing their experiences with verbal story

problem solving. They continued writing and solving number

sentences for the remainder of the week.

Postassessment

The following posttests were administered during the

seventh week of the study:

1. measure of ability to solve addition and

subtraction number sentences (task one of the

measure of symbol understanding Hypotheses 1

and 4),

2. measure of ability to produce addition and

subtraction number sentences (task two of the

measure of symbol understanding -- Hypotheses 2

and 4),









3. measure of ability to interpret addition and

subtraction number sentences (task three of the

measure of symbol understanding -- Hypotheses 3 and

4),

4. measure of overall symbol understanding (combined

scores of each of the three tasks on the measure of

symbol understanding),

5. assessment of ability to solve addition and

subtraction story problems (Hypothesis 5), and

6. revised clinical interviews identifying first

graders' problem solving processes.

Tests were scored and analyzed by the investigator to

determine if there were any differences among treatment

groups. Clinical interviews were analyzed by the

investigator to identify the problem solving strategies

employed by subjects both before and after instruction.

Four weeks following the completion of the study, a

second set of posttests was administered identical to the

first set administered directly after the treatment period.

Results were again analyzed by the investigator to determine

if there were any differences among treatment groups and to

see if those differences were similar to those found when

analyzing the first set of posttests.



Data Analysis



The research design of this study was a nonequivalent

Pretest-Posttest Control Group Design (McMillan &









Schumacher, 1984) using repeated measures (Glass & Hopkins,

1984). Although complete randomization of subjects is

ideal, it was neither possible nor practical for this study.

As a result, it was necessary to use groups as they had

already been organized into classes by using stratified

randomization procedures. The school principal had divided

students into groups on the basis of sex, race, and reading

achievement and had randomly selected students from each

group to be assigned to classrooms. Although this procedure

had been followed for most students, there were some

exceptions such as with students who were repeating a grade.

Those students were assigned to classrooms they were not in

the previous year.

Pretests were analyzed using an analysis of variance

procedure (Glass & Hopkins, 1984) to determine if

any significant differences existed among groups before the

treatment period. An analysis of covariance procedure was

used to analyze both sets of posttests with the pretest

serving as the covariate. This procedure was chosen in

order to partially compensate for any differences which

might have existed in treatment groups before the treatment

period (McMillan & Schumacher, 1984). The Bonferroni

technique of multiple comparisons was performed as a

follow-up procedure to detect where specific mean

differences existed.

Audio-tapes of interviews were analyzed to determine

which problem solving strategies were used by subjects both






51


before and after instruction. Groups were then compared to

see if subjects in one treatment preferred one strategy more

than another as well as to see if problem solving strategies

changed more in one treatment group than another.














CHAPTER 4
ANALYSIS AND INTERPRETATION OF THE DATA



The purpose of this study was to examine the effect of

three instructional approaches designed to introduce

addition and subtraction symbols to first graders. In one

instructional approach, first graders were introduced to

addition and subtraction symbols in the traditional manner

by first teaching them to solve number sentences and then

presenting them with related story problems. In the second

instructional approach, pupils were first introduced to

story problems. Immediately after each story problem was

solved a number sentence was introduced to symbolically

represent the story problem. In the third and final

instructional approach, pupils were presented with story

problems to solve orally for 4 weeks. In the fifth week,

number sentences were introduced as representations of the

story problems solved orally.

First grade classrooms were used at Duval Elementary

School in Gainesville, Florida. Each class contained 22

pupils who participated in the study. The study began in

September, 1986 and lasted for 7 weeks. During the first

week of the study, students were preassessed as to their

ability to solve addition and subtraction number sentences,

solve addition and subtraction story problems, and produce









and interpret addition and subtraction number sentences. A

measure of their total understanding of addition and

subtraction symbols was derived by examining their combined

ability to solve, produce, and interpret addition and

subtraction number sentences. In addition, clinical

interviews were conducted to determine what processes or

strategies. first graders used to solve story problems. The

same assessments were administered during the last week of

the study following the treatment period. Four weeks later,

students were reassessed as to their ability to solve,

produce, and interpret number sentences as well as solve

story problems.

The purpose of this chapter is to present the

data collected as a result of the assessments both before

and after the treatment period. These data were

statistically analyzed and used to test the five null

hypotheses posed in Chapter I. There is also a discussion

of the results of the clinical interviews conducted to

determine children's processes used to solve addition and

subtraction story problems before and after instruction as

well as a presentation of the results of the second set of

posttests administered 4 weeks following the completion of

the study.



