DOCUMENT RESUME SE 020 237 McMaster, Mary Jane … › fulltext › ED116968.pdfWhOle7-NuMber Solution Cells 'Code 2 \ Page 47 51 53 Code 3. %. 53 Code 4 53 Code 5 53 No Whole-Number

$Page 1: DOCUMENT RESUME SE 020 237 McMaster, Mary Jane … › fulltext › ED116968.pdfWhOle7-NuMber Solution Cells 'Code 2 \ Page 47 51 53 Code 3. %. 53 Code 4 53 Code 5 53 No Whole-Number$
DOCUMENT RESUME

ED 116 968 SE 020 237

AUTHOR McMaster, Mary JaneTITLE Differential Performance of Fourth- Through

Sixth-Grade Students in Solving Open Multiplicationand Division Sentences. (Part 2 of 2 Parts).Technical Report No. 375.

INSTITUTION Wisconsin Univ., Madison. Research and DevelopmentCenter for Cognitive Learning.

SPONS AGENCY National Inst. of Education (DHEW), Washington,D.C.

REPORT NO WRDCCL-TR-375-Pt-2PUB DATE Dec 75CONTRACT NE-C-00-3-0065NOTE 77p.; Report from the Project on Conditions of School

Learning and Instructional Stra egies; For theaccompanying Part 1 report, seg SE 020 236

EDRS PRICE MF-$0.76 HC-$4.43 Plus P9s ageDESCRIPTORS Basic. Skills; *Division; Doctoral Theses; Elementary

Education; *Elementary School Mathematics;Mathematics Education; *Multiplication; *NumberConcepts; *Research

IDENTIFIERS *Open Sentences; Research Reports

ABSTRACTThis volume provides the appendices to the report

appearing as SE 020 236, which is a study of the effects of students*grade level (tOur, five, or six) and of five variables related totypes of open arithmetical sentences on students' ability to solveopen-sentence problems. (SD)

***********************************************************************Documents acquired by ERIC include many informal unpublished

* materials not available from other sources. ERIC makes every effort ** to obtain the best copy available. Nevertheless, items of marginal *

* reproducibility are often encountered and this affects the quality *

* of the microfiche and hardcopy reproductions ERIC makes available *

* via the ERIC Document Reproduction Service (EDRS). EDRS is not* responsible for the quality of the original document. Reproductions ** supplied by EDRS are the best that can be made from the original.***********************************************************************

U.S. DEPARTMENT OF HEALTH,EDUCATION & WELFARENATIONAL INSTITUTE OF

EDUCATION

THIS DOCUMENT HAS BEEN REPRO-DUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIGIN-ATING IT POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRE-SENT OFFICIAL NATIONAL INSTITUTE OFEDUCATION POSITION OR POLICY

Techni7A1 Report No. 375(Part 2 of 2 Parts)

DIFFERENTIAL PERFORMANCE OF FOURTH- THROUGH SIXTH-GRADE STUDENTSIN SOLVING OPEN MULTIPLICATION AND DIVISION SENTENCES

by

;Mary Jane McMaster

Report from the Project onConditions of School Learning and Instructional Strategies

Wayne OttoPrincipal Investigator

Wisconsin Research and DevelopmentCenter for Cognitive LearningThe University of Wisconsin

Madison, Wisconsin ,o.

rv)og

December 1975

O 2

Gh .

This Technical Report is a doctoral dissertation reporting research supported by

the Wisconsin Research and Development Center for Cognitive Learning. Since it has

been approved by a University Examining Committee, it has not been reviewed by the

,Center. It is published by the Center as a record of some of the Center's activities

and as a service to the student. The bound original is in the University of

Wisconsin Memorial Library.

Published by the Wisconsin Research and Development Center for Cognitive Learning,

supported in part as a research and development center by funds from the National

Institute of Education, Department of Health, Education, and Welfare. The opinions

expressed herein do not necessarily reflect the position or policy of the National

Institute of Education and no official endorsement by that agency should be inferred.

Center Contract No. NE-C-00-3-0065

ii

3

WISCONSIN RESEARCH AND DEVELOPMENTCENTER FOR COGNITIVE LEARNING

MISSION

The mission of the Wisconsin Research and Developmen t Center

for Cognitive Learning is to help learners develop as rapidlyand effectively as possible their potential as human beingsand as contributing members of society. The R&D Cr *er is

striving to fulfill this goal by

conducting research to discover more abouthow children learn

developing improved instructional strategies,processes and materials for school administrators,teachers, and children, and

offering assistance to educators and Citizenswhich will help transfer the outcomes of researchand development into practice

PROGRAM

The activities of the Wisconsin R&D Center are organizedaround one unifying theme, Individually Guided Education.

FUNDING

The Wisconsin R&D Center is supported with funds from theNational Institute of Education; the Bureau of Education forthe Handicapped, U.S. Office of Education; and the University

of Wisconsin.

4iii

ACKNOWLEDGEMENTS

The author wishes to acknowledge and to thank the many people who

contributed in so mary ways to the preparation of this dissertation

and to the author's professional development.

I would like to express my gratitude and sincere appreciation

to my advisor, Professor J. Fred Weaver, for his continued guidance,

encouragement, and support throughout the thesis and my graduate work.

I would also like to thank Professors Thomas A. Romberg and M. Vere

DeVault for their assistance in editing as well as for serving on my

committee, and Professors Elizabeth Fennema and Lawrence Hubert for

serving on my committee.

Gratitude is expressed to the Waukesha Public Schools for thLLr

participation in this study. The cooperation of the teachers and

students who were involved in this study was greatly appreciated.

To my family and friends, I extend my heart-felt appreciation

for their moral encouragement and support. Finally, a special thank

you to my parents who have always been encouraging and supportive

throughout my college career. Their pride in my accomplishments-

is a reward as meaningful as the degree itself.

I

II

TABLE OF CONTENTS

Acknowledgements

List of Tables vi

List of Figures viii

Abstract ix

Page

ii

Introduction

Previous Research

1

Importance of Open Sentenced 2

Purpose of the Study 4

Definitions of Terms 41 6

Open Sentence 6

Number Puzzles, Problems, Number Equations 6

Symmetric Property--Operation-Left, Operation-Right. . . 7

Placeholder Positions a, b, and c 7

Canonical Form 8

Specific Purpose o£Atudy 8

Background of the Problem--Related Research 11

Results of the Weaver Study 13

Grouws' Research With Respect to Difficulty of VariousOpen Sentences 17

Suppes' Research Involving Open Sentence Difficulty 19

Conclusion 22

Principal Questions Investigated in the Study 23

Questions of Secondary Interest 24

The Study 25

The Tests 26

The Number Puzzles Test (NPT) 26

Basic Multiplication and Division Test (BMDT) 32

The Instructions to Subjects 34

The Pilot Study 36

The Sample 37

Mathematical Background of the Children in the Sample . 38

Assignment of Tests to Subjects 43

Test Administration 44

The Experimental Design 45

Data Analysis 45

,1\

TABLE, OF CONTENTS (Cont. )

IFI Descriptive/Data Analysis

WhOle7-NuMber Solution Cells'Code 2 \

Page

47

51

53Code 3. % . 53Code 4 53Code 5 53No Whole-Number Solution Cells. . . . . . , . . 57Summary Conjectures Based on the Datatinalyele;:-.-:. 59Conjectures Based on the No Whole-Number Solution Data

Analysis *63

Summary 66

IV Statistical Analysis of the Data 67

A. Plan for Data-Analysis 69B. Open Sentences Having Whole-Number Solutions 75

C. Open Sentences Having No Whole-Number Solutions 84Summary 93

V Conclusion to the Thesis .95

Introduction 95Summary 95Conclusions and Implications 98Recommendations for Future Research 107

Appendix A: Number Puzzle Tests 1-4

Appendix B: Number Puzzle Tests 1-4

Appendix C: Basic Multiplication and Division TestCombinations

Appendix D: Pilot Study Results

Appendix E: Teacher Questionnaire

Appendix F: Frequency of Coded Responses 1-7 for Whole-,.

Number Solution Cells

1-4

vi

111

114

119

121

123

125

Appendix G:

Appendix H:

Appendix I:

Appendix J:

Appendix K:

Appendix L:

TABLE OF CONTENTS (Cont.)

Percent of Code 3, 4, 6, and 7 Responseion Number Puzzle Test (Whole Number Solution).

No Whole-Number Solution Cells

Percent of Codes 1-7 Responses for the No-WholeNumber Solution Open Sentences

Cell Entries--Raw Data Matrix

Transformation Matrix

ANOVA -- Results Using a vs b and !b vs c asCompared with b vs c and be vs a

References

Page

. . 143

vii 1,(///'

8

148

152

155

166

168

170

LIST OF TABLES

Table Page

1.1 Open Sentence Types Investigated 5

1.2 Examples of Twenty Open Sentence Types Employedin Weaver Study 12

1.3 Mean Correct Responses on 32-Item Inventory(Weaver Study) 13

1.4 Student Means for Addition and SubtractionSentences (Weaver Study) 14

1.5 Student Means for Operation-Left and Operation-RightSentences (Weaver Study) 14

1.6 Mean Correct Responses on Open Sentences HavingPlaceholders in Position a, b, and c (Weaver Study). 15

1.7 Mean Correct.Responses on Open Sentences That DoHave Whole-Number Solutions and Others That Do NotHave Whole-Number Solutions (Weaver Study) 16

1.8 Percent of Correct Responses for Each Open SentenceType Investigated (Grouws Study) 18

1.9 Average Probability Correct--Grade 1 (Suppes Study).

2.1 Generic Open Sentence Types Investigated

2.2 Open Sentence Types for Number Puzzles Test (NPT).

2.3 Sample Composition by School and Grade

2.4 Frequency of Occurrence of Open Sentence Types AsFound in Mathematics Textbooks (Harcourt, Brace andWorld)

2.5 Student Experience With Each Open Sentence Type(Rated by Teacher Opinion Questionnaire)

21

26

28

39

41

42

3.1 Number of Students Per Test Form 47

3.2 Open Sentence Types Assigned to Each of 12 Cellsfor Number Puzzle Test 48

9

Table

3.3

LIST OF TABLES (Cont.)

