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THESIS

El
((00. ("\
llBRARY
Michigan State
University

This is to certify that the

dissertation entitled

HOW WELL DO HONG KONG GRADE 5 STUDENTS
WITH DIFFERENT SOCIAL BACKGROUNDS SOLVE
ROUTINE AND NON—ROUTINE
MATHEMATICS WORD PROBLEMS
presented by

Tat Ming Sze

has been accepted towards fulfillment
of the requirements for

PhD. degree in Educational Psychology

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MS U is an Affirmative Action/Equal Opportunity Institution 012771

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HOW WELL DO HONG KONG GRADE 5 STUDENTS
WITH DIFFERENT SOCIAL BACKGROUNDS SOLVE ROUTINE
AND NON-ROUTINE MATHEMATICS WORD PROBLEMS

By

Tat Ming Sze

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Counseling, Educational Psychology, and Special Education

2001

 

 

ABSTRACT

HOW WELL DO HONG KONG GRADE 5 STUDENTS
WITH DH-TFERENT SOCIAL BACKGROUNDS SOLVE ROUTINE
AND N ON-ROUTINE MATHEMATICS WORD PROBLEMS

By

Tat Ming Sze

The goal of this study is to explore and investigate whether Hong Kong elementary
grade 5 students develop mathematical problem solving competence at school. Problem
solving competence has become more and more important in developed countries such as
Hong Kong. The job market has fewer and fewer routine job openings in the let Century.
To survive, students must develop problem-solving competence in school and apply these
skills in the workplace. But the existing Hong Kong curriculum and examination system
focuses heavily on solving practiced problems. Students may not develop problem solving

competence in schools.

In this study, grade 5 students (n = 123) worked a lO—item quiz comprised of
routine and non-routine problems; a subset of 27 students participated in the second phase
of this study in an interview setting. In the interviews, students participated in problem

sorting activities and solved one to three other non-routine problems.

The results indicate that students from different social backgrounds have difficulty
solving mathematical, non-routine problems in general. Their difficulties seem to be related
to conceptual mistakes relative to two specific mathematical concepts—area and proportion.
Comparing two groups of students who came from different social backgrounds, students

from middle class families outperformed their counterparts on solving non-routine

 

problems. A more interesting result is the fact that even very successful students (grades
and exams) struggled with non-routine problems. And, some students who were not strong

performers in school showed surprising strength on some problems.

 

Copyright by
Tat Ming Sze
2001

Dedicated to Dr. William M. Fitzgerald

ACKNOWLEDGMENTS

My current cognitive structures and ethical stands are linked with my developmental
history. Many people helped me develop my perspective and disposition to education and
psychology. Here, I would like to acknowledge those who, throughout my life, fostered

and guided me towards the development of my attitudes and competency.

First, I would like to acknowledge my debt to my family, especially my parents,
Mr. Sze Shiu Leung and Mrs. Cheung Yui. They offered me a wonderful environment in
which to grow psychologically. Few children from a low-income family could enjoy such
luxurious learning experiences in some of the world's finest universities. Without their
support, all would be impossible for me. Also, I would like to acknowledge my Aunt,
Mrs. Tsui Sze Po Ling, who has always offered her moral support and concern for my

academic work.

Second, I would like to thank and acknowledge my two high school teachers, Dr.
Yat-shing Shaw and Mr. Kam-yung Choi. Dr. Shaw was my Chinese teacher and was the
first person who awakened within me a hunger for philosophy. Dr. Shaw's philosophical
stance and attitude towards truth had a great influence on me. He also encouraged me to
read and understand my own culture. Mr. Choi was my biology teacher. I enjoyed his
lessons and he fostered my interest in man as a physical being. His attitudes on teaching
and ways of teaching impacted me on reflecting how to teach. Combined, these two
teachers laid a path for me that drew me to psychology, which is a field combining

philosophy and biology.

In my college days, two teachers had a great impact on me in education and

psychology. Dr. Chun-hsing Chang was very enthusiastic about education and had an

 

altruistic personality. His attitudes towards education and society impacted me greatly. He
was one of my role models as an educator. Dr. Sieh-Hwa Lin was another teacher who
changed my life and pushed me to think about the importance of providing an exciting
learning environment and opportunities for students to explore on their own. He
encouraged me to explore and to learn about computers. I will always remember what he
once told me in his computer laboratory. He said that the equipment he brought to the
department was for students to explore. He did not want students to worrying about
breaking the machines. Dr. Lin felt that schools are the place for students to explore and
learn. He believed it was his duty to provide an exciting environment in which students
could explore. Dr. Lin graciously offered his services as a consultant on the statistical

analysis in this study.