Preassessment Results



During the first week of the study, all subjects

included in the study were given the following protests:









1. measure of ability to solve addition and

subtraction number sentences,

2. measure of ability to produce addition and

subtraction number sentences,

3. measure of ability to interpret addition and

subtraction number sentences,

4. total measure of symbol understanding, and

5. measure of ability to solve addition and

subtraction story problems.

Mean scores are summarized in Table 1. In all three

treatment groups, mean scores were higher for the measure of

ability to solve story problems than for the measure of

ability to solve number sentences. Across all treatment

groups, students were more able to produce number sentences

correctly than to interpret number sentences.

Using the scores from the protests, an analysis of

variance was performed to determine if there were any

significant differences in treatment groups before the

treatment period. Because it was crucial not to make a Type

II error and find no significant differences among the

treatment groups when indeed there were significant

differences, the level of significance was set at .10 rather

than .05. F tables can be found in the appendices for both

the protests and posttests.

When examining the ANOVA tables for each pretest, no

statistically significant differences were found in mean

pretest scores among treatment groups. See Table 2.









Table 1. Mean Scores on Pretests for Each Treatment Group


Treatment Group


Pretest Traditional Immediate Delayed


Solving + and -
Number Sentences 2.90 3.27 3.50

Producing + and -
Number Sentences 3.27 3.77 3.36

Interpreting + and -
Number Sentences 1.31 2.14 1.45

Total Symbol
Understanding 7.50 9.18 8.51

Solving + and -
Story Prolems 5.27 6.13 5.95









As is indicated in Table 2, none of the calculated F

values were found to be significant at the .10 level of

significance when comparing scores among treatment groups

for each pretest. All probability levels were calculated to

be higher than the set criterion of .10. As a result, it

was concluded that no significant differences existed among

treatment groups before the treatment period on any of the

protests administered.

These findings were particularly important due to the

fact that complete randomization of subjects was not

possible and intact classrooms were used as treatment

groups. The fact that the treatment groups were not found

to differ significantly before the treatment period

indicates that any differences found among treatment groups

on the posttests would probably not be due to initial

differences in the groups but to other factors.

Clinical interviews were also conducted during the

first week of the study to determine what processes subjects

used to solve addition and subtraction word problems before

instruction. The results of the interviews are discussed in

a separate section later in the chapter.



Postassessment Results



Two sets of posttests were administered to subjects

during the course of this study. One set was given

immediately following the treatment period. A second set









Table 2. Summary of the Analysis of Variance
Significance Using Pretest Scores


Test of


Pretest F value PR > F


Solving + and -
Number Sentencs .24 .78


Producing + and -
Number Sentences .10 .90


Interpreting + and -
Number Sentences .77 .46


Total Symbol
Understanding .20 .81


Solving + and -
Story Problems .19 .82









was given 4 weeks following the treatment period to see if

differences in the treatment groups still existed.

Posttests were identical to the protests administered 6

weeks prior and were administered, scored, and analyzed

using an analysis of covariance procedure with pretest

scores as the covariate. While no statistically significant

differences in pretest scores were noted, there were

similarities among pretest and posttest scores indicating

there may have been initial differences in the groups which

could have contributed to differences in posttest scores.

For this reason, posttest scores were analyzed using an

analysis of covariance procedure, rather than an analysis of

variance procedure, because it would partially control for

initial differences in treatment groups. The criterion set

for statistical significance was .05 (rather than .10 as was

the case when analyzing protests) because it was crucial not

to make a Type I error in which significant differences

would be found when in reality there were none.

Results of the First Set of Posttests

The first set of posttests were given during the

seventh week of the study immediately after the treatment

period. The following is a presentation of the findings of

each of the posttests administered at that time.

Solving addition and subtraction number sentences

Immediately following the 5-week treatment period, a

posttest was administered to all subjects which measured

their ability to solve addition and subtraction number









sentences. Students were given 10 addition and 10

subtraction number sentences and asked to write the answer

for each. Table 3 shows the mean scores for addition,

subtraction, and combined addition and subtraction.

The data showed that the majority of students in each

treatment group solved more addition number sentences

correctly than subtraction. This is consistent with past

research findings revealing that in general, children find

subtraction problems more difficult to solve than addition

(Baroody, 1984).

The question which was being investigated was whether

or not the time of introduction of addition and subtraction

symbols would have an effect on students' ability to solve

addition and subtraction number sentences. The null

hypothesis (Hypothesis 1) was that the timing of the

introduction of addition and subtraction symbols would have

no effect on first graders' ability to solve addition and

subtraction number sentences. In other words, there would

be no statistically significant differences among the mean

posttest scores for each treatment group. The following

statistical null hypothesis was tested:

Ho: 1= 2 = 3

with 1 representing the mean for the traditional group, 2

representing the mean for the immediate group, and 3

representing the mean for the delayed group.