Page

Number of Open Sentences Assigned to Each Cell onNumber Puzzle Test 49

3.4 Percent of Code 1 Responses on Number Puzzle Test . 52

3.5 Percent of Code 2 Responses on Number Puzzle. Test . . 54

3.6 Percent of Code 5 Responses on Number Puzzle Test . . 56

3.7 Percent of Code 1, 6, & 7 Responses for the No-WholeNumber Solution Open Sentences on Number Puzzle Test. 58

4.1 Number of Students by Grade and Test Form in theFirst and Second Categories 68

4.2 12 Class Means for Each of the 53 Classes--Used forANOVA 71

4.3 Transformed Means of Whole-Number Solution CellsPut on a 1-0 Metric 76

4.4 12 Cells Means by Grade--Whole-Number Solution CellsPut on a 1-0 Metric 77

4.5 Analysis of Variance for Grade Level (4, 5, 6). . . 78

4.6 Analysis of Variance for Operation (Mult., Div.). . . 79

4.7 Anglysis of Variance for Symmetric Property (Oper-Left, Oper-Rt.) 80

4.8 Analysis of Variance for Placeholder Positions(a vs. b and ab vs. c) 81

4.9 Analysis of Variance Interactions 83

4.10 Cells for Each Grade 85

4.11 Eight Cell Means by Grade No Whole Number SolutionCells Put on a 1-0 Metric 86

4.12 Means of No-Whole Number Solution Cells Pu_ ona 1-0 Metric 87

X

LIST OF TABLES (Cont.)

Table Page

Wilcoxen Test--Comparison of Correct Responses Givento Eight Cells for Which Whole-Number SolutionsExisted and the Same Eight Cells for Which No-Whole-Number Solutions Existed-[-Put on a 1-0 Metric . . . . 89

4.14 Means by Grade for the 8 No-Whole-Number-SolutionCells Compared to the Comparable 8 Whole-Number-Solution Cells 90

4.15 No Whole-Number-Solution Open Sentences BasicFact Products Compared to Not a Basic Fact Products-Number Correct Transformed to a 1-0 Metric- -Wilcoxen Matched Pairs Signed Ranks Test 92

LIST OF FIGURES

Figure Page

2.1 Cover Page for Number Puzzle Tests 35

2.2 Representation of the Experimental Design 46

4.1 Schema for Data Analysis 70

Ve

ABSTRACT*

Differential Performance of Fourth- Through Sixth - Grade Students

in Solving Open Multiplication and Division Sentences

Mary Jane McMaster

Under the Supervision of Professor J. Fred Weaver

The Problem

The purpose of this study was to find out whether differences

exist in pupils' performance when solving selected types of open

multiplication and division sentences derived from the form a o b = c.

Procedure,:,

Specifically, this investigation sought to find out the differences

in students' responses to open number sentences when the followling .

factors were varied: (A) school grade (4, 5, and 6), (B) the symbol4

for the operation specified in a s,entence (x or e), (C) sentence typeN--

as determined by the symmetric property of the equality relation

(a o b = c versus c = a o (D) the position of the placeholder

in a sentence (a, b, or c), (E) the existence or non-existence of

an open sentence solution within the set of whole numbers am x b = 20

versus x 5 = 21), and (F) the largest number being a basic Lact

product or not a basic fact product in, open sentences which have no

whole number solution (3 x = 25 versus 3 x = 23).

12

Two distinct kinds of multiplication and divisionopen sentence

tests were constructed and administered to 1298 fourth7, fifth-, and

sixth-grade students from eight schools. Each student. was administered

a 28-item wen sentence number puzzle test (NPT) and a 14-item basic

multiplication and diVision test (BMDT).

The data furnished by all 1298 subjects were corrected and coded

by the investigator. Theinformation was then key punched for computer

analysis and analyZed by the Fortap Statistical Package. This process

yielded descriptive statistical results. The data furnished by

6

students,who responded correctly to at least four of the five

nontrivial multiplication items and four of the five nontrivial

items on the BMDT, were further analyzed by ANOVA and Wilcoxen

Signed Ranks Test.

Results

1. The performance level of subjects onopen sentences having whole

number solutions was significantly different between grade

levels.

2. The performance level of subjects on open multiplication sentences

was significantly' different from the performance level of subjects

on open division sentences.

3. The performance level of subjects on operation-left open sentences

was significantly different from the performance level of subjects

on operation-right open sentences.

13xiv

4. The performance level of subjects on open sentences was signi-

ficantly different for placeholder positions a, b, and c.

5. Significant interactions existed among the following factors:

grade level, operation, symmetric property, and placeholder

position.

6. The performance level of subjects on open number sentences

which have no whole number solutions was significantly different

from the performance level of subjects on open sentences which

have whole number solutions.

7. Relative to the open sentences with no whole number solutions,

there was no significant difference between students' performance

level on open sentences in which the largest number was a basil

fact product, and students' performance level on open sentences

in which the Largest number was not a basic fact product.

Conclusion

The analysis was complex to interpret because of the significant

interactions. There appeared to be a very high interaction between

operation division and placeholder position a. Significant Inter-

actions also existed between the following factors: (1) grade and

operation; (2) grade and symmetric factor; (3) grade and placeholder

position; (4) operation and symmetric factor;. (5) operation,

symmetric factor, and grade; (6) operation and placeholder

position; (7) symmetric factor and placeholder position; and

(8) operation, symmetric factor, and placeholder position. Caution

14xv

tI

must be exercised, therefore, in taking an overly simplistic inter-

pretation of significant differences between levels of principal

factors.

Nevertheless, the evidence warrants the belief that much

greater attention needs to be given to principal factors B, C, D,

and E in preparation of text materials'and in instruction pertaining

to open multiplication and division sentences.

xvi

1

Appendix A

NUMBER PUZZLE TESTS 1 - 4

16

111

112

Number Puzzle Test 1

Number fact Generic form:

Number Puzzle Test: 2

Number fact Generic form

1) 2 x 3 = 6 W- 1 ''.2 x 4 = 8 W-.3

2) 2 x 7 = 14 W -2 2 x 8 = 16 W -4

3) 3 x 4 = 12 W -3 3 x 5 = 15 7,-N-...5 *

4) 3 x 8 = 24 W -4 3 x 9 = 27 W- 6*

5) 4 x 5 = 20 W -5 4 x 6 = 24 W -1

6) 4 x 9 = 36 W -6 5 x 2 = 10 W -2

7) 5 x 6 = 30 W -1 5 x 7 = 35 W -3

8) 6 x 2 = 12 W -2 .6 x 3 = 18 W -4

9) 6 X 7 = 42 N- 3 # 6 x 8 = 48 W -5

10) 7 x 3 = 21 N- 4 * 7 x 4 = 28 N -646

11) 7 x 8 = 56 W -5 7 x 9 = 63 1,1. 1

12) 8 x 4 = 32 W -6 8 x 5 = 40 W -2

13) 8 x 9 = 72 W -1 9 x 2 = 18 W -3

14) 9 x 5 = 45 W -2 9 x 6 = 54 W -4

15) 10 4. 5 = 2 W -7 12 + 6 = 2 W -9

16) 18 i- 9 = 2 W- 8 6= 2 = 3 N-10 #

17) 18 + 6 = 3 W- 9 21 4. 7 = 3 W-11

18) 8 ÷ 2 = 4 W-10 12 .1- 3 = 4 W-12

19) 28 + 7 = 4 W-11 32 + 8 = 4 W- 7

20) 15 + 3 = 5 W-12 20 + 4 = 5 W- 8

21) 40 + 8 = 5 N- 7 # .45 + 9 . 5 W -9

22) 24 + 4 = 6 N -8 * 30 + 5 = 6 W-10

23) 54 + 9 = 6 W- 9 14 + 2 = 7 W-11

24) 35 ÷ 5= 7 W-10 42 i. 6= 7 W-12

25) 16 + 2-= 8 W-11 24 + 3 = 8 W- 7

26) 48 i- 6 = 8 W-12 56 -:- 7 = 8 W- 8

27) 27 + 3 = 9 W- 7 36 i.- 4 = 9 N- 9*

28) 63 + 7 = 9 W- 8 72 + 8 = 9 W-10

* product (or dividend) is a basic fact# product (or dividend) is not a basic fact

1 7

Number Puzzle Test 3 Number Puzzle Test 4


1) 2 x 5' = 10 W -5

2) 2 x 9 = 18 W -6

3) 3 x 6 = 18 W -1

4) 4 x 2 = 8 W -2

5) 4 x.. 7 = 28 N7 3*

6) 5 x 3 .. 15 W -4

7) 5 x 8 = 40 W- 5

8) 6 x 4 24 W -6

9) 6 x 9 = 54 W -1

10) 7 x 5 = 35 W -2

11) 8 x 2 = 16 W -3

12) 8 x 6 = 48 N- 4 #

13) 9 x 3 = 27 W -5

14) 9,x 7 = 63 W -6

15) 6 4 3 = 2 W-11

16) 14 + 7 = 2 W -12

17) 12 + 4 = 3 N- 7 #

18) 24 + 8 = 3 W- 8

19) 20 + 5= 4 W- 9

20) 36 4' 9= 4 W-10

21) 30 + 6 = 5 W-11

22) 12 1 . 2 = 6 W-12

23) 42 + 7 = 6 W -7

24) 21 + 3 = 7 N- 8 *

25) 56 ÷ 8 = 7 W- 9

26) 32 + 4= 8 W-10

27) 72 + 9 = 8 W -11..

