Problem Representation and Mathematical Problem Solving of Students of Varying Math Ability

152 Pages·2015·2.15 MB·English

by  Jennifer Lee Krawec

University of Miami
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2010-07-27
Problem Representation and Mathematical
Problem Solving of Students of Varying Math
Ability
Jennifer Lee Krawec
University of Miami, [email protected]
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UNIVERSITY OF MIAMI

PROBLEM REPRESENTATION AND MATHEMATICAL PROBLEM SOLVING OF
STUDENTS OF VARYING MATH ABILITY

By

Jennifer Lee Krawec

A DISSERTATION

Submitted to the Faculty
of the University of Miami
in partial fulfillment of the requirements for
the degree of Doctor of Philosophy

Coral Gables, Florida

August 2010
©2010
Jennifer Lee Krawec
All Rights Reserved
UNIVERSITY OF MIAMI

A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy

PROBLEM REPRESENTATION AND MATHEMATICAL PROBLEM SOLVING OF
STUDENTS OF VARYING MATH ABILITY

Jennifer Lee Krawec

Approved:

________________ _________________
Marjorie Montague, Ph.D. Terri A. Scandura, Ph.D.
Professor of Teaching and Learning Dean of the Graduate School

________________ _________________
Batya Elbaum, Ph.D. Wendy Cavendish, Ph.D.
Associate Professor of Teaching Assistant Professor of Teaching
and Learning and Learning

________________
Soyeon Ahn, Ph.D.
Assistant Professor of Educational
and Psychological Studies
KRAWEC, JENNIFER (Ph.D., Teaching and Learning)
Problem Representation and Mathematical (August 2010)
Problem Solving of Students of Varying
Math Ability.

Abstract of a dissertation at the University of Miami.

Dissertation supervised by Professor Marjorie Montague.
No. of pages in text. (141)

The purpose of this study was to examine differences in math problem solving
among students with learning disabilities (LD), low-achieving (LA) students, and
average-achieving (AA) students. The primary interest was to analyze the problem
representation processes students use to translate and integrate problem information as
they solve math word problems. Problem representation processes were operationalized
as (a) paraphrasing the problem and (b) visually representing the problem. Paraphrasing
accuracy (i.e., paraphrasing relevant information, paraphrasing irrelevant linguistic
information, and paraphrasing irrelevant numerical information), visual representation
accuracy (i.e., visual representation of relevant information, visual representation of
irrelevant linguistic information, and visual representation of irrelevant numerical
information), and problem-solving accuracy were measured in eighth-grade students with
LD (n = 25), LA students (n = 30), and AA students (n = 29) using a researcher-modified
version of the Mathematical Processing Instrument (MPI). Results indicated that
problem-solving accuracy was significantly and positively correlated to relevant
information in both the paraphrasing and the visual representation phases and
significantly negatively correlated to linguistic and numerical irrelevant information for
the two constructs. When separated by ability, students with LD showed a different
profile as compared to the LA and AA students with respect to the relationships among
the problem-solving variables. Mean differences showed that students with LD differed
significantly from LA students in that they paraphrased less relevant information and also
visually represented less irrelevant numerical information. Paraphrasing accuracy and
visual representation accuracy were each shown to account for a statistically significant
amount of variance in problem-solving accuracy when entered in a hierarchical model.
Finally, the relationship between visual representation of relevant information and
problem-solving accuracy was shown to be dependent on ability after controlling for the
problem-solving variables and ability. Implications for classroom instruction for students
with and without LD are discussed.
TABLE OF CONTENTS

Page

LIST OF FIGURES …………………………………………………………………. v

LIST OF TABLES ………………………………………………………………….. vi

CHAPTER 1
INTRODUCTION …………………………………………………………………... 1
Mathematical Problem Solving…………………………………………….... 4
Problem Representation ……………………………………………... 6
Paraphrasing ………………………………………………… 8
Visual Representation ……………………………………….. 9
Students with Learning Disabilities …………………………………………. 12
Math LD ……………………………………………………………... 14
Computation …………………………………………………. 15
Problem Solving …………………………………………….. 16
Summary …………………………………………………………….. 18
Statement of the Problem …………………………………………………… 18

CHAPTER 2
REVIEW OF THE LITERATURE …………………………………………………. 21
Students with LD and Low-Achieving Students ……………………………. 21
Reading ……………………………………………………………… 23
Mathematics ………………………………………………………… 25
Paraphrasing ………………………………………………………………… 28
Visual Representation ………………………………………………………. 34
Categories of Visual Representation ………………………………... 35
Overview ……………………………………………………………………. 38
Statement of Purpose ……………………………………………………….. 40
Research Questions …………………………………………………………. 40
Hypotheses ………………………………………………………………….. 41

CHAPTER 3
METHODS ………………………………………………………………………….. 44
Participants ………………………………………………………………….. 44
Measures …………………………………………………………………….. 46
Mathematical Processing Instrument ……………………………….. 47
Scoring ………………………………………………………. 49
Accuracy of Paraphrasing …………………………… 49
Accuracy of Visual Representation …………………. 52
Problem-Solving Accuracy …………………………. 52
Test of Mathematical Ability ……………………………………….. 52
Scoring ……………………………………………………… 53
iii
Procedures …………………………………………………………………... 55
Consent ……………………………………………………………… 55
Background Information …………………………………………….. 55
Assessment …………………………………………………………... 55
Administration ……………………………………………….. 56
Coding ……………………………………………………………….. 56
Reliability ……………………………………………………. 57
Design and Analysis …………………………………………………………. 57

CHAPTER 4
RESULTS …………………………………………………………………………... 63
Preliminary Analyses ………………………………………………………. 64
Question 1: Relationships among Problem-Solving Variables …………….. 68
Average-Achieving Students ……………………………………….. 71
Low-Achieving Students …………………………………………… 73
Students with LD …………………………………………………… 74
Summary ……………………………………………………………. 75
Question 2: Differences in Problem-Solving Variables by
Ability Group ……………………………………………………………….. 76
Question 3: Effects of Problem-Solving Processes on Accuracy …………... 82
Interactions ………………………………………………………….. 84
Summary …………………………………………………………….. 88

CHAPTER 5
DISCUSSION ………………………………………………………………………. 91
Question 1: Relationships among Problem-Solving Variables …………….. 92
Question 2: Differences in Problem-Solving Variables by
Ability Group ……………………………………………………………….. 97
Question 3: Effects of Problem-Solving Processes on Accuracy …………... 102
Limitations ………………………………………………………………….. 105
Implications for Instruction …………………………………………………. 107
Directions for Future Research ……………………………………………... 111

APPENDICES …………………………………………………………………….... 113
REFERENCES ……………………………………………………………………... 132

iv
LIST OF FIGURES

Page

CHAPTER 1

Figure 1.1 Conceptual Framework: Math Problem-Solving Process ………… 6
Figure 1.2 FCAT Math Extended Response …………………………………. 7

CHAPTER 3

Figure 3.1 Modification of the original MPI………………………………….. 49
Figure 3.2 Paraphrasing and Visual Representation Scoring Example ………. 51
Figure 3.3 Pictorial – Schematic Continuum ………………………………… 54

CHAPTER 4

Figure 4.1 Significant Relationships by Ability Group ………………………. 76
Figure 4.2 Interaction Between Ability and Visual Representation
of Relevant Information ………………….……………………….. 88
Figure 4.3 Hierarchical Multiple Regression Model …………………………. 89

CHAPTER 5

Figure 5.1 Effects of Misinterpretation of Information ………………………. 94

v