Mean posttest scores differed numerically, with the

immediate group (12.86) having a higher mean score than both









Table 3. Adjusted Mean Posttest Scores for the Measure of
Ability to Solve Addition and Subtraction Number
Sentences Given Immediately Following the Treat-
ment Period


Treatment Groupa


Test Traditional Immediate Delayed


Addition 6.58 7.68 6.19


Subtraction 4.14 5.18 4.08


Combined 10.72c 12.86b 10.27c

a Means with the same letter are not significantly different
at the criterion set for statistical significance (alpha =
.05).










the traditional group (10.72) and the delayed group (10.27).

An analysis of covariance was performed to determine if

these differences were statistically significant. The level

of significance was set at .05.

The computed F value was 4.92 and the null hypothesis

was rejected at the .01 level of significance. The

conclusion was, therefore, that at least one pair of mean

scores was significantly different. Thus, the time of

introduction of symbols did seem to have an effect on first

graders' ability to solve addition and subtraction number

sentences.

In order to detect specifically where significant

differences existed, the Bonferroni technique of multiple

comparisons was performed. Findings are indicated in Table

3.

No statistical differences were detected when comparing

mean posttest scores of the traditional group and the

delayed group. Statistical differences were noted, however,

when comparing mean posttest scores for the immediate group

with those for the traditional and delayed groups. The mean

posttest score for the immediate group was found to be

significantly higher than scores for both the traditional

and delayed groups.

Producing addition and subtraction number sentences

As part of the measure of symbol understanding, first

graders were asked to produce number sentences to go with

story problems that the researcher told and demonstrated









with manipulatives. The question under investigation was

whether or not the timing of the introduction of addition

and subtraction symbols would affect first graders' ability

to produce number sentences when presented with story

problems. The null hypothesis (Hypothesis 2) was that there

would be no effect on first graders' ability to produce

addition and subtraction number sentences. Consequently,

mean posttest scores for each treatment group would not be

significantly different. The statistical null hypothesis

was in the following form:

Ho: 1 2 = 3
with 1 representing the mean for the traditional group, 2

representing the mean for the immediate group, and 3

representing the mean for the delayed group. The posttest

was administered the week following the treatment period to

all subjects. Mean scores for each treatment group can be

found in Table 4.

As was the case with the scores on the measure of

ability to solve addition and subtraction number sentences,

scores were lower for addition number sentences than for

subtraction number sentences across all treatment groups.

Posttest mean scores differed numerically with the

traditional group having an adjusted mean score (4.18) lower

than adjusted mean scores for both the immediate (8.00) and

delayed (5.40) groups. An analysis of covariance was

performed to determine if these differences were

statistically significant.









Table 4. Adjusted Mean Posttest Scores for the Measure of
Ability to Produce Addition and Subtraction Number
Sentences Given Immediately Following the Treat-
ment Period


Treatment Groupa


Test Traditional Immediate Delayed


Addition 2.02 4.36 3.26


Subtraction 2.16 3.64 2.14


Combined 4.18c 8.00b 5.40c

a Means with the same letter are not significantly different
at the criterion set for statistical significance
(alpha = .05).









The F value was computed at 7.56 (p = .001). Thus, the

null hypothesis was rejected indicating that for at least

one pair of mean scores, the difference was statistically

significant.

In order to detect specifically where significant

differences existed, the Bonferroni technique of multiple

comparisons was performed. As Table 4 indicates, the mean

score of the traditional group and the mean score of the

delayed group were not significantly different. Mean scores

of the traditional and immediate groups were found to

differ significantly, however, as were the mean scores of

the immediate and delayed groups. Thus the immediate group

scored significantly higher on the posttest measuring first

graders' ability to produce addition and subtraction number

sentences than did the traditional group and the delayed

group. The performance of the traditional and delayed

groups was statistically equal.

Interpreting addition and subtraction number sentences

Subjects were asked to interpret addition and

subtraction number sentences as part of the measure of

symbol understanding. Students were individually presented

with addition and subtraction number sentences and given a

choice of telling a story to go with the number sentences or

"acting out" the number sentences with manipulatives.

Although, on the pretest, many subjects chose to "act out"

number sentences with manipulatives, on the posttest, all

subjects chose to tell a story to go with the number









sentences. The results of this task were used to test the

null hypothesis (Hypothesis 3) that the time of introduction

of addition and subtraction symbols would have no effect on

first graders' ability to interpret addition and subtraction

number sentences. The following statistical null hypothesis

was tested:

Ho: 1 2 3

with 1 representing the mean of the traditional group, 2

representing the mean of the immediate group, and 3

representing the mean of the delayed group.

Adjusted posttest mean scores can be found in Table 5.