28) 45 + 5 = 9 W-12

113


2 x 6 = 12 W -1

3 x_2 = 6 W -2

3 x 7 = 21 W -3

4 x 3 = 12 W -4

4 x 8 = 32 W -5

5 x 4 ... 20 N- 6 *

5 x 9 45 W- 1

6 x 5 = 30 W -2

7 x 2 = 14 W -3

7 x 6 = 42 W -4

8 x 3 = 24 N- 5 #

8 x 7 = 56 W -6

9 x 4 = 36 W -1

9 x 8 = 72 W -2

8 4. 4 = 2 W -7

16 -:- 8 = 2 W- 8

15 + 5 = 3 W -9

27 + 9 = 3 N-10 *

24 + 6= 4 W-11

10 4. 2= 5 W-12

35 -1. 7 = 5 W- 7

18 + 3 = 6 W- 8

48 4. 8 = 6 N7 9 #

28 + 4 = 7 W-10

63 + 9 = 7 W-11

40 I 5= 8 W-12

18 4, 2 = 9 W- 7

54 +.6 = 9 W- 8

* product (ordividend) is a basic fact# product (or dividend) is not a basic fact

18

114

Appendix B

NUMBER PUZZLE TESTS 1 - 4

Number fact


Generic form

1) 40 4. 8 = 5 N- 7

2) 18 4. 6 = 3 W -9

3) 4 x 9 = 36 W- 6

4) 8 ÷ 2 = 4 W-10

5) 28 7 = 4 W-li

6) 9 x 5 = 45 W- 2

7) .5 x 2 = 12 W -2

8) 3 x 8 = 24 W -4

9) 5. x 6 = 30 W -1

10) 48 '4. 6 = 8 W-12

11) 6x7 =42 N- 3

12) 10 + 5 = 2 W- 7

13) 8 x 9 = 72 W- 1

14) 16 :.2 =. 8 W-11

15) 54 ÷ 9 = 6 W- 9

16) 63 7 = 9 W -8

17) 35 4- 5 = 7 W-10

18) 7 x 3 = 21 N -4

19) 7 x 8 = 56 W- 5

20) 8 x 4 = 32 W -6

21) 2 x 3 = 6 W -1

22) 4 x 5 = 20 W- 5

23) 27 3 = 9 W- 7

24) 3 x 4 = 12 W- 3

25) 15 + 3 = 5 W-12

26) 18 ÷ 9 = .2 W -8

27) 24 4. 4 = 6 N -8

28) 2 x 7 = 14 W- 2

20

115

Opensentence formed

41 + 8 =018 4. - 3

36 = fx 9

4= 8 :El1 ÷ 7= 4El= 9 x 5

0 6x 2

24= 3xLJ5x 6 = 08 = a: 6

6 x0- 43

10 + 5 = El8 x 9= 0

2 = 8

54 ÷ = 6

[J =63 4- 7

7 = 35 ÷

22= 7xDEl x 8 = 56

32 = El x 4

2x 3 =DD.x 5 = 20

27 4, 3 =El3 x E= 12

5 = El ÷ 3

El = 18 ÷ 9

El = 25 + 4

El= 2 x 7'

Number fact


Generic form

1)

2)

3)

14 ÷ 2 =

7 x 9 =

24 -4 3 =

7

63

8

W-11

W -1

W -7

4) 5 x 2 = 10 W -2

5) 72 4- 8 = 9 W-10

6) 20 + 4 = 5 W -8

7) 9 x 2 = 18 W -3

8) B x 5 = 40 W -2

9) 7 x 4 = 28 N- 6 #

10) 3 x 5 = 15 N- 5 *

11) 5 x 7 = 35 W -3

12) 6: 2 = 3 N-10 #

13) 32 + 8 = 4 W- 7

14) 2 x 8 = 16 W -4

15) 45 ÷ 9 = 5 W -9

16) 30 4- 5 = 6 W-10

17) 3 x 9 = 27 W- 6

18) 12 4- 6 = 2 W -9

19) 12 4- 3= 4 W-12

20) 6 x 8 = 48 W -5

21) 6 x 3 = 18 W- 4

22) 21 ÷ 7 = 3 W-11

23) 2 x 4 = 8 W -3

24) 36 -4. 7 = 8 W -8

25) 4 x 6 = 24 W -1

26) 9 x 6 = 54 W -4

27) 42 4- 6= 7 W-12

28) 36 + 4 = 9 N- 9 *

* indicates basic fact1! indicates not a basic fact

116

Opensentence formed

El ÷ 2= 7

7x 9 = 024 4- 3 = 0

0= 5x 2

9 =.72 .40

0 = 20 + 4

9 x 0 = 18

0= 8x 529 = n x 4

0-x 5 = 16

5 x 0= 35

3= 7: 032 ÷ 8 = 0

16= 2 x 0

45 4- El= 5

6 = 30 .4.

27 = 0x 9

12 + 0 = 2

4= El+ 3

El x 8 = 48

18 = 6 x 0

0= 7= 3

2 x El = 8

0 = 56 4% 7

4x 6 = 054= 9 x 0

7= D: 6

35 ÷ 0= 9

Number fact


Generic form

1) 8 x 2 = 16, W- 3

2) 45 .4 5.= 9 w-lq

3) 2 x 9 = 18 W- 6

4) '21 -4 3 = 7 N- 8 *

5) 56 -1 8 = 7 W- 9

6) 8 x 6 = 48 N- 4 #

7) 72 i 9 = 8 W-11

8) 3 x 6 = 18 W- 1

9) 6 x 9 = 54 W -1

10) 5 x 8 = 40 W- 5

11) 2 x 5 = 10 W- 5

12) 30 .1 6 = 5 W-11

13) 42 4. 7 = 6 W -7

14) 6 x 4 = 24 W -6

15) 4 x 7 = 28 N- 3 *

16) 20 -1 5 = 4 W- 9

17) 4 x 2 = 8 W -2

18) 14 -1 7= 2 W-12

19) 24 -1 8= 3 W- 8

20) 36 4. 9 = 4 W-10

21) 32 4- 4 = 8 W-10

22) '9 x 3 = 27 W- 5

23) 7 x 5 = 35 W -2

24) 12 4 2= 6 W-12

25) 9 x 7 = 63 W -6

26) 5 x 3 = 15 W -4

27) 12 .4 4 = 3 N- 7 #

28) 6 -1- 3 = 2 W-11

* indicates basic factI indicates not a basic fact

22

117

Opensentence formed

8 x 0 = 16

9 = 0 -1 5

18 = 0 x 9

0 = 20 4. 3

56 4 0 = 7

47= 8 x 0

0 4 9 = 8

3 x 6 =Q6x 9 = 0

0x 8 = 40

0 x 5 = 10

4- 6= 5

42 -4 7 = 0

24 = El x 4

4 x 0 = 27

20 4 0 = 4

0 = 4x 22 = -1 7

0= 2 4 8

4 = 36 4 0

8 = 32 4- 0

0 x 3-= 27

El= 7x 56= 0 .4 2

63 = 0 x 7

15= 5 x 0

13 4. 4 = E:i

Di ÷ 3= 2

Number fact


Generic form

1)

2)

3)

4 ) '

5) .