I would like to acknowledge some of my teachers and members of my dissertation
committee who shared their insights and helped me grow academically. Dr. Ed Wolfe
offered statistical consultation when I did my data analysis. His advice was very helpful in
guiding me towards the interpretation of my results. Dr. Larabee extended his helpful hand
when I unexpectedly lost two dissertation committee members. Although Dr. David
Larabee joined my committee only recently, his suggestions were invaluable and helped me
clarify the value of this study. Dr. Mun Tsang offered his cultural expertise on the
committee and acted as a bridge between me and my committee members who desired to
better understand contextual and cultural information about a study completed in Hong
Kong. The help offered by Dr. Tsang was uncountable. In the last two years, Dr. Tsang
sat in as an external member to help me finish my dissertation work. If there was another
word that could better express in more depth the meaning of the word, thanks, I would use
that word here to express my gratitude to him. Dr. Ralph Putnam is another committee
member who helped me frame this study at the beginning. His suggestions can be found

everywhere in this study. Dr. Glenda Lappan's suggestions can be found in the most micro

vii

of elements—my grammar and word usage, to the most macro of elements—the data
analysis and its implications. Without her suggestions and advice, what I learned in
graduate school would have been totally different. In addition to the knowledge domain,
Dr. Lappan is a role model for me as an educator and as an honorable human being. Dr.
Jack Smith was my academic advisor and the chair of my dissertation committee during the
last 7 years. His ways of thinking and morals have had a great impact on me. The result of
his help and advice is clearly evident throughout this study—from formation of ideas to the
final draft. I can say that he is the co-author of this dissertation. In addition to academic
advice, Jack consistently offered psychological support towards my daily quality of life.
His heartfelt caring about people has had a great impact on me. I will always consider him

to be a role model for my life.

I enjoyed my life at Michigan State University. I found several good friends who
offered a lot of help towards my daily life and academic work. Dr. Cheong Yuk Fai and
Dr. Chiu Wing Tat helped me score the 123 student quiz sheets. The nightmare of those
days and nights is still very clear in my mind. I would like to acknowledge their generous
help. In addition, my friends Chris Dilley and Billie Keusch asked me several times
whether I needed help completing my grammatical edit. Although I did not take them up on
their offers, I would like to thank them for their generosity. Elizabeth Phillips offered her
moral support and daily help during the last seven years. Jade Corsé let me enjoy a very
warm friendship in Michigan. Dr. Mark Hoover always gave me insightful ideas about my
studies. I can frankly say that I owe him a lot for those enlightening talks. Judith Miller is
my best friend in the States. Her care made me feel as if I had a family member in
Michigan. Judi's help on editing the written work was enormous. Without her grammatical
polish, no one could understand my Chinese-English in this dissertation. Finally, I need to

thank my other-half, Dr. Chen Cheng Hui. Without her support, I doubt whether I could

viii

have finished this work. During the last 9 years, she offered her psychological support

every day and night. She also gave me a lot of suggestions and insights into this study.

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................... xiii
LIST OF FIGURES ...................................................................................................... xv
INTRODUCTION ..................................................................................................... 1
Chapter 1
MOTIVATION OF THE RESEARCH. . . . . . . . . . . . .. . 3
Psychological and Educational Concern ............................................... 3
Teaching and Learning Concern ............................................................. 3
Social and Economic Concern ................................................................... 4
Personal Experience .................................................................................... 5
Research Questions ................. , .................................................................... 6
Chapter2
CONTEXTUAL BACKGROUND AND SCHOOLING In HONG KONG ..... 7
Geographical Information ......................................................................... 7
Historical Context .......................................................................................... 7
Schooling in Hong Kong .............................................................................. 9
General Education and Student Population ........................... 9
Primary Schools ................................................................................ 11
Secondary Schools ............................................................................ 13
Junior Secondary Schools .................................................. 13
Senior Secondary Schools .................................................. 13
Pre-University Sectors ................................................................... 14
Chapter 3
LITERATURE REVIEW ......................................................................................... 16
Non-Routine Mathematics Word Problems ....................................... l6
Philosophical Perspective ............................................................. 17
Psychological Perspective ............................................................. 18
Visibility of Goal State ........................................................ 19
Problem Space..........- ............................................................. 20
Ill-defined Problems and Non-Routine
Problems ........................................... 2 1
Educational Perspective ................................................................. 22
Understanding ............................................................................................... 25
Perceptual Understanding ............................................................ 27
Mental Models ................................................................................... 28
Johnson-Laird's Mental Models ...................................... 29
Mayer and His Colleagues' on Conceptual
Models ....................................................................................... 3 1
Mental Models Suggested by Hegarty, Mayer
and Monk ................................................................................. 32
Schema .................................................................................................. 33
Problem-Schema ................................................................... 34

Pragmatic Schema ................................................................ 3 5

Principle-based Understanding .................................................. 3 7
Counting Principles .............................................................. 3 7
Subtraction Principles ........................................................ 3 9
Multiplicative Reasoning ........................................................................... 4 3
Proportion ................................................................................ 4 4
Speed ......................................................................................... 45
Area and Perimeter ............................................................. 4 7
Students from Low SES Families ............................................................ 4 9
Chapter 4
METHOD ................................................................................................................. 53
Subjects ............................................................................................................. 5 3
Tools ................................................................................................................... 5 4
Problem Development .................................................................... 5 4
Quiz Section ............................................................................. 5 5
Interview Section ................................................................. 5 6
Design and Procedures ............................................................................... 5 7
Quiz Section ......................................................................................... 5 7
Interview Section ........................ '. .................................................... 5 8
Scoring ............................................................................................................... 6 0
Quiz Section ......................................................................................... 6 O
Scoring Scheme .................................................................................. 6O
Inter-rater Reliability ................................................................................ 6 4
First Round Inter-rater Reliability ........................................... 6 4
Second Round Inter-rater Reliability ...................................... 6 6
Observations with Disagreement ............................................... 6 8
Instance 1 ................................................................................ 6 9
Instance 2 ................................................................................ 7 0
Chapter 5
RESULT ................................................................................................................. 71
Quiz Result ....................................................................................................... 7 1
Table Description .............................................................................. 75
A Special Example ............................................................................ 9 3
Example .................................................................................... 9 3
Interview Result ........................................................................................... 9 5
Card-Sorting ........................................................................................ 9 5
Students' Performance on the Interview Problems .......... 9 6
The Train Problem ............................................................... 9 6
Classroom Board Problem ................................................. 102
Chapter 6
NATURE OF CONCEPTUAL MISTAKES AND THEIR IMPACT ON
PERFORMANCE ................................................................................................... .106
Two Focus Problems .................................................................................... 107