Once again, scores were lower for subtraction number

sentences than for addition number sentences. The mean

score for the traditional group (5.86) was lower than for

both the immediate (8.31) and delayed (9.04) groups. In

order to determine if this difference was significant, an

analysis of covariance was performed with the level of

significance set at .05.

The F value was computed at 3.37 (p = .041) thus

rejecting the null hypothesis and indicating that the

difference between at least one pair of mean scores was

statistically significant. The Bonferroni method of

multiple comparisons was performed in order to determine

which pair of means differed statistically. The results are

in Table 5.

As is shown in Table 5, the only statistically

significant difference was between the mean score for the









Table 5. Adjusted Mean Posttest Scores for the Measure of
Ability to Interpret Addition and Subtraction Number
Sentences Given Immediately Following the Treatment
Period


Treatment Groupa


Test Traditional Immediate Delayed


Addition 3.49 5.15 5.43


Subtraction 2.37 3.16 3.61


Combined 5.86c 8.31bc 9.04b

aMeans with the same letter are not significantly different at
the criterion set for statistical significance (alpha = .05).









delayed group and the mean score for the traditional group

with the mean score for the delayed group (9.04) being

significantly higher than the mean score for the traditional

group (5.86). Mean scores for the delayed (9.04) and

immediate (8.31) groups did not significantly differ nor did

mean scores for the traditional (5.86) and immediate (8.31)

groups.

Total measure of symbol understanding

As was noted earlier, it is assumed that students who

can perform all three tasks on the measure of symbol

understanding (solving addition and subtraction number

sentences, producing addition and subtraction number

sentences, and interpreting addition and subtraction number

sentences) have a better understanding of addition and

subtraction symbols than do students who can only do one or

two of the above. In order to determine which treatment

groups seemed to have had a better understanding of symbols

overall, scores of each of the three tasks on the measure of

symbol understanding were combined and statistically

analyzed. These scores were used to test the null

hypothesis (Hypothesis 4) that the time of introduction of

addition and subtraction symbols would have no effect on

first graders' ability to understand those symbols. In

other words, mean scores on the measure of symbol

understanding would not significantly differ among treatment

groups. The statistical null hypothesis tested was in the

following form:


H
o: 1 2= 3









with 1 representing the mean for the traditional group, 2

representing the mean for the immediate group, and

representing the mean for the delayed group. Table 6

contains posttest mean scores for the measure of symbol

understanding.

Because mean scores were found to differ numerically,

an analysis of covariance procedure was used to determine if

those differences were statistically significant with the

level of significance set at .05. The computed F value was

3.85 (p = .026). The null hypothesis was rejected

indicating that there was a statistically significant

difference between at least one pair of mean scores.

The Bonferroni method of multiple comparisons revealed

that the immediate group (29.17) and the delayed group

(24.73) had significantly higher mean scores than the

traditional group (20.76) (Table 6). In addition, the mean

score for the delayed group did not significantly differ

from the mean score for the immediate group.

Solving addition and subtraction story problems

Subjects were also assessed as to their ability to

solve addition and subtraction story problems. In each

treatment group, students were read 20 story problems by the

researcher. Students were given this task in small groups

(two groups of 7 and one group of 8). After reading each

problem, pupils were asked to write their answer on an

answer sheet. Five kinds of story problems were included:

combine addition problems, compare addition problems,









Table 6. Adjusted Mean Posttest Scores for the Measure of
Total Symbol Understanding Given Immediately
Following the Treatment Period


Treatment Groupa


Test Traditional Immediate Delayed


Solving 10.72 12.86 10.27


Producing 4.18 8.00 5.40


Interpreting 5.86 8.31 9.04


Total 20.76c 29.17b 24.73b

a Means with the same letter are not significantly
different at the criterion set for statistical signfi-
cance (alpha = .05).









separate subtraction problems, combine subtraction problems,

and compare subtraction problems.

The hypothesis being tested (Hypothesis 5) was that the

time of introduction of addition and subtraction symbols

would have no effect on first graders' ability to solve

addition and subtraction story problems presented orally.

The following statistical hypothesis was tested:

o 1 2 = 3
with 1 representing the mean for the traditional group, 2

representing the mean for the immediate group, and 3

representing the mean for the delayed group. Adjusted mean

posttest scores are summarized in Table 7.

The mean posttest score for the traditional group

(9.09) was much lower than for both the immediate (13.54)

and delayed groups (13.68). An analysis of covariance was

performed to determine if this difference was statistically

significant at a .05 level of significance. The F value,

calculated to be 29.47, was significant at the .0001 level

of significance. The null hypothesis was rejected and it

was concluded that for at least one pair of mean scores, the

difference was significant.