18 4 2 =

3 x 2 =

48 + 8 =

15 4. 5 =

18 4 3

9

6

6

3

6

W- 7

W -2

N- 9 #

W- 9 a

W -8

6) 8 x 3 = 24 N- 5 #

7) 3 x 7 = 21 W -3

8) 5 x 9 = 45 W- 1

9) 6 x 5 = 30 W -2

10) 54 4. 6 = 9 W- 8

11) 9 x 8 = 72 W -2

12) 4 x 8 = 32 W -5

13) 7 x 6 = 42 W -4

14) 16 4 8 = 2 W -8

15) 8 : 4 = 2 W- 7

16) 7 x 2 = 14 W -3

17) 2 x 6 = 12 W -1

18) 40 4 5 = 8 W-12

19) 28.4 4= 7 W-10

20) 9 x 4 = 36 W -1

21) 4 x 3 = 12 W -4

22) 63 4. 9= 7 W-11

23) 24 4. 6= 4 W-11

24) 10 4 2 = 5 W-12

25) 8 x 7 = 56 W -6

26) 27 .4 9 = 3 N-10 *

27) 5 x 4 = 20 N- 6 *

28) 35 4 7 = 5 W- 7

* indicates basic fact# indicates not a basic fact

118

Opensentence formed

18 4- 2 = ED

0= 3x 247 : 0 = 6

15 .4 0 = 3

0 = 18 : 3

0x 3 = 23

3 x 0 = 21

5 x 9 =00= 6x 5ID = 54 + 6

0= 9x 8El x 8 = 32

42 = 7 x 0

El = 16 4. 8

8: 4 =07 x 0 ' 14

2x 6 = 08 = El + 5

7= 28 + 0

9x 4 = 012= 4 x 0El + 9= 7

0 .1- 6= 45= 0: 256 = 0 x 7

3 = 28 + 0

21 = ID x 4

35 4. 7 ' 0

7

Appendix C

BASIC MULTIPLICATION AND DIVISION TEST COMBINATIONS

2 zi

119

120

TEST 1 TEST 2 TEST 3 TEST 4

1) 4 x 9 6 x 3 7 x 5 4 x 3

2) 8 x 1 7 ÷ 1 4 ÷ 1 3 x 1

3) 16 ÷ 2 72 +8 4 x 7 54 + 6

4) 2 x 3 21 + 7 2 x 9 63 + 9

5) 8 x 9 3 x 1 32 + 4 6 x 5

6) 3 x 8 4 x 6 1 x 3 4 ÷ 1

7) 8 ÷ 2 56 + 7 30 ÷ 6 48 ÷ 8

8) 1 x 3 9 x 6 12 ÷ 2 7 + 1

9) 15 + 3 4 + 1 20 + 5 1 x 8

10) 4 ÷ 1 5 x 7 8 x 1 7 x 2

11) 3 x 4 30 ÷ 5 9 x 7 8 x 3

12) 18 ÷ 9 8 x 5 72 + 9 5 x 4

13) 7 ÷ 1 32 ÷ 8 7 ÷ 1 16 + 8

14) 54 + 9 1 x 8 8 x 6 15 ÷ 5

2 5

121

Appendix D

PILOT STUDY RESULTS

26

Factor and Level

Factor E

Factor C

FaCtor B

Factor D

Nurbyr11

of

Items

,

Number of

Incorrect

Responses

a2

.5

aob= c

b1

0

Mult.

,c

3.

2

a2

4

c =. a o b

b1

4

c3

1A Solution

Exists

a2

32

'

Within W

- aob= c

b2

7

c2

8Division

a2

36

c = a o

luoi

b2

7.

c2

7

No Solution

Mult.

aob= c

c= a 6 b

b(NBF)

b(BF)

14

Exists

Within W

Division

aob= c

c =aob

c(NBF)

c(BF)

1 I

3 3

Sample Size = 22 Students (fifth-grade)

W = Whole Numbers

NBF = Not a Basic Fact / BF = Basic Fact

March 1974

123

Appendix E

TEACHER QUESTIONNAIRE

2 a

MATH TEACHER

124

SCHOOL GRADE

Please circle the appropriate letter to indicate the experience your

students have had with each type of equation. Since all the basic

facts will be used in this study, please focus on the format of the

equation and not on the size of the nUmerals. (L indicates little

experience, M indicates moderate experience, E indicates extensive

experience)

. a. 7 x 8 =0

b. 0 = 5 x 6

c. 6 x0= 42

d. 35 = 5 x

e. f-lx 9 = 63

f. 40=0 x 8

g. 07 9= 8

h. 6 =[:]-:- 8

i. 30 41:-J= 5

j. 6 = 42 [:]

k. 56 7 =

1. El = 35 -1. 7

L M E

L M E

L M E

L M E

L M E

L M E

L M

L M E

L M E

L M E

L M E

L M E

m. equations which have no whole number solutions

for example,

x 9 = 29

3 = 28 I1=I

25

L M E

L M . E

125

Appendix F

FREQUENCY OF CODED RESPONSES 1-7 FOR WHOLENUMBER SOLUTION CELLS

126

Explanation of responses 1-7

1. Correct solution

2. Subject responded with N, indicating no whole number solution.

3. Subject used the inverse operation, i.e., multiplied while solving

a division puzzle or vice versa.

4. Subject responded with a numeral quite close to the correct

solution.

5. Subject 'performed the operation across the equality sign.' For

example, the puzzle 8 x = 16 received 128 as the correct solu-

tion, indicating the subject might have changed the puzzle to the

form 8 x 16 =

6. Subject responded with some numeral not falling into any of the

above categories.

7. The subject performed 'the inverse operation across the equality

sign'. For example the puzzle 15 4. 0= 3 received 45 for the

solution, indicating the subject might have changed the puzzle to

the form 3 x 15 =

Note: - A zero coding was given to those dazzles to which the student

did not respond or to which the response was illegible. It was

necessary to employ the zero coding very rarely. If one sums the

seven percentage scores on any puzzle, the total will be close to

99%. The difference between 99% and 100% is caused by the zero

category being eliminated. In cases where no zero codes were

employed, the sum of the seven percentages should be 100%.

3.1

12?

Frequency of Coded Responses 1 - 7 For Whole Number Solution Cells

Cell 1MLA

Elx 8 =

5 =

x 8 =

x 5 =

E x 3 =

ResponseNumber Givingthis Response Percent

56 1 286 87.202 15 4,573 0 0.004 11 3.355 0 0.006 9 2.747 0 0.00

20 1 310 94.51

2 3 0.913 0 .00

4 3 0.91

5 1 0.306 6 1.837 0 0.00

48 1 287 85..16

2 1.6 4.753. 0.004 7 2.085 0 0.006 21 6.237 0 0.00

1 298 93.122 10 3.133 0 0.004 3 0.94

5 0 0.006 8 2.507 0 0.00

10 1 307 95.942 1 0.313 0 0.004 0 0.005 4 1.256 7 2.197 0 0.00

27 1 275 85.942 19 5.943 0 0.004 6 1.875 0 0.006 16 5.007 D 0.00

32

Cell 1(Cont.)

= 32

128

Number GivingResponse This Response Percent

Cell 2MLB

3 x = 12

3 x = 21

7 x u = 14.

9 x = 18

5 x = 35

3 3

1

2

3

4

5

6

7

288

10

0

5

1

8

0

92.013.190.001.600.322.56

' 0.00

1 306 93.292 5 1.526 0 0.004 1 0.305 1 0.306 10 3.057 0 0.00

1 293 93.612 6 1.923 0 0.004 2 0.645 0 0.006 10 3.197 0 0.00

1 298 95.212 0 0.003 0 0.004 0 0.005 1 0.326 11 3.517 0 0.00

1 318 94.362 2 0.593 0 0.004 1 0.305 0 0.006 15 4.457 0 0.00

1 312 92.582 5 1.483 0 0.004 12 3.565 0 0.006 7 2.087 0 0.00

Cell, ,2

(Cont.)

8 xI 1= 16

129


1 315 93.47

2 2 0.59

3 0 0.00

4 2 0.59

5 7 2.08

6 8 2.37

7 0 0.00

1 307 95.94

2 3 0.94

3 0 0.00

4 1 0.31

5 0 0.00

6 9 2.81

7 0 0.00

Cell 3MLC

5 x 6 = 1 319 97.26

2 3 0.91

3 0 0.00

4 1 0.305 0 0.00

6 3 0.91

7 0. 0.00

8 x 9 = 1 296 90.24_2 8 2.44

3 0 0.00

4 10 3.05

5 0 0.00

6 11 3.35

7 0 0.00

2 x 3 = E: 1 314 95.73

2 4 1.22

3 0 0.00

4 2 0.C1

5 . 0 0.00

6 5 1.52

7 0 0.00

7 x 9 = [2 1 291 86.352 4 1.19

3 0. 0.00

4 24 7.12 ,

5 0 0.00

6 16 4.75

7 0 0.00

130

ResponseNumber GivingThis Response Percent

Cell 3 4 x 6 =L-3. 1 303 89.91(Cont.) 2 8 2.37

3 0 0.004 9 2.675 0 0.006 14 4.157 0 0.00

3 x 6 = 1 300 93.752 0 0.003 3 0.944 5 1.565 0 0.006 12 3.757 0 0.00

6 x 9 = 1 283 88.442 3 0.943 0 0.004 24 7.505 0 0.006 9 2.817 0 0.00

5 x 9 = E 1 ' 298 95.212 1 0.323 0 0.004 9 2.885 0 0.006 '4 1.287 0 0.00

2 x 6 =11-: 1 287 91.692 1 0.323 20 6.394 0 0.005 0 0.006 3 0.967 0 0.00

1 292 93.299 x 4 =Li2 1 0.323 0 0.004 7 2.245 0 0.006 12 3.837 0 0.00

3 ti

131


Cell 4 36 = D x 9 1 286 87.20

MRA 2 13 3.963 0 0.004 4 1.22

5 0 0.006 22 6.717 0 0.00

32 =D x4 1 286 87.202 15 4.573 0 0.004 10 3.055 0 0.00

6 12 3.66

7 0 0.00

27 = Dx 9 1 305 90.502 14 4.153 , 0 0.004 6b 1.78

5 0 0.006 11 3.267 0 0.00

18 = Lix 9 1 303 94.692 1 0.31

3 0 0.004 0 0.005 0 0.00

6 16 5.007 0 0.00

24 = D x 4 1 290 90.632 9 2.813 0 0.004 4 1.25

5 0 0.006 16 5.007 0 0.00

63 =I Ix7 1 275 85.942 21 6.563 0 0.004 8 2.505 0 0.006 13 4.067 0 0.00

36

Cell 4(Cont.)