The "Extended Square" Problem ................................................ 108

Summary .................................................................................. 1 1 8
The "Ratio" Problems ...................................................................... 1 24
Summary .................................................................................. 1 3 0
General Summary ............................................................................. 1 3 3
Other Non-Routine Problems ....................................................... 134
Chapter 7
STUDENTS' CONCEPTUAL STRENGTHS AS EVIDENCED IN NOVEL
SOLUTIONS ............................................................................................................... .136
Students' Typical Solutions on the Two Ratio Problems .............. 1 3 7
Students' Unexpected Solutions on the Two Ratio
Problems .......................................................................................................... 1 4 O
Intuitive Function Strategy .......................................................... 140
Summary and Conclusion .............................................................. 143
A Special Case—Bobby ................................................................................ 145
The Routine Problems in the Quiz ............................................. 146
The "Multiplication" and "Unknown Digit"
Problem .................................................................................... 1 4 6
The Non-routine Problems in the Quiz .................................... 147
The "Lake" Problem ............................................................. 146
The "Balance" Problem ....................................................... 149
The "Glass House" Problem ............................................... 15 1
The "Coins" Problem ............................................................ 156
Non-routine Problems in the Interview ................................. 15 7
The "Train" Problem ............................................................ 157
The "Classroom Board" Problem ..................................... 16 3
Summary and Implications .......................................................... 17 1
Chapter 8
CONCLUSION .................................................................................................... 174
APPENDICES .................................................................................................... 187
APPENDIX A .................................................................................................... 1 8 8
APPENDIX B .................................................................................................... 2 0 8
APPENDIX C ..................................................................................................... 2 2 3
BIBLIOGRAPHY .................................................................................................... 225

xii

LIST OF TABLES

Table 1.1 Comparison of Projected Manpower Supply & Demand in 1996 & 2001
Table 2.1 Hong Kong Student Population in Different Educational Institutions in

1999
Table 4.1 Students Distribution in Quiz Section
Table 4.2 Students Distribution in Interview Section
Table 4.3 Problem Distribution on Two Forms
Table 4.4 Problems for the Interview
Table 4.5 First Round Inter-rater Reliability in Form A
Table 4.6 First Round Inter-rater Reliability in Form B
Table 4.7 Second Round Inter-rater Reliability in Form A
Table 4.8 Second Round Inter-rater Reliability in Form B
Table 5.1 Performance from Both Schools on Form A
Table 5.2 Performance from Two Schools on Form B
Table 5.3 Students with Different Achievement Levels Distribution
Table 5.4 Performance from Low SES School on Form A
Table 5.5 Performance from Low SES School on Form B
Table 5.6 Performance from Middle Class School on Form A
Table 5.7 Performance from Middle Class School on Form B
Table 5.8 High Achievers Performance on Form A
Table 5.9 High Achievers Performance on Form B
Table 5.10 Mid Achievers Performance on Form A
Table 5.11 Mid Achievers Performance on Form B
Table 5.12 Low Achievers Performance on Form A
Table 5.13 Low Achievers Performance on Form B
Table 5.14 Frequencies from Students' Card-Sorting Results

Table 5.15: Brief Interview Performance on the Train Problem

xiii

 

54
54
56
57
65
65
67
68
78
79
80
81
82
83
84
86
87
88
89
90
91
95
98

Table 5.16: Brief Interview-Performance on the 4-Squares Problem
Table 5.17: Brief Interview Performance on the Classroom Board Problem
Table 6.1 Performance on the Extended Square Problem on Form A
Table 6.2 Performance on the Extended Square Problem on Form B
Table 6.3 Incorrect Responses Frequency on the Extended Square Problem
0 Table 6.4 Performance on the Photo-Robber Problem at Two Schools
Table 6.5 Performance on the Photo-You Problem at Two Schools
Table 6.6 Incorrect Responses Frequency on the Two Ratio Problems
Table 6.7 Incorrect Responses Frequency on the Cake Problem
Table 7.1 A Brief Summary of Bobby's Performance
Table 7.2 The First Case's Response on the "Glass House" Problem
Table 7.3 The Second Case‘s Response on the "Glass House" Problem
Table 8.1 Performance on Routine and Non-Routine Quiz Problems on Form A

Table 8.2 Performance on Routine and N on-Routine Quiz PrOblems on Form B
Table 8.3 Incorrect Responses Related to Conceptual and Computational Errors on

the Two Ratio Problems
Table 8.4 Incorrect Responses Related to Conceptual and Computational Errors on

the Extended Square Problems
Table 8.5 Two Schools Performance on the Non-Routine Problems
Table 8.6 Incorrect Responses Frequency on the Two Ratio Problems at Two