The Bonferroni method of multiple comparisons was used

to determine which pairs of mean scores were significantly

different. Table 7 summarizes the results of this analysis.

While the mean scores for the delayed and immediate groups

were not found to differ significantly, the mean score for

the traditional group was found to be significantly lower










Table 7.


Adjusted Mean Posttest Scores for the Measure of
Ability to Solve Addition and Subtraction Story
Problems Given Immediately Following the Treatment
Period


Treatment Groups


Mean Posttest Scoresa


Traditional 9.09c


Immediate 13.54b


Delayed 13.68b



a Means with the same letter are not significantly different
at the criterion set for statistical significance (alpha =
.05)


_~









than mean scores for both the immediate and delayed groups.

It can be concluded, therefore, that the immediate and

delayed groups did significantly better in solving story

problems than did the traditional group and that the

traditional and delayed groups had mean scores that were

statistically equivalent.

Table 8 shows which problem types were more frequently

solved correctly for each treatment group. Combine addition

story problems were more frequently solved correctly than

any other problem type in all three treatment groups

followed by separate subtraction problems. In all

treatment groups, combine subtraction problems were

correctly solved more frequently than both compare addition

and compare subtraction story problems but less frequently

than combine addition and separate subtraction problems.

There were fewer compare subtraction problems solved

correctly than any other problem type in all three treatment

groups.

Summary of results

After analyzing posttest mean scores the following

findings can be reported:

1. The immediate group had a significantly higher mean

posttest score on the measure of ability to

solve addition and subtraction number sentences than

both the delayed and traditional groups. Thus the

null hypothesis (Hypothesis 1) was rejected

indicating that the time of introduction of addition

and subtraction symbols had a significant effect on









Table 8. Number of Each Type of Story Problem Solved
Correctly Within Treatment Groups


Treatment Groups


Problem Type Traditional Immediate Delayed


Combine Addition 71 83 81


Separate Subtraction 64 81 77


Combine Subtraction 45 74 75


Compare Addition 21 61 89


Compare Subtraction 9 30 25


Total Number Correct 210 329 347









first graders' ability to solve addition and

subtraction problems.

2. The immediate group had a significantly higher mean

score on the measure of ability to produce addition

and subtraction number sentences than both the

traditional group and delayed group. Thus the null

hypothesis was rejected (Hypothesis 2) indicating

that the time of introduction of addition and

subtraction had a significant effect on first

graders' ability to produce addition and subtraction

number sentences.

3. The delayed group had a significantly higher mean

score on the measure of ability to interpret

addition and subtraction number sentences than the

traditional group. Thus the null hypothesis was

rejected (Hypothesis 3) indicating that the time of

introduction of addition and subtraction symbols had

a significant effect on first graders' ability to

interpret addition and subtraction number sentences.

4. Both the immediate and delayed groups scored

significantly higher on the measure of total symbol

understanding than the traditional group. Thus the

null hypothesis was rejected (Hypothesis 4)

indicating that the time of introduction of addition

and subtraction symbols had a significant effect on

first graders' ability to understand the meaning of

addition and subtraction symbols.









5. Both the immediate group and the delayed group

scored significantly higher on the measure of

ability to solve addition and subtraction story

problems. Thus the null hypothesis was rejected

(Hypothesis 5) indicating that the time of

introduction of addition and subtraction symbols had

a significant effect on first graders' ability to

solve addition and subtraction story problems.

Results of the Second Set of Posttests

Four weeks following the completion of the study, all

subjects were given a second set of posttests identical to

both the protests and first set of posttests administered

immediately following the treatment period to determine if

differences found to exist immediately after the treatment

period were maintained. The following is a presentation of

the results of each assessment. Data again were analyzed

using an analysis of covariance with the pretest as a

covariate.

Solving addition and subtraction number sentences

Table 9 contains the adjusted mean scores for the

measure of first graders' ability to solve addition and

subtraction number sentences. Mean scores on this posttest

were numerically higher for all treatment groups than mean

scores on the posttest administered immediately following

the treatment period. Thus, subjects solved more number

sentences correctly on the second set of posttests given 4

weeks after the treatment period than on the first set of

posttests.









Table 9. Adjusted Mean Posttest Scores for the Measure of
Ability Solve Addition and Subtraction Number
Sentences Given Four Weeks Following the Treatment
Period


Treatment Groupa


Test Traditional Immediate Delayed


Addition 8.22 9.18 8.13


Subtraction 7.13 7.54 6.73


Combined 15.35b 16.72b 14.86b

a Means with the same letter are not significantly different
at the criterion set for statistical significance
(alpha = .05).