56 = x 7

132


1 264 84.352 14 4.473 0 0.004 19 6.075 0 0.006 15 4.797 0 0.00

Cell 5MRB

24 = 3 x

16 = 2 x

18 = 6 x

54 = 9 x E

15 = 5 x

1 280 85.372 21 6.403 0 0.004 3 0.915 1 0.306 21 6.407 0 0.00

1 315 93.472 2 0.59-3 0 0.004 2 0.595 2 0.596 14 4.157 0 0.00

1 271 80.422 24 7.123 0 0.004 32 9.505 0 0.006 8 2.377 0 0.00

1 289 85.762 16 4.753 0 0.004 18 5.345 0 0.006 12 3.567 0 0.00

1

2

3

4

5

6

7

3091

0

1

1

7

0

96.560.310.000.31

0.312.190.00

3 *i

Cell 5(Cout.)

42 = 7 x D

12 = 4 x 0

133


Cell 6MRC

1:::1= 9 x 5

E:I= 2 x 7

= 5 X 2

1

2

3

4

5

6

7

281

10

0

5

1

14

\ 0

89.783.190.001.600.32

4.470.00

1 281 89.782 6 1.923 0 0.004 5 1.605 9 2.886 10 3.197 0 0.00

1 302 92.072 10 3.053 0 0.004 6 1.835 0 0.006 8 2.447 0 0.00

303 92.382 3 0.913 15 4.574 1 0.305 0 0.006 5 1.527 0 (LOU

1 307 93.602 11 3.353 0 0.004 0 0.005 0 0.006 7 2.137 0 0.00

1 325 96.442 6 1.783 0 0.004 0 0.005 0 0.006 5 1.487 0 0.00

3a

Cell 6(Cont.)

"E=1 = 8 x 5

4 x 2

7 x 5

= 3 x 2

= 6 x 5

9 x 8

134


3

1

2

3

4

5

6

7

313

6

0

12

0

5

0

92.881.78

0.003.560.001.480.00

1 302 94.372 2 0.623 10 3.134 2 0.625 0 0.006 3 0.947 0 0.00

1 299 93.442 8 2.503 0 0.004 7 2.195 0 0.006 5 1.567 0 0.00

1 300 95.852 5 1.603' 0 0.004 3 0.965 0 0.006 5 1.607 0 0.00

1 303 96.812 3 0.963 0 6.004 1 0.325 0 0.006 5 1.607 0 0.00

1' 294 93.932 3 0.963 0 0.004 5 1.605 0 0.006 9 2.887 0 0.00

ResponseNumler GivingThis Response

135

Percent

1

2

3

4

5

6

7

122

167

0

0

0

360

37.2050.91

0.000.000.00

10.980.00

1 162 49.392 23 7.01

3 0 0.00

4 0 0:005 128 39.026 11 3.357 0 0.00

1 150 44.512 154 45.703 0 0.00

4 0 0.00

5 0. 0.006 28 8.31

7 0 0.00

1 173 51.342 132 39.17

3 0 0.00

4 1 0.30

5 0 0,006 26 7.72

7 0 0.00

1 121 37.812 173 54.063 ,0 0.00

4 4 1.25

5 0 0.006 18 5.62

7 0 0.00

;

1 132 41.252 170 53.13

3 . 0 0.004 0 0.00

5 0 0.00

6 16 5.007 0 0.00

4 U

Cell 7(Cont.)

Do+ 3 = 2

0

÷ 9= 7

÷ 6=4.

136


Cell 8 18 1 = 3

DLB

54 + iii- = 6

45 ÷ = 5

1 179 55.942 124 38.753 0 0.004 3 0.945 0 0.006 12 3.757 0 0.00

1 137 43.772 138 44.093 0 0.004 4 1.28

5 0 0.006 31 9.907 0 0.00

1 161 51.442 1i9 38.023 0 Q.004 2 0.645 0 . 0.00

6 28 B.957 0 0.00

1 297 90.552 16 4.88

3 0 0.004 0 0.00

5 0 0.006 12 3.667 0. 0.00

1 265 80.792 29 8.843 0 0.00

4 8 2.44

5 0 0.006 20 6.107 1 0.30

1 304 90.21

2 13 3.863 0 0.004 10 2.975 0 0.006 9 2.677 0 0.00

41

Cell 8(Cont.)

12 -I I= 2

56 + = 7

20: =4

15

137


1 309 91.692 11 3.263 0 0.004 1 0.305 0 0.006 12 3.567 2 0.59

1 273 85.31

2 16 5.003 0 0.00

4 18 5.625 0 0.006 12 3.757 0 0.00

1 287 89.692 14 4.373 0 0.004 12 3.755 0 0.00

6 4 1.257 2 0.62

= 3 1 298 95.212 4 1.283 0 0.004

.0 0.00

5 0 0.00

6 7 2.247 3 0.96

Cell 9DLC

10 + 5 =L'

27 + 3 =

1 277 84.452 3 0.91

3 15 4.574 0 0.005 0 0.006 30 9.157 0 0.00

1 262 79.882 29 8.843 0 0.004 4 1.22 .

5 0 0.006 26 7.937 0 0.00

42

Cell 9(Cont.)

24 ÷ 3 = [:3

32 ÷ 8=

42 ÷ 7=

18 + 2 = {=1

8 ÷ 4 =

35 7

ResponseNumber GivingThis Response

13à

Percent

1

2

3

4

5

6

7

297

11

2

10

0

16

0

88.13

3.260.592.970.004.750.00

1 308 91.392 7 2.083 1 0.304 5 1.485 0 0.006 14 4.157 0 0.00

1 287 89.692 7 2.193 0 0.004 8 2.505 0 0.006 16 5.007 0 0.00

1 267 85.302. 1 0.323 5 1.604 9 2.885 0 0.006 31 9.907 0 0.00

1 270 86.262 2 0.643 20 6.394 0 0.005 0 0.006 a 17 5.437 0 0.00

1 286 91.372 8 2.563 1 0.324 2 0.645 0 0.006 12 3.837 0 0.00

43

Cell 10DRA

8 = ÷6

5 =E3-1. 3

4 = 3

9=0+,5

= 7

139


1

2

3

4

5

6

7

87

1990

2

0

36

0

26.52

60.670.000.610.00

10.980.00

1

2

3

4

5

6

7

102180

0

0

0

40

0

31.1054.880.000.000.00

120:2000

I

2

3

4

5

6

7

117185

0

0

0

29

0

34.7254.900.000.000.008.610.00

1

2

3

4

5

6

7

102200

0

1

0

29

0

30.2759.350.000.300.008.610 00

1

2

3

4

5

6

7

89197

0

2

0

29

0

27.8161.560.000.620,009.060.00

1

2

3

4

5

6

7

108

1890

0

0

19

0

33.7559.060.00

0.000.005.940.00

4 4

Cell 10(Cont.)

6 =Q- 2

8 = -I- 5

5 = 21 2

140


Cell 11DRB

4=8 +0

7 = 35 ÷ El

9= 72

1 120 37.502 28 8.753 0 0.004 0 0.00

160 50.006 9 2.817 0 0.00

1 83 26.522 199 63.583 0 0.004 1 0.325 0 0.006 26 8.317 0 0.00

1 116 37.062 162 51.763 0 0.004 0 0.005 0 0.006 32 10.227 0 0.00

1 227 69.212 45 13.723 0 0.004 0 0.005 0 0.006 23 7.017 30 9.15

1 282 85.982 17 5.183 0 0.004 4 1.225 0 0.006 14 4.277 5 1.52

1 297 88.132 17 5.043 0 0.004 13

,3.86

5 0 0.006 7 2.087 0 0.00

141

Cell 11 6=(Cont.)

30 -1. DResponse

Number GivingThis Response Perms cent

1

2

3

4

5

6

7

29621

0

4

0

11

1

87.836.230.001.190.003.260.30

4 = 36 + 1 265 82.812 26 8.12

3 0 0.004 11 3.44

5 0 0.00

6 15 4.697 1 0.31

8 = 32 .1. El 1

2

27718

86.565.62

3 0 0.004 4 1.25

5 0 0.006 20 6.25

7 0 0.00

7 = 28 1

2

27322

87.227.03

3 0 0.00

4 6 1.92

5 0 0.00

6 8 2.56

7 1 0.32

Cell 12 0DRC

= 24 8 1 285 89.062 17 5.313 0 0.004 6 1.87

5 0 0.006 10 3.13

7 0 0.00

= 18 4- 3 1 272 86.902 22 7.033 5 1.604 1 0.32

5 0 0.00

6 12 3.837 0 0.00

4

Cell 12 El = 54 7:- 6

(Cont.)

El= 16 + 8

El= 63 4. 7

D = 18 + 9

/

El = 20 + 4

= 56 + 7


1

2

3

4

5

6

7

26025

3

7

0

16

0

83.077.990.962.240.005.110.00

1 271 86.582 13 4.153 2 0.644 2 0.645 0 0.006 23 7.357 0 M.00

1 255 77.742 44 13.413 l 0.304 12 3.665 0 0.006 11 3.357 0 0.00

1 264 80.492 22 6.713 1 0.304 1 0.305 0 0.006 34 10.377 0 0.00

1 299 88.722 17 5.043 4 1.194 6 1.785 0 0.006 10 2.977 0 0.00

1 274 81.312 25 7.423 0 0.004 22 6.535 0 0.006 10 2.977 0 0.00

143

Appendix G

PERCENT OF CODE 3, 4, 6, AND 7 RESPONSES ON NUMBERPUZZLE TEST (WHOLE NUMBER SOLUTION)

48

Cell

3. axb=

E6.