Schools

101
104
120
121
122
131
132
132
135
146
153
153
175
176
180

181
185

186

 

 

LIST OF FIGURES

Figure 2.1 Hong Kong Schooling and Examination System
Figure 3.1 Tower of Hanoi Problem
Figure 3.2 Distance-centered Conception of Speed
Figure 3.3 Distance-Time Conception of Speed
Figure 4.1 Scoring Procedure
Figure 4.2 A Student’s Work on the Balance Problem
Figure 4.3 A Student's Work on the Extended Square Problem
Figure 5.1 The Cake Problem on Form B
Figure 5.2 A Student's Response on the Cake Problem
Figure 5.3 The Train Problem
Figure 5.4 The 4-Square Problem
Figure 5.5 The Classroom Board Problem
Figure 6.1 The Extended Square Problem in Form A
Figul‘e 6.2 The Extended Square Problem in Form B
Figure 6.3 The Square-root Confusion
Fi gul‘e 6.4 Type A Conceptual Mistake on the Extended Square Problem
Figure 6.5 Type B Conceptual Mistake on the Extended Square Problem
Figure 6.6 Type C Conceptual Mistake on the Extended Square Problem
Fi gul‘e 6.7 Type D Conceptual Mistake on the Extended Square Problem
Fi glue 6.8 Type E Conceptual Mistake 1 on the Extended Square Problem
Figure 6.9 Type E Conceptual Mistake 2 on the Extended Square Problem

Pi gure 6.10 The Photo-Robber Problem

Pi glue 6.11 The Photo-You Problem
Pi gure 6.12 Type A Mistake Case 1 on the Photo-Robber Problem

F ‘
l glue 6.13 Type A Mistake Case 2 on the Photo-You Problem

10
19
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47
63
69
70
93
94
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100
102
108
107
109
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125
125
126
127

 

Figure 6.14 Type B Conceptual Mistake on the Photo-Robber Problem
Figure 6.15 Type C Mistake Case 1 on the Photo-Robber Problem
Figure 6.16 Type C Mistake Case 2 on the Photo-Robber Problem
Figure 7.1 The Photo-Robber Problem on Form A
Figure 7.2 A Typical Solution for the Photo-Robber Problem on Form A
Figure 7.3 The Photo-You Problem on Form B
Figure 7.4 A Typical Solution for the Photo-You Problem on Form B
Figure 7.5 An Example of Functional Thinking on Proportion
Figure 7.6 An Example of Additive Thinking on Proportion
Figure 7.7 The "Unknown Digit" Problem
Figure 7.8 The Lake Problem and Its Solution on Form A
Figure 7.9 Bobby's Work on the "Lake" Problem
Figure 7.10 The "Balance" Problem and Its Solution on Form A
Figure 7.11 Bobby's Work on the "Balance" Problem
Figure 7.12 The "Glass-House" Problem and Its Solution on Form A
Figure 7.13 Bobby's Work on the "Glass House" Problem
Figure 7.14 The "Coin" Problem and Its Solution on Form A
Figure 7.15 Bobby's Work on the "Coins" Problem
Figure 7.16 Covarying Relationship Between Time and Distance
Figure 7.17 The "Train" Problem in the Interview
Figure 7.18 Bobby's Work on the "Train" Problem
Fi gure 7.19 Two Similar Speed Models for Two Speed Problems
Fi glue 7.20 The "Classroom Board" Problem in the Interview
IFigure 7.21 Bobby's Work on the "Classroom Board" Problem

128
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INTRODUCTION

Solving problems is a competence that we should help students develop in schools.
The reasons for developing such competence are many. Nowadays, the most obvious and
direct one is an economic one. Any developed country, like Hong Kong, needs more and
more intelligent problem solvers in the workplace. The main focus of this study is to
investigate whether Hong Kong students develop such competence in elementary schools.

Non-routine problems are considered to be true problems in this study; they are

Used to investigate whether students develop problem solving competence in schools. The
underlying assumption about problem solving competence is that students can solve any
non-routine problems under the constraint that they have the related knowledge and
understand the concept(s) which are required for the problems. So, the competence is
related to conceptual understanding. In this study, a special case, Bobby, revealed the
validity of the assumption. Bobby had a very firm conceptualization on the concept, speed.
He knew that the co-varying relationship between two entities was important to the
concept. So, he had no difficulty solving the two speed problems, "Lake" and "Train,"

which had totally different structures and story contents. He also was the only students able

to think about the "Lake" problem with a mathematical perspective.
This study focuses on the multiplicative reasoning among students in grade 5.

Multiplicative reasoning is an important mathematical competence in upper elementary

gr ades. It is related to many important mathematical concepts—area, proportion, speed,

rati 0. rate, and fraction—which students learn or develop in middle and upper elementary
g1Tades. This thinking is quite different from the additive drinking that students learn and

deVelop in the lower elementary grades. However, it also poses some difficulties for

s . . .
tudents to develop understanding on the related concepts of area, proportion, and fraction

This study explores the possible reason(s) for the difficulties that students have in

:lathematical reasoning.