An analysis of covariance revealed that mean scores on

the measure of ability to solve addition and subtraction

number sentences did not significantly differ among

treatment groups when administered the second time. The

calculated F value was 1.50 at a probability level of .23

which is greater than the criterion for statistical

significance set at .05. This differed from the results of

the posttest given just after the treatment period in which

there were statistically significant differences among mean

scores with the immediate group performing significantly

better than both the delayed and traditional groups.

Producing addition and subtraction number sentences

Results of the measure of first graders' ability to

produce addition and subtraction number sentences

administered 4 weeks following the completion of the study

are summarized in Table 10. Once again, mean scores were

higher across all treatment groups for this posttest than

for the same posttest administered 4 weeks prior. Mean

scores for the second posttest differed among treatment

groups with the mean score for the delayed group being

higher than mean scores for both the immediate and

traditional groups and the mean score for the immediate

group being higher than the mean score for the traditional

group. An analysis of covariance was performed to determine

if these differences were significant.

The computed F value of 7.05 was found to be

significant (p = .0017). Thus at least one pair of mean










Table 10. Adjusted Mean Posttest Scores for the Measure of
Ability to Produce Addition and Subtraction Number
Sentences Given Four Weeks Following the Treatment
Period


Treatment Groupa


Test Traditional Immediate Delayed


Addition 5.55 6.95 7.45


Subtraction 4.45 6.09 6.77


Combined 10.00c 13.04b 14.22b

a Means with the same letter are not significantly different at
the criterion set for statistical significance (alpha = .05).









scores was found to be significantly different. The

Bonferroni technique of multiple comparisons showed that the

delayed (14.22) and immediate (13.04) groups performed

significantly better on the measure of ability to produce

addition and subtraction number sentences than the

traditional group (10.00) and the delayed and immediate

groups did not significantly differ.

Results from the analysis of the posttest measuring

first graders' ability to produce addition and subtraction

number sentences given just after the treatment period

differed from the results of the posttest given 4 weeks

later. As was described in Table 4, the mean score for the

immediate group (8.00) was significantly higher than mean

scores for both the traditional (4.18) and delayed (5.40)

groups just after the treatment period. In addition, the

mean score for the delayed group was not significantly

different from the mean score for the traditional group.

While on the first posttest, the delayed group did not

perform as well as the immediate group and equal to the

traditional group, the delayed group performed significantly

better than the traditional group and statistically equal to

the immediate group on the posttest administered 4 weeks

later.

Interpreting addition and subtraction number sentences

Mean scores of the measure of ability to interpret

addition and subtraction number sentences given 4 weeks

after the completion of the treatment period can be found in

Table 11.









Table 11. Adjusted Mean Posttest Scores for Measure of Ability
to Interpret Addition and Subtraction Number
Sentences Given Four Weeks Following the Treatment
Period


Treatment Groupa

Test Traditional Immediate Delayed


Addition 4.77 6.82 6.54


Subtraction 3.82 5.18 5.64


Combined 8.59c 12.00b 12.18b

a Means with the same letter are not significantly different
at the criterion set for statistical significance
(alpha = .05).









Mean scores for all treatment groups were higher than were

mean scores for the posttest administered 4 weeks prior.

Mean scores on this posttest differed among treatment groups

with the traditional group having a lower mean score (8.59)

than both the immediate group (12.00) and the delayed group

(12.18). An analysis of covariance was performed to

determine if there were any statistically significant

differences.

The computed F value of 6.43 was found to be

significant (p = .0029) indicating that there was a

significant difference among at least one pair of mean

scores. The Bonferroni technique of multiple comparisons

revealed that the mean scores for the delayed and immediate

groups were significantly higher than the mean score for the

traditional group (Table 11). Mean scores for the delayed

group and immediate group did not significantly differ.

These results differed from the results of the posttest

administered 4 weeks prior. The only significant difference

noted just after the treatment period was between the

delayed group and the traditional group with the mean score

for the delayed group (9.04) being significantly higher than

the mean score for the traditional group (5.86).

Total measure of symbol understanding

The results of the second set of posttests measuring

total symbol understanding are summarized in Table 12. Mean

scores were again higher for all treatment groups for the

posttests given 4 weeks after the treatment period than for









Table 12. Adjusted Mean Posttest Scores for the Measure of
Total Symbol Understanding Given Four Weeks
Following the Treatment Period


Treatment Groupa

Test Traditional Immediate Delayed


Solving 15.35 16.72 14.86


Producing 10.00 13.04 14.22


Interpreting 8.59 12.00 12.18


Total 33.95c 41.77b 41.27b

a Means with the same letter are not significantly
different at the criterion set for statistical signfi-
cance (alpha = .05).









the posttests given 4 weeks prior. The immediate and

delayed groups produced mean scores on the posttest given 4

weeks after the treatment period that were higher than mean

scores for the traditional group. An analysis of covariance

found this difference to be a statistically significant

difference (F263 = 4.94, p = .01).