=axb

9.

ab =

12. 0 = a

b

Example

12

34

56

78

910

00

00

01

00

60

05

00

0.

30

00

0

50

02

60

02

11

00

Code 3 indicated the subject used the inverse

operation, i.e., multiplied while solving

a division open sentence or vice versa.

Cell

Example

24

56

89

10

1. Qx b= c

31

2c1

02

2-

-

2 a x ri = c

01

01

0-

3. axb= 0

03

17

32

83

02

4. c=E,xb

13

20

12

6-

--

S .

c,= a x

11

10

50

22

--

-

6. 11=axb

20

00

41

21

02

00

00

10

11

1-

7.

i__

+b = c

8.

a 4- Eil = c

02

30

64

0-

--

9.

a.-4 b=0

01

31

23

0.

1-

-

10.

c = E] + b

10

00

10

00

0-

11.

c = a 4- ril

14

13

12

--

-

12. 0=a4-13

20

21

40

27

--

Code 4 indicated the subject responded with a numeral quite close to thecorrect

solution.

Exam le

Cell

1.

x b = c

4.

c=

74

35

45

42

42

43

8.

aI=1= c

9.

ab =

10.

c = G b

11

12

Code 6 indicated the subject responded with some numeral not falling into any of

the other six categories.

Cell

Example

12

45

67

89

10

8.

a -EJ =c

00

10

11

11. c=asll

00

-0

0

Code 7 indicated the subject performed the inverse o eration across

the

equality sign.

For example the open sentence 15

= 3 received 45

for the solution, indicating the subject might have c anged the open

sentence to the type 3 x 15 = 0.

148

Appendix H

NO WHOLE-NUMBER SOLUTION CELLS

149''


Cell 1 1 300 89.02= 16MLA 2 0 0.00

3 0 6.004 21 6.235 0 0.006 9 2.67

7 0 C.00

= 23 1 256 81.792 0 0.003 0 0.00

4 20 6.395 1 0..32

6 34 10.867 0 0.00

Cell 2 4 x F =.27 1 249 77,.81

MLB 2 0 0.003 0 0.00

4 24.

5 0 0.00

6 43 13.447 0 0.00

6 x LT] = 43 1 269 82.012 0 0.00

3 0 0.00

4 33 10.065 0 0.00

6 18 5.497 0 0.00

Cell 4 21 = n x 4 1 '267 85.30

MRA 2 0 0.003 0 0.004 19 6.075 0 0.00

6 25 7.997 0 0,00

29 = u x 4 1 284 84.272 0 0.003 0 0.00

4 24 7.125 1 0.30

6 21, 6.23

7 0 0.00

54

Cell 5 20

MRB

'47

Cell 8 35

ALB

47

Cell 9 13

ALC

150

NumberResponse This Response Percent

=7x0

= 8 x

9

= =6

+ 4 =

41 + 8 =

283 86.28'2 0 0.003 0 0.004 16 4.885 0 0.006 21 6.4Q7 0 0.00

1 262 81.872 0 0.003 0 0.004 26 8.125 0 0.006 28 8.757 0 0.00

1 284 84.272 0 0.003" 0 0.004 8 2.375 0 0.006 38 11.287 1 0.30

1 258 82.432 0 0.003 0 0.004 17 5.435 0 0.006 37 11.827 1 0.32

1 274 85.622 0 0.003 0 0.004 22 6.875 0 ".006 16 5.007 0 0.00

1 265 80.792 0 0.003 0 0.004 23 7.015 0 0.006 29 8.847 0 0.00

151

Nulber GivingResponse This Response Percent

Cell 11 3 = 28 + 1 245 78.27

DRB 2 0 0.00

3 0 0.00

4 32 10.22

5 0 0.00

6 29 9.27

7 3 0.96

3 = 7 + 1 203 60.242 0 0.00

3 0 0.004 8 2.37

5 0 0.006 20 5.93

7 101 29.97

Cell 12DRC

=20+3

= 25 + 4

1 259 80.942 0 0.003 9 2.81

4 9 2.81

5 0 0.00

6 39 12.19

7 0 0.00

1 269 82.01

2 0 0.00

3 4 1.22

4 20 6.105 0 0.00

6 23 7.01

7 0 0.00

152

Appendix I

PERCENT OF CODES 1 - 7 RESPONSES FOR THE NO WHOLE NUMBER

SOLUTION OPEN SENTENCES

5 7

Perc

ent o

f C

ode

1Pe

/cen

t of

Cod

e 2

Res

pons

esR

espo

nses

Perc

ent o

f C

ode

3Pe

rcen

t of

Cod

e 4

Res

pons

esR

espo

nses

Cel

lE

xam

ples

12

Exa

mpl

es1

2E

xam

ples

12

Exa

mpl

es1

2

1. L

xb=

c89

820

00

06

6

2.a

x E

l = c

7882

00

00

810

4.c

=Fi

x b

8584

00

00

67

5. c

=ax

p86

820

00

05

8

8.

a D

. c

8482

00

00

25

9.a÷

b =

8681

00

00

77

7860

00

00

102

11.

c =

a ÷

II

12. J

I= a

= b

8182

00

31

36

Percent of Code 5

Responses

Percent of Code 6

Percent of Code 7

Responses

Responses

Cell

Examples

12

Examples

12

Examples

12

1.

jlx b

=c

00

311

00

2.

a x

Ej =

c0

013

50

0

4.

c=

jx b

00

86

00

5.c=ax

00

06

90

0

8.

air

1=

c0

012

12

00

9.

a-1- b

=0

05

90

11.

00

96

130

c = a

[

÷

12.

Q=

a+ b

03

12

70

0

155

Appendix J

CELL ENTRIES - RAW DATA MATRIX

GO

Class

141

- 22 Students

AB

C

Class

143 -

23 Students

AB

CD

Class 151

- 26 Students

AB

CD

Cell

1112

214

7Cell

18

720

4Cell

113

621

5

26

11

514

25

20

78

27

18

710

318

79

21

315

13

12

11

320

12

12

12

412

415

74

97

21

4.

.4

14

621

5

55

12

514

54

20

77

56

19

710

617

710

21

615

12

14

12

618

13

14

15

717

710

21

78

28

67

27

10

2

88

78

88

914

14

48

13

14

14

4

910

88

79

91.4

79

914

12

714

10

12

74

20

10

32

43

10

13

53

11

42

25

11

914

12

411

12

12

14

5

12

11

74

20

12

714

712

12'

11

10

613

No. of Ob.

64

57

No. of0b.

57

71

No. of Ob.

77

75

Class

142 - 21 Students

Class 144 - 19 Students


AB

CABCD

AB

CD

Cell

110

612

4Cell

18

514

4Cell

114

617

8

25

17

412

25

15

47

27

18

616.

315

12

818

314

89*

11

321

11

12

23

410

512

54

94

15

44

14

616-

8

54

16

412

55

13

5 t'1/47-77

56

14

515

615

10

818

615

10

812

621

12

924

76

37

37

22

42

78

812

10.

88

12

76

89

99

48

14

12

10

8

99

12

415

98

10

511

9.12

11

621

10

21

52

10

12

43

10

42

34

11

812

75

9'

99

411

13

12

10

8

12

612

415

12

10

94

11

.12

10

12

623

No. of Ob.

56

46

No. of Ob.

55

54

No. of Ob.

76

68

Class 153

23 Students

AB

C

Cell

116

617

28

17

6

324

11

12

416

617

58

18

6

622

12

11

77

79

816

12

11

914

12

R6

10

74

7

11

11

12

11

12

15

12

6


AB

CD

3Cell

112

10

24

3

62

633

86

7-3

18

19

16

8

34

11

11

24

3

65

630

86

96

18

21

16

9

27

78

11

2

38

12

21

15

3

99

10

21

88

010

77

71

311

11

22

15

2

812

12

18

89

No. of Ob.

S.

66

Class 161.- 26 Students

AB

C

Cell

117

316

29

'

12

6

327

712

418

417

58

11

6

625

812

78

66

816

812

917

74

10

66

3

11

17

710

12

17

75

3No. of. Ob.

711

83

Class 163

32 Students

DA

BC"

D

7Cell

111

722

11

13

26

21

720

20

318

14

15

33

64

12

720

10

14

56

20

820

21

618

14

15

32

87

'8

816

12

78

12

14

15

11

18

911

14

831

610

56

11

9

711

11

14

14

lr

19

12

10

14

831

No. of Ob.

94

67

No. of Ob.

67

811

Class 241 -

19 Students

AB

CD

Class 251 -

22 Students

AB

CD

Class .261

27 Students

AB

CD

Cell

1S

518

2Cell

111

616

4Cell

112

815

7

24

17

64

25

18

68

26

23

613

311

11

12

63

18

11

12

12

317

14

12

20

48

617

24

10

618

44

12

718

6

53

15

64

56

17

68

56

20

613

69

12

10

66

18

11

10

12

617

16

12,

21

72

64

17

77

12

67

89

73

86

11

12

28

11

12

12

48

11

16

11

7

95

11

66

910

12

69

910

16

621

10

02

31

10

66

96

10

36

42

11

812

'11

211

10

12

12

411

11

15

11

7

12

611

65

12

11

11

611

12.

11

13

620

No. of Ob.

5,

66

2.No. of,Ob.

66

64

No. of Ob.