‘

The difficulties in solving non-routine mathematical problems can be caused by
many variables. It can be a developmental one, a linguistic one, a conceptual one, one
related to social background, or an integrated one. This study found that the difficulties

Hong Kong students have are related to the conceptual understanding of specific

mathematical concept—area, perimeter, and proportion.
The social background is another focus of this study. Although there are lot of

studies in the United States that reveal that social background and socio-economic status,
are the main factors that influence students' learning of mathematics, Hong Kong educators
are not interested in investigating the effect social background has on learning in any school

Subjects. This study will explore the influence of this factor on the way Hong Kong

students learn mathematics.
There were 2 phases of this study. In the first phase, 123 Hong Kong grade 5

students participated. Students came from two different schools located in two different
Communities, one in a middle class community and one in low SES community. Students

Were asked to work on a quiz containing 10 problems, 4 were routine and 6 were non-

r ontine problems, in one hour. In the second phase, a subset of 27 students participated in

the interview setting. They were asked to participate in two activities, problem sorting and

Problem solving.
The result of this study revealed a very interesting finding. All of the self-developed

rhoClels or rules for thinking about ratio were developed by the mid-achievers in two
different schools. Most of the high achievers in both schools were able to solve the ratio

p r0blems with the cross-multiplication method, but none used any self-developed rules or

models for solving the problems.

Chapter 1

MOTIVATION OF THE RESEARCH

Psychological and Educational Concern
A lot of educators and psychologists (Duncker, 1945; Schoenfeld, 1985;

Thompson, 1985; Wertheimer, 1959) are interested in knowing how individuals or
Students solve problems. Although they may have different definitions for the term,
problem, and may be concerned with different issues concerning problem-solving, there
may be common reasons behind their concerns about human problem-solving. The most
obvious reason may be that problem—solving is a crucial component of intelligence
(Holyoak, 1995; Resnick & Glaser, 1976; Stemberg, 1982). An intelligent individual is
one able to adjust oneself and make use of prior knowledge to deal with strange or new
situations. My concern focuses on whether our schools or our educational systems help our
Students develop such competence, and whether they help or hinder the development of
Stlldents' disposition to the use of their knowledge to tackle challenging problems. If we

examine student assessments and observe classroom practices, we will find that schools

PUt our students at a disadvantage relative to problem solving.

Teaching and Learning Concern

This study, like previous studies (diSessa 1982; Carpenter, Fennema, Peterson,
Chiang & Leof, 1989), hopes to offer useful information to teachers and teacher educators.
If teachers want to help their students learn a school subject with understanding, then
teaChers need to better understand student learning. In stating this, what I mean is how they
learn, what are the cognitive difficulties they have when they learn mathematical concepts,
and what are the possible cognitive structures and representations that help students
understand various mathematical concepts. This study will inform teachers and teacher

duCators about some cognitive obstacles associated with mathematical concepts. For

 

‘

example, what are the difficulties that grade 4 and 5 students have relative to area and

perimeter. This study will also target at issues about poor mathematical education practices

at schools in Hong Kong.

Social and Economic Concern
While scholars and educators with idealistic goals and enthusiasm push toward

educational improvement, education is also controlled by limited resources and societal
demands. One societal demand is our labor market. In a capitalistic society, a strong
influence on education is the need to educate workers to meet the demands of the
workplace. Five years ago, the Education and Manpower Branch of the Hong Kong
Government estimated the manpower supply and demand for the coming 7 years. In year
the 2001, the job market will not need as many semi-skilled laborers; these are the workers
that are high school graduates and below. But, the market will ask for more and more
workers with technical knowledge and problem solving competence, most of which are

graduates with post-secondary diplomas or degrees (T able 1.1). From the table, we can
examine trends in the Hong Kong job-market in the let Century. There will be fewer and

fewer labor-intensive job vacancies in the next century. More and more opportunities will

Open to workers who have a good technical knowledge-base and problem-solving

Competence.

Table 1.1 Comparison of Projected Manpower Supply & Demand in 1996 &

 

 

2001
E1 '
ducatron Level/ Supply Demand Balance
Supply and
Demand 1996 2001 1996 2001 1996 2001
Lower secondary&
Below 1,226,900 1,108,600 1,177,300 1.068300 +49~600 +4930“
Upper Secondary 703,900 647,600 7 15,400 633,200 -6,500 +14,000
F .
orm Six 130,100 105,900 140,900 123,900 -10,800 -l8.000
Craft 11,900 11.000 18.700 16.100 41300 '5’100
Techmcran 3 8,700 33,100 43,500 37,900 -4,800 -4,800
8
Meg” 87,800 74,000 92.300 77.500 4500 “3300
F
“we?“ 118.200 89,600 132.300 111,600 44100 42100"
P
ostgraduate 10,500 8,000 12,600 9,400 -2,100 -1.400

 

Source: Government of Hong Kong (1994). Manpower 2001 revisited projection of manpower supply and
requirements for 2001. Education and Manpower Branch. Hong Kong, p. 44.

Personal Experience
In Hong Kong, my generation did not face issues related to social class in our

schools. Most students came from low socioeconomic status (SES) families. Most of our

Parents left Mainland China in the early 1950's, and most of them were poor. That may be

the reason why there have been no serious studies about how students from that social
Class learned and developed at school. Last year I spoke with one Hong Kong mathematics
edUCator about my study. He told me that social class is an issue only for Americans; Hong
Kong does not have such an issue in their system. As a result of his comments, I began to
have a totally different idea about social class issues.