The Bonferroni technique of multiple comparisons found

the difference in mean scores for the immediate (41.77) and

delayed groups (41.27) not to be statistically significant.

Mean scores for the traditional group (33.95) and the

immediate group, however, as well as mean scores for the

traditional group and the delayed group, were significantly

different. Mean scores for the immediate and delayed groups

were found to be statistically higher than the mean score

for the traditional group (Table 12). These results were

consistent with the results of the first set of posttests

measuring total symbol understanding administered

immediately following the treatment period.

Solving addition and subtraction story problems

The results of the measure of ability to solve story

problems given 4 weeks after the treatment period are

summarized in Table 13 and were similar to the results of

the posttest administered immediately following the

treatment period. While mean scores were lower across all

treatment groups for the second posttest than for the first,

the immediate (8.81) and delayed (9.81) groups continued to

produce higher mean scores than the traditional group










Table 13. Adjusted Mean Posttest Scores for the Measure of
Ability to Solve Addition and Subtraction Story
Problems Given Four Weeks Following the Treatment
Period


Treatment Groups


Mean Posttest Scoresa


Traditional 4.27c


Immediate 8.81b


Delayed 9.81b



a Means with the same letter are not significantly different
at the criterion set for statistical significance (alpha =
.05).










(4.27). The analysis of covariance indicated that a

significant difference existed between at least one pair of

mean scores (F2,63 = 8.77, p = .0004). Using the Bonferroni

technique of multiple comparisons, it was determined that

both the immediate and delayed groups had significantly

higher scores than the traditional group (Table 13). Nc

statistically significant differences were found to exist

between mean scores for the immediate and delayed groups.

This was true as well when results from the first posttest

given 4 weeks prior were analyzed.

Summary of results

After analyzing mean scores of the second set of

posttests administered 4 weeks following the treatment

period, the following findings can be reported:

1. On the measure of ability to solve addition and

subtraction number sentences, there were no

significant differences among treatment groups. On

the posttest administered immediately after the

treatment period, however, the immediate group had

a significantly higher mean score than the delayed

group and the traditional group.

2. On the measure of ability to produce number

sentences, the mean scores of the delayed and

immediate groups were statistically equal and

significantly higher than the mean score of the

traditional group. On the posttest administered 4

weeks prior, the immediate group had a signifi-

cantly higher means than both the delayed









group and the traditional group whose means were

statistically equivalent.

3. On the measure of ability to interpret number

sentences, both the immediate and delayed groups

had significantly higher mean scores than the

traditional group. This differed from the results

of the posttest given just after the treatment

period in which the delayed group had a

significantly higher mean score than the

traditional group.

4. Results of the measure of total symbol

understanding were the same for both the posttest

given 4 weeks after the treatment period and the

posttest given immediately after the treatment

period with the immediate group and the delayed

group performing significantly better than the

traditional group.

5. Results of the measure of ability to solve story

problems were the same for both the posttest given

4 weeks after the treatment period and the posttest

given immediately after the treatment period. The

immediate and delayed groups had significantly

higher mean scores than the traditional group.

6. All mean scores for the second set of posttests

were higher than scores for the first set of

posttests given 4 weeks prior except for the

measure of ability to solve addition and









subtraction story problems. Mean scores on this

measure were lower for the second posttest than for

the first posttest.



Clinical Interviews



During the first week and the last week of the study,

subjects were individually interviewed. At the time of the

first interview, no formal instruction had occurred. At the

time of the second interview, children had received 5 weeks

of instruction. The kind of instruction each subject

received varied depending upon the treatment group

(traditional, immediate, or delayed) to which each subject

had been assigned. During the interviews, subjects were

asked to solve addition and subtraction story problems and

explain or demonstrate how they had arrived at their

solutions. There were five problem types included: combine

addition, combine subtraction, separate subtraction, compare

addition, and compare subtraction. Results of the

interviews before instruction are summarized in Table 14 and

after instruction in Table 15.

Solving Combine Addition Problems

Combine addition problems, as pointed out earlier,

involve combining or joining two or more separate sets such

as in the problem, "John had 4 cookies. His mom gave him 3

more cookies. How many cookies does John have altogether?"

Table 16 shows the problem solving strategies used both










Table 14. Problem Solving Strategies Employed Before Instruction

Problem

Trmt Combine Combine Separate Compare Compare
Strategy Grp Add. Sub. Add. Add. Sub.