'6

86

7




AB

CD

AB

CD

AB

CD

Cell

18

415

5Cell

112

520

6Cell

113

721

6

24

14

511

26

.4

714

27

21

612

312

910

18

318

:0

14

19

321

14

13

17

48

515

34

12

519

54

14

718

5

53

14

57

56

15

712

56

19

712

612

10

10

16

617

10

14

21

621

13

14

18

73

17

67

78

13

57

89

11

8

88

10

85

Er

11

914

68

12

12

13

6

98

10

515

99

10

721

98

14

718

10

00

55

10

66

72

10

68

78

11

78

75

111

10

11

611

13

12.

14

6

12

89

416

12

.9

97

19

12

12

14

618

No. of Ob.

45

56

No. of Ob.

65

.

77

No. of Ob.

77

76

Class 341

19 Students

AB

CD

Class 351

13 Students

AB

CD

Class 361

28 Students

AB

CD

Cell

110

413

4Cell

16

315

2Cell

116

421

6

25

12

59

23

95

42

818

614

312

69

15

39

69

63

24

12

13

19

410

214

44

63

15

24

15

619

5

54

11

57

53

85

45

817

713

611

89

15

69

610

66

24

12

14

20

76

40

67

32

11

27

11

614

3

810

79

58

66

10

28

13

12

14

7

99

85

13

94

64

59

12

10

710

10

23

13

10

30

72

10

86

12

2

11

88

73

11

66

92

11

15

12

13

7

12

66

57

12

46

56

12

14

12

720

No. of Ob.

54

55

No. of.0b.

35

2No. of Ob.

'8

67

7

Class

342

11 Students


Class 362

25 Students

AB

CD

AB

CD

AB

CD

Cell

16

46

2Cell

118

417

9Cell

110

723

5

23

12

23

29

12

618

24

21

710

39

74

63

26

711

26

315

14

14

13

45

46

14

17

415

94

10

721

5

53

10

24

59

12

518

55

20

710

69

84

66

27

812

27

614

14

16

15

73

21

17

10

57

16

73

419

7

86

84

28

17

811

98

913

14

5

95

82

59

17

86

26

9'

914

614

10

22,

10

10

11

58

12

10

40

14

6

11

48

41

11

18

810

811

913

14

5

12

68

25

12

18

86

23

12

814

515

No. of Ob.

34

22

No. of Ob.

94

69

No. of Ob.

78

5

%.0


AB

C

Cell

110

10

24

5

25

29

812

315

19

16

18

410

924

6

55

29

812

615

20

16

18

75

14

20

8

89

19

16

5

910

20

817

10

318

17

6b

1.1

919

16

6

12

10

18

818

No. of Ob.

510

86


AB

CD


AB

CD

Class 542- 26 Students

AB

CD

Cell

18

45

2Cell

110

312

4Cell

112.

716

7

24

11

24

25

84

82

620

614

312

83

63

15

68

12

317

13

12

19

46

33

24

10

312

44

12

617

5

53

11

24

55

94

65

519

614

612

82

66

15

68

12

6.18

14

10

20

77

14

17

42

64

76

610

9

87

64

28

10

5.

34

812

13

10

7

96

82

59

10

64

12

912

14

617

10

20

11

10

32

72

10

23

52

11

58

32

11

96

6. 4

11

714

11

6

12

66

25

12

96

412

12

513

617

No. of Ob.

44

22

No. df Ob.

53

44

No. of Ob.

67

67




AB

CD

AB

CA

BC

Cell

14

33

3Cell

112

519

5Cell

19

412

1

22

92

62

622

89

25

15

32

36

63

93

13

16

14

15

314

10

73

44

34

34

12

822

54

10

511

1

52

91

65

5-.21

810

55

14

42

65

61

96

16

16

16

15

615

10

73

73

42

57

411

75

72

89

2

84

63

38

12

16

12

58

10

10

51

92

61

99

11

15

713

97

10

32

10

05

15

10

49

67

10

34

50

11

44

23

11

716

14

511

810

81

12

45

18

12

11

15

715

12

69

43

No. of Ob.

23

23

No. of Ob.

68

85

No. of Ob.

55

1

Class 552 -

22 Students

AB

CD

Class 561 -

26 Students

AB

CD

Class 642

- 21 Students

AB

CD

Cell

110

717

4Cell

114

521

6Cell

111

610

4

25

21

68

27

17

712

26

18

49

315

14

11

12

32.1

11

12

17

317

97

14

49

616

34

IS

520

54

11

610

4

55

21

68

57

15

711

56

18

48

614

14

11

12

621

10

14

17

618

11

614

73

10

96

78

913

57

25

60

810

14

12

48

14

11

14

68

12

12

65

910

13

612

914

12

715

910

10

314

10

38

75

10

69

10

610

33

30

11

913

12

411

13

11

14

611

811

65

12

10

14

69

12

13

10

717

12

12

11

412

No. of Ob.

57

64

No. of Ob.

76

76

No. of Ob.

66

4




AB

CD

AB

CD

AB

C

Cell

110

714

7Cell

16

39

4Cell

18

516

5

25

21

514

24

12

410

24

15

610

315

13

920

312

'8

715

312

10

11

15

410

715

74

84

12

34

65

18

5

55

19

513

54

11

49

53

12

610

615

14.

10

21

612

68

15

69

912

15

76

67

87

34

44

72

54

8

89

12

10

78

64

65

88

10

11

5

910

14

520

97

73

14

98

95

15'

10

32

84

10

20

10

10

26

56

11

914

10

611

E2

64

11

39

12

5

12

10

12

521

12

66

413

12

69

S15

No. of Ob.

57

57

No. of Ob.

44

45

No. 'of Ob.

45

65

Class 651 -

26 Students

AB

C

Class 653

- 23 Students

AB

CD

Class 662 -

23 Studenti

AB

C

Cell

116

614

6Cell

110

5.

16

7Cell

115

621

7

28

20

512

24

614

27

17

713

324

14

918

315

911

19

324

10

13

20

414

514

64

10

516

64

16

619

6

58

18

511

55

13

614

58

17

714

624

14

10

18

615

012

21

623

12

14

21

79

34

77

1$

77

77

10

11

9

815

13

96

810

09

78

13

12

13

7

9.

14

12

516

99

96

20

914

12

521

10

41

43

10

'0

27

110

810

69

11

12

12

10

611

69

11

11

16

12

13

7

12

14

13

516

12

79

620

12

15

12

719

C7)

.Cr)

.No. of Ob.

87

56

No. of Ob.

55

67

No. of Ob.

86

i7



Class 663 -

22.Students

A'B

CA

BA

BC

D

Cell

112

515

4Cell

111

712

6Cell

112

319

5

26

21

57

26.

21

312

26

10

610

317

14

912

317

12

718

318

612

15

410

715

44

11

711'

6-4

10

420

5

56

20

58

55

20

411

56

97

10

618

14

10

12

617

14

818

616

814

15

76

86 4

79

11

3-

-8

74

710

7-

811

14

84

811

14

86

8'

10

813

S-

912

14

58

911

14

318

911

77

14

10

35

51

10

76

66

10

03

92

11

914

10

311

12

13

76

11

98

13

5

12

913

412

12'

.10

13

418

12

10

77

12

No. of Ob.

67

5No. of Ob.

67

46

No. Of Ob.

64

7

C'D

Class 741

9 Students

AB

CD

Class 761 -

21' Students

AB

CD

Class 842

24 Students

AB

CD

Cell

1 2

3 2

1,

2

3

.1

2 2

Cell

1.

2

10 5

3

11

13 5

7

14

Cell

1 2

10 5

5

14

18 6

8

14

62

23

314

610

21

315

10

11

20

41

13

14

9,3

13

34

85

16

8

52

21

25

45

95

514

614

65

22

36

14

810

18

613

10

12

24

72

23

07

52

42

74

46

11

82

12

18

88

67

87

10

12

8

92

2'

12

98

.75

19

99

10

622

10

00

00

10

51

42

10

10

37

11

22

21

11

96

611

99

12

8

12

12

13

12

94

15

12

88

623

No. of Ob.

21

12

No. of Ob.

54

57

No. of Ob.

55

'

68




AB

CD

AB

CD

AB

C

Cell

14

25

2Cell

14

92

Cell

114

617

4

22

61

42

312

36

28

17

68

36

43

63

c5

9T

24

12

12

14

43

24

14

68

24

14

617

5

51

62

45

312

35

57

16

68

66

44

66

96

96

24

12

12

1S

71

43

47

16

17

81

36

84

43

28

68

53

814

12

12

5

92

40

59

68

39

912

11

612

10

04

21

10

02

30

10

60

33

11

34

21

11

56

52

11

10

12

ll

4

12

24

24

12

68

37

12

12

11

515

No. of Ob.

23

22

No. of Ob.

34

33-'

No.-of Ob.

86

65

6

Class 863 -

Cell

1 2 3 4 5 6 7s

8 9

10

11

12

No. of Ob.

21 Students

AB

CD

Class 852 -

Cell

1 2 3 4 5 6 7 8 9

10

11

12

No. of Ob.

21 Students

AB

CD

Class S61 ,

Cell

1 2 3 4 5 6 7 8 9

10 11

12

No. of Ob.