In the past 30 years, the Hong Kong economy has produced a new social class, the

“fiddle class. And, they are quickly growing and becoming the majority in the society. At
the Sallie time, this social group intentionally or unintentionally tried to scramble social

lEso‘ll‘ees. Such social change may put the low SES class in an even more disadvantaged

position. Often, families in this social class are not as active in politics. Therefore, they can
easily be neglected or sacrificed. Under such conditions, access to education becomes very
irnportant. The question is, "Can we lay out a plan so that all students have fair access to
educational resources?" I think we need to rethink our old sabbatical-based educational
System to push forward this social change. This study will highlight the issue of low socio-
economic status in Hong Kong society, although I know that it cannot change the society
or the educational system. Also, from a series of studies (Griffin, Case, & Siegler, 1994;
Saxe, Guberman & Gearhart, 1987; Siegler, 1993), we know that students from low SES
families do not perform as well as students from middle class families in school
mathematics. This study may help teachers understand how they can help students learn
and develop intellectually in school, regardless of social class.
All of these concerns and questions lead me to think about a study that would

investigate them. My research questions are also based on those concerns and questions.

Research Questions

1. Do Hong Kong students have difficulties solving non-routine mathematical

problems?
2. If students have difficulty, what are the general characteristics of their

difficulties—conceptual, computational, linguistic, or developmental?

3. If students have difficulties, are these issues related to social classes?

Chapter 2

CONTEXT UAL BACKGROUND AND SCHOOLING IN HONG KONG

Geographical Information
Hong Kong is physically a small city. The total area is about 1,100 square

kilometers (Phoenix, AZ, has about the same area, 1,125 sq. km), but there are more than
6.3 million people living in Hong Kong. So, the population density is 5,780 people per
Square kilometer. Obviously, the density of population is very high, especially in the urban
districts. About half of the population (3.1 millions) live in public housing apartments. The
average wage of a Hong Kong worker was about US $1,250 per month in 1996. About
95% of the residents of Hong Kong are Chinese. Most of them entered Hong Kong after
the Chinese Communist Party took control of China in 1949. In the early fifties and late
sixties, there were two major waves of immigrants who established the current population
Structure. In the coming years, Hong Kong will receive many new immigrants from
Mainland China. The most conservative figure is that 60,000 new immigrants will come
from the Mainland to Hong Kong in the coming five years. Most new immigrants will be
SChool-age children and their working class parents. Also, Ming Pao reported in 1997 that

the poor population was increasing and that about 13% of the total population was living in

POVer-ty.

Historical Context
Hong Kong was a British colony for more than 100 years. After the Opium War,
1 840‘1 842, the British government focused their efforts into turning this small fishing
Village into an East Asian trade city. The British Government was very successful at
Sculptin g this village into a modern city. The Government also imported a lot of western
culture to this eastern land. The Hong Kong Chinese may be the most westemized Chinese

111 all Of China (Hong Kong is part of China now). Although the British government did

‘

not establish a democratic system in the Hong Kong, they did establish a western legal
System. This system has influenced the Hong Kong Chinese legal spirit. I think it is a fact
that when we compare these three Chinese societies (Hong Kong, Mainland China, and
Taiwan), we will find that the Hong Kong Chinese enjoy a fair legal system. I think no one
Will deny the British contribution to establishing a fair legal system. In 1997, the Hong
Kong government switched from a colonial British Government to the current Chinese
Government. Politically, there is no noticeable change. It is still a capitalistic, non-
democratic, but free society. In fact, most of the Hong Kong people are more concerned
about whether they will lose the legal system they had before than they are worried about
any change in the capitalistic ideology or the political system overseeing the society.
Although there have been no big changes in the political system, the transition of
sovereignty from Britain to China will undoubtedly bring a lot of changes to schooling in
Hong Kong. First, the decolonization of education is taking place in Hong Kong.
Certainly, no one will be able to guess the result of this shift from colonization to
decolonization. 0n education, will the school administrators and teachers adjust themselves
easily to this new age? Does the new government have enough foresighted policy makers in
Place to plan and carry the new policies forward for the new age? The answer to these
Questions of the future will only be able to be read in the 2lst Century. Second, there will
be Some new curricula introduced into schools. We know that the Government will
intrOduce a civil education curriculum into the schools in the very near future. Some
eduGators and scholars have suggested that a general problem-solving skills curriculum,
that is experimental, is used in some chosen schools. Third, there will be more than
60‘1 00,000 new immigrants, most of them school—age, moving to Hong Kong within the

me)“ 5 to 10 years. How does the education system serve this population? These will be the

b' . . . .
lg edllcatron issues of the socrety 1n the next 5 to 10 years.

Schooling in Hong Kong

General Education and Student Population
The education system in Hong Kong is composed of five sectors. They are, 1) pre-

school, 2) general education, 3) technical education/vocational education, 4) higher
education, and 5) adult education. The students who participated in this study all belonged
to the general education sector. This section will provide some data on general education in
Hong Kong. Most of the data will be based on the paper, The Education System, recently
Written by Dr. Cheng Kai Ming (1997). First, lets look at the student population (Table
2.1) in the general education and higher education sectors in 1999. Generally, primary
school education in Hong Kong, as in the United States, comprised of grades 1 through 6.
Secondary school education in Hong Kong, as in the United States, comprised of grades 7
through 12. The exception is that the secondary schools in Hong Kong cover 7 years of
study . This general education scheme is a copy of the school system in England. Students

only spend 3 years in college if they enroll in a full-time degree program. A description of

general education in Hong Kong will be elaborated upon in the following sections.