Counting All
With Models



Counting All
Without Models



Counting Up
From First



Adding On




Separate
From



Matching




Basic Facts




Other


T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total






89


Table 15. Problem Solving Strategies Employed After Instruction


Problem

Trmt Combine Combine Separate Compare Compare
Strategy Grp Add. Sub. Sub. Add. Add.


Counting All
With Models



Counting All
Without Models



Counting Up
From First



Adding On




Separate
From



Matching




Basic Facts




Other


T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total

T
I
D
Total










Table 16. Problem Solving Strategies Used When Solving Combine
Addition Problems Before and After Instruction


Treatment Group


Strategy Timea Traditional Immediate Delayed Total # Correctb


Counting All B 12 15 15 42 22
With Models A 11 3 4 18 14

Counting All B 0 0 0 0 --
Without Models A 1 2 2 5 1

Counting Up B 7 7 4 18 10
From First A 6 12 13 31 26

Adding On B 0 0 0 0 --
A 0 0 0 0 -

Separate From B 0 0 0 0 --
A 0 0 0 0 -

Matching B 0 0 0 0 --
A 0 0 0 0 -

Basic Facts B 2 0 0 0 2
A 4 3 5 12 11

Other B 1 0 3 4 0
A 0 0 0 0 -

a
aB = before instruction
A = after instruction


Total Correct Before Instruction = 34
Total Correct After Instruction = 52









before and after instruction in solving combine addition

problems.

Out of the 66 subjects interviewed, 42 used the

counting all with models strategy to solve the combine

addition problem before receiving instruction. Eighteen

used the counting up from first strategy, 2 used the basic

facts strategy, and 4 used an unidentifiable strategy such

as guessing or "thinking". Thus most subjects chose to use

the counting all with models strategy across all treatment

groups to solve the combine addition problem before

receiving instruction. Those that did not use the counting

all with models strategy, primarily used the counting up

from first strategy. This again was true in all treatment

groups. The only other strategy used, basic facts, was used

by 2 subjects in the traditional group. They reported that

their parents had taught them to add.

Thirty-four subjects solved the combine addition

problem correctly. So just over half of the subjects tested

solved the combine addition problem correctly before

receiving instruction. Of those 34 subjects, 13 used the

counting up from first strategy. Only 5 subjects using this

strategy did not solve the problem correctly. Both subjects

using the basic facts strategy solved the problem correctly

and 14 out of the 42 subjects using the counting all with

models strategy solved the problem correctly.

The results of the interviews conducted after

instruction differed somewhat. All subjects used an









identifiable strategy unlike before when 4 subjects had not.

The strategy used by more subjects in the immediate and

delayed groups was the counting up from first strategy with

13 subjects in the delayed group and 12 subjects in the

immediate group using this strategy. Only 6 subjects in the

traditional group used this strategy.

The strategy used by more subjects in the traditional

group continued to be the counting all with models strategy

with 11 subjects using this strategy. Only 3 subjects in

the delayed group used this strategy and 4 in the immediate

group.

The basic facts strategy was used by more subjects in

the second interview than in the first. Four subjects in

the delayed group, 3 subjects in the immediate group, and 5

subjects in the traditional group used the basic facts

strategy to solve the combine addition problem during the

second interview compared to 2 subjects in the first

interview who used the basic facts strategy.

A strategy not previously used was the counting all

without models strategy. Two subjects in the delayed group

and 2 in the immediate group used this strategy during the

second interview. There was 1 subject in the traditional

group who used this strategy.

The number of subjects solving the combine addition

problem correctly after instruction (52) was higher than

before instruction (34). Fourteen out of the 18 subjects

using the counting all with models strategy, 1 out of the 5









using the counting all without models strategy, 26 out of

the 31 using the counting up from first strategy, and 11 out

of the 12 subjects using the basic facts strategy solved the

problem correctly after instruction.

To summarize, when solving combine addition problems

before instruction, three identifiable strategies were

employed: counting all with models, counting up from first,

and basic facts. When solving combine addition problems

after instruction, four identifiable strategies were

employed in the delayed and immediate groups: counting all

with models, counting up from first, basic facts, and

counting all without models. The traditional group

continued to use only three identifiable strategies.

Overall, more subjects began using the counting up from

first strategy and the basic facts strategy after receiving

instruction. This was especially true in the immediate and

delayed groups. In addition, more subjects solved the

problem correctly after instruction than before instruction

with subjects using the basic facts strategy producing the

highest number of correct responses.

Solving Combine Subtraction Problems

Combine subtraction problems are problems in which

there is no direct or implied action such as in the

following problem: "There are 6 kids on the playground. Two

are girls. How many are boys?" Table 17 shows the problem

solving strategies used before and after instruction when

solving the combine subtraction problem.




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