8 Students

AB

CD

9 5

15

10 4

14 5

10 7 5

10

10 5

4-

12 8 4

11 8 6 8 7 5 8 7 4

20 7

13

21 7 13

10

12 7 7

11 7 7

5 10

13 4

10

15 1 5

14 2 5

13 5

12 6

17

12 4

16 3

11

12 4 9

10 6

4

15

10 5

15

10 6 9

10 3 8 8 5

18 6

12

18 6

11 7

11 4 5

10 6 6

4 8

11 4 8 10 2 4

12

.1 3

12 4

8 4

12 8 4

11 6 8 7 6 8 8

.4

2 6 4 2 6 4 4 3 4 4 4 3 2

3 1 2 3 1 2 0 2 1 1 2 1.

1

1 2 3 1 2 3 1 1 2 1 1 3 1

Class 851 -

16 Students

AB

CD

Class 853 -

18 Students

AB

CD

Class 862 -

23 Students

AB

CD

Cell

15

511

4Cell

110

215

5Cell

18

518

7

23

15

48

25

95

10

24

17

613

39

97

12

315

.5

913

312

11

11

21

46

512

34

92

15

54

65

18

6

53

14

48

54

75

95

417

614

69

10.

812

613

610

15

6.

12

12

11

19

72

712

57

52

69

75

78

7

86

97

48

95

10

48

812

12

7

96

10

412

910

45

14

97

12

620

10

06

12

410

20

44

10

35

10

5

11

610

84

11

55

10

411

612

12

6

12

69

412

12

73

513

12

712

517

No. of Ob.

35

44

No. of Ob.

53

55

No. bf Ob.

46

67

166

Appendix K

TRANSFORMATION MATRIX

71

167

CellTest Form

A B C D

Cell 1 3 6 2 6

Cell 2 6 2 6 3

Cell 3 2 3 3 2

Cell 4 3 6 2 6

Cell 5 6 2 6 3

Cell 6 2 3 3 2

Cell 7 3 3 2 3

Cell 8 3 3 3 6

Cell 9 3 3 6 2

Cell 10 3 3 2 3

Cell 11 3 3 3 6

Cell 12 3 3 6 2

Number of Sentence Types Per TestForm Contributing to Each Cell

CellTest Form

A B C D

Cell 1 2 1 3 1

Cell 2 1 3 1 2

Cell 3 3 2 2 3

Cell 4 2 1 3 1

Cell 5 1 3 1 2

Cell 6 3 2 2 3

Cell' 7 2 2 3 2

Cell 8 2 2 2 1

Cell 9 2 2 1 3

Cell 10 2 2 3 2

Cell 11 2 2 2 1

Cell 12 2 2 1 3

168v

Appendix L

ANOVA--RESULTS USING a VS b AND ab VS :cAS COMPARED WITH b VS c AND be VS a

73

Placeholdera vs bab vs cb vs cbe vs a

Grade and Placea vs bab vs cb vs cbc vs a

MD x PlaceMD x a vs bMD x ab vs cMD x b vs cMD x bc vs a

MD x Place x GMD x a vs bMD x ab vs cMD x b vs cMD x bc vs a

LR x PLR x a vs bLR x ab vs cLR x b vs cLR x bc vs a

LR x P x GradeLR x a vs bLR x ab vs cLR x b vs cLR x be vs a

MD x LR x PAbove with abAbove x ab vs cAbove x bcAbove x bc vs a

MDxLRxPxGAbove x abAbove x ab vs cAbove x bcAbove x bc vs a

169

DF MS F P

2 164.1808 903.3201 .00011 249.8057 1325.8050 .00011 78.5559 480.8352 .00011 .0515 .7873 .37921 328.3101 1146.2927 .0001

4 1.3578 7.7182 .00122 1.4582 7.7392 .00122 1.2575 7.6972 .00132 .1745 2.6694 .07922 2.5412 8.8726 .0006

2 138.0383 667.4437 .00011 211.6520 944.6439 .00011 64.4246 390.2435 .00011 .1043 4.5030 .22601 275.9722 863.1252 .0001

4 .7256 3.7465 NS2 ,8140 3.6332 .03372 .6372 3.8598 .02772 .0580 .8360 .43952 1.3932 4.3574 .0181

2 4.2760 57.1078 .00011 3.0432 20.0365 .00011 5.5089 94.1791 .00011 1.3466 29.1088 .00011 7.2055 43.9048 .0001

4 .0593 .9044 NS2 .0208 .1371 .87232 .0978 1.6716 .19832 .0671 1.4515 .24402 .0515 .3135 .7323

2 2.2694 26.5586 .00011 3.9553 44.3866 .00011 .5835 8.7305 .00481 .1108 2.1485 .1490_1 4.4280 42.4215 .0001

4 .1158 1.6164 NS2 .0619 .6943 .50422 .1697 2.5386 .08922 .0546 1.0588 .35462 .1769 1.6951 .1940

170

REFERENCES

Cromer, F. E. Structural models for predicting the difficulty of

multiplication problems. Journal for Research in Mathematics

Education, 1974, 5: 155-166.

Davis, R. B. Discovery in mathematics. Reading, Massachusetts:

Addison-Wesley, 1964.

Finn, J. D. Multivariance: Fortran program for univariate and

multivariate analysis of variance and covariance. Buffalo,

New York: Department of Educational Psychology, State

University of New York at Buffalo, 1967.

Grouws, D. A. Differential performance of third-grade children

in solving open sentences four types. (Doctoral dissertation,

University of Wisconsin) Madison, Wisconsin: 1971.

Grouws, D. A. Open sentences: Some instructional considerations

from research. Arithmetic Teacher, 1972, 19, 595-599.

Hays, W. L. Statistics for psychologists. New York: Holt, Rinehart

and Winston, 1965.

Heddens, J. W. Todays mathematics. Chicago: Science Research

Associates, 1968.

Kirk, R. E. Experimental design: Procedures for the behavioral

sciences. Belmont, California: Brooks/Cole, 1968.

Osburn, Worth J. Levels of difficulty in long division. Elementary

School Journal, 1950, 46, 441-447.

171

Suppes, P. & Morningstar, M. Computer-assisted instruction at

Stanford, 1966-68: Data, models, and evaluation of the

arithmetic programs. New York: Academip Press, 1972.

Suydam, M. N. & Weaver, J. F. Multiplication and division with whole

numbers. Set-B, H 2.

Van Engen, H. Why use frames in arithmetic? Arithmetic Teacher, 1966,

. 13, 315-316.

Van Engen, H. & Gibb, E. G. General mental functions associated with

division. Educational Service Studies, No'. 2. Cedar Falls: Iowa.

State Teachers College, 1956.

Weaver, J. F. Some factors associated with pupils' performance levels on

simple open addition and subtraction sentences. Arithmetic Teacher,

1971, 18, 513-519.

Weaver, J. F. The symmetric property of the equality relation and young

children's ability to solve open addition and subtraction sentences.

Journal for Research in Mathematics Education, 1973, 4: 45 -56.

Weaver, J. F. The ability of first-, second-, and third-grade pupils to

identify open addition and subtraction sentences for which no solution

exists within the set of whole numbers. School Science and Mathematics,

1972, 72, 679-691.

Matteis.' Eva Wail.. Om =kW*

Francis S. Chase, Committee ChairmanEmeritus Professor, Department of EducationUniversity of 'Chicago

Helen BainPut PresidentNational Education Association

Lyle E. Bourne, Jr.Institute for the Study of Intellectual BehaviorUniversity of Colorado

Sue BuelDissemination and Installation ServicesNorthwest Regional Educational Laboratory

Roald CampbellEmeritus Professor, Department of EducationalAdministrationThe Ohio State University

George E. DicksonCollege of EducationUniversity of Toledo

L. R. GouletDepartments of Educational Psychology and PsychologyUniversity. of IllinoisChampaign-Urbana

Chester W. HarrisDepartment of EducationUniversity of CaliforniaSanta Barbara

W. G. KatsenmeyerDepartment of EducationDuke University

Hugh J. ScottDepartment of EducationHoward University

Barbara ThompsonSuperintendent of Public InstructionState of Wisconsin

Joanna WilliamsDepartment of Psychology and EducationTeachers' College, Columbia University

Exeutiv. Committee

Richard A. Rossini Iler, Committee ChairmanDirectorR & D Center

William R. B shDeputy rectorR & D enter

M. Vere De ultProfessor of Curriculum and InstructionSchool of Education

Dale D. JohnsonAssistant DeanSchool of Education

Karlyn KammDevelopmental SpecialistIt & 1) Center

Faculty of Principal Investigators

Vernon L. AllenProfessorPsychology

B. Dean BowlesProfessorEducational Administration

Marvin J. FruthProfessorEducational Administration

John G. HarveyAssociate ProfessorMathematics

Frank H. HooperProfessorChild and Family Studies

Herbert J. KlausmeierV. A. C. Henmon ProfessorEducational Psychology

Gisela Labouvie-ViefAssistant ProfessorEducational Psychology

Joel R. LevinProfessorEducational Psychology

L. Joseph LinsProfessorInstitutional Studies

Herbert J. KlausmeierPrincipal InvestigatorR & D Center

Joel R. LevinPrincipal InvestigatorR & D Center

James M. MoserSenior Research ScientistR & D Center

Len VanEuAssociate Vice ChancellorUniversity of Wisconsin Madison

James LiphamProfessorEducational Administration

Wayne OttoProfessorCurriculum and IP. truction

Robert PetsoldProfessorCurriculum and Instruction

Thomas A. RombergProfessorCurriculum and Instruction .

Richard A. RoumillerProfessorEducational Administration

Dennis W. SpuckAssistant ProfessorEducational Administration

Richard L. VeneskyProfessorComputer Sciences

Larry M. WilderAssistant ProfessorChild and Family Studies