Table 2.1 Hong Kong Student Population in Different Educational Institutions

 

 

 

_ in 1999
Number of Number or Pupil-Teacher
__ Schools Enrollments Ratio
Kindergartens 744 175,073 13.0: 1
Primary Schools 830 476,802 22.7 : 1
(17,799)*
Secondary 471 455,872 19.3: 1
Schools (3,141)*
Special Schools 74 9513 5.9 : 1
Govt-Funded 8 45,523** N/A
30 Colle es
:1. “fie: Hong Kong Education Department Website, http://www.infogovhk/edl
ese are the numbers of new immigrant students

in; Th
eSe are the enrollments for the lst degree programs.

In addition to these general statistics, Figure 2.1 shows how the student population

changes at different levels, and how examinations function at different levels as gatekeepers

to track students as they venture down different educational paths.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Age Schooling System Examination System
3 years colleges / universities
(50 - 60% pre-u sector students can attain
19 . colleges) ‘— Hong Kong_Adyanced
2 year Pre-university sector Level Exarmnatron
17 _ (1/3 seniors go to the pre-u sector) ‘— Hong Kong Certificate
for Education
I 2 years Senior Secondary Schools Examination
. . . *
15 7 (about 90% of junrors go to senror level) I Allocation Test
(government calls it,

 

 

3 years Junior Secondary Schools exercise)

1 2 r
‘— 2 Aptitude Tests
‘ in grade 5 and 6

 

 

6 years Primary Schools

 

 

 

* another 5 % juniors go to technical institutes
Figure 2.1 Hong Kong Schooling and Examination System

Formal schooling in Hong Kong starts at age 6. Nine years of school education,
from primary 1 (P1) to junior secondary education (Form 3), which is equal to grades 1
through 9 in the United States, is compulsory. By law, all children are required to attend
thfise 9 years of schooling. Home schooling is not accepted as an alternative. After 3 years
of j uni or secondary education, there are two more sectors of secondary education.
Following junior secondary education, there is a 2-year sector of senior secondary

ed . . . .
“can on. Between these two stages, there rs an allocatron exercrse that screens and

10

channels 5% of the junior secondary students to technical institutes for a technical education
and another 5% is channeled directly to the workplace. After a 2-year senior secondary
education, about one-third of the seniors will go on to attend another 2-year pre-university
Sector that will prepare them for a college education. Between these two senior secondary
education sectors, there is a key examination in a Hong Kong students' life, the Hong
Kong Certificate for Education Examination. This is the main gatekeeper examination that
screens students for either further college education or the job market. After the 2-year pre-
University sector, students need to take another examination for attending college, the Hong

Kong Advanced Level Examination. About 50—60% of the students in the pre-university

sector can attend colleges or the universities.

Primary Schools
Primary schools follow a uniform curriculum comprised of eight to nine subjects,

including Chinese language, mathematics, English language, integrated science (integrating
social and natural science into a school subject), Mandarin, art, music, physical education,
religious studies (offered in schools sponsored by religious organizations), and civil
education, which will be introduced in the schools very soon. Around 90% of the primary
80110015 are running half-day sessions for economic and geographic (not enough land for
bllilding schools) reasons. So, two schools share the same location, one operates in the
morning and another in the afternoon. But, the two schools are run by two different groups
0f SChool administrators and teachers. Although the government has promised that there
Will be a gradual conversion to whole-day schooling in primary schools, there are some
unexpected obstacles—economic difficulties for government finances, and the sudden
i“creaSe in the student population of new immigrant students from Mainland China. The
Class Size is about 35 students per class. Some schools adopt the "activity approach"
(learning is based mainly on designed activities in which students actively participate, a

klnd 0f learning-by-doing philosophy) for instruction instead of lecturing. Class size is

11

about 30 students per class. In grades 5 and 6, students take a public examination called
Aptitude Tests. The test results are the main point of reference for allocating students to
different secondary schools after they finish their primary educations. The mechanism for
allocating students to different secondary schools is quite complicated. Basically, it is a
school-based referenced allocation mechanism. In other words, if your school has a lot of
high achievers and they perform very well on the aptitude tests, then your school receives a
hi gher priority in choosing secondary school seats for your students. Every year in Hong
Kong, when it is time to apply to primary schools, a lot of parents actually line up outside
some good primary schools in the very, very early morning (about 5 or 6 am.) to receive
application forms. A good primary school will bring more guarantees that their children
will be admitted to a good secondary school. Because the popularity of a school is
determined by the students' performance on the aptitude tests, schools thus spend a lot of
time coaching their students on how to achieve high scores in the examination.

Most of the primary schools use Cantonese as the instructional language. The
Chinese written systems were united two thousand years ago. The modern written Chinese
SYStem was modified one hundred years ago. All Chinese still use the unified written
SYstem at the grammatical level (Although Mainland uses the simplified character system,
and Hong Kong and Taiwan use the traditional system, their grammars are the same. The
only difference is the shape of the characters). However, because the Chinese did not have
a unified spoken system until a hundred years ago, Mandarin was used as the official
Spoken language. Cantonese is a dialect in southern China. In fact, Cantonese is an old
Chinese language. One thousand and five hundred years ago, it was the main tongue in
China. So, Cantonese is still the best dialect in which to read old and classical Chinese
poems. The textbooks for different subjects are written in Chinese (the traditional Chinese
written system), and Cantonese is the instructional spoken language. There are some

xccTvptrons, some elrte and intematronal schools use English as their instructional medium.

12

 

Secondary Schools
u ' S h 1
Junior secondary schools also use a similar uniform curriculum comprised of 12 to
13 subjects, including Chinese language, English language, mathematics, Mandarin,
integrated science, computer studies, Chinese history, world history, economic and public
affairs, geography, home-economics, art, music, physical education. Religious studies are
only offered in schools sponsored by religious groups. Students need not take all the
subjects; most of them take 10 or 11. The class size is 40 students per class, although there
is a current policy for reducing this to 35 students per class. In the third year of junior
secondary education, there is another public examination—Allocation Exercise. About 90%
of the students successfully complete this examination and attend the senior secondary
schools. About 5% of the students will be tracked to technical institutes for craftsman

education, and the other 5% will leave the general education system and go into the

workplace.

Wis

Senior secondary schools offer two different kinds of classes, science and arts. The
different groups use different curricula and different schools offer their own subject
cOmbinations for their arts and science classes. Also, students can build their own
Combination of subjects. But, all the curricula for different subjects follow the same
Syllabus assigned by the Hong Kong Examination Authority. This government
organization manages most of the public examinations in Hong Kong. Generally, subjects
included in the arts classes are, Chinese language, Chinese literature, Chinese history,
English language, English literature, some other foreign languages (French, German,
’ ‘ ' etc), mathematics, Chinese history, European history, American history, economics,
public affairs, geography, biology, religious studies, and home-economics. For the science

claSSes, most of the schools offer the following subjects: Chinese language, English

13

language, mathematics, advanced mathematics, biology, chemistry, computer studies,
economics, geography, geography, and religious studies. Some schools offer special
subjects. For example, technical drawing is offered by most of the technical-oriented
secondary schools. Most of the students will choose 6 to 9 subjects on which to focus.
These 6 to 9 subjects are needed to get into the most important public examination of their
lives, the Hong Kong Certificate for Education Examination, taken in their 2nd year of their
senior high school. Most of the students take 6 to 9 papers based on school subjects in the
exaxnination. After this examination, only one-third of the students stay in the general
education system and go on to attend the pre-university sector for another 2 years of
general education. Another two-thirds take some non-degree technical courses offered by
the technical institutes, or will go into the work-market directly. Most of the classrooms in
the senior secondary schools have 40 students.

The language used for instruction is mixed in secondary schools. There were only a
few Chinese-instruction schools in Hong Kong in the past. Most of the schools were
known as Anglo-Chinese schools. The medium of instruction in those schools is a rrrix of
English and Cantonese. Their textbooks are written in English and the spoken language in
the Classes is Cantonese. Beginning in 1999, the Hong Kong government began forcing all
secondary schools to become Chinese-medium schools. There are about 100 schools are
the CXCeptions to this rule. All of these 100 schools are elite schools in Hong Kong, and

coIltinue to use English as their instruction medium.

I)l‘e-University Sectors

Most of the senior secondary schools have their own pre-university sector, but
mOSt of the students switch to other schools for their pro-university educations. The
condition for admission for most pre-university sectors accepting students is based on the
students' performance in the Hong Kong Certificate for Education Examination. Most of

the Dre-university sectors offer three different kinds of classes to their

14

students—mathematics, biology, and the arts. Most of the students take 3 to 6 subjects in
this 2-year sector. All the curricula in these 2 years are examination-oriented, too. Students
in the pre-university sectors are preparing for the university-entrance assessment, Hong
Kong Advanced Level Examination. The students in the mathematics classes are planning
to attend a college of engineering or natural science. The students in the biology classes are
planning to attend a college of medicine or natural science. Some students in the
mathematics and biology programs takes classes offered by the arts program. An example
would be taking econorrrics in the pre-university sectors when they plan to apply to the
college of business. Students in the arts programs are planning to attend the college of arts
and business. The class size in this sector is 30 students per class. Only about 50 percent of
the students will be able to attend the degree courses offered by the universities or colleges.
The other students will attend non-degree programs. For example the teaching diploma
offered by the Hong Kong Institute of Education. Or, some other non-degree programs

offered by some universities and colleges. Or these students will go directly into the

Workplace.

There are a lot of non-degree courses offered by the technical and vocational
education sectors. Students who cannot attend the senior secondary schools are likely to
attend some craftsman courses offered by the technical institutes. Students who are not able
to attend the pre-university sectors are likely to attend some technician courses offered by
the teehnical colleges. Students Who cannot attend the degree courses offered by the
colleges or universities will likely attend non-degree courses or attend some part-time
degree courses offered by Hong Kong Open University or other colleges and universities.

Bun this sector does not belong to the general education sector, so a detailed description

Will not be given here.

15

Chapter 3

LITERATURE REVIEW

There are three main sections in this chapter. In the first section, I will try to clarify

what I mean by non-routine problems. Although I cannot offer you a final definition of the
terrn, I will try to use different perspectives—philosophical, psychological, and
educ ational—to characterize the concept. In the second section, I will talk about another
vague, but important concept, understanding. Using different psychological constructs,
mental models, mental schemes, and principle-based understanding, I will attempt to
outline the characteristics of understanding. In the third section, I will attempt to give a
review of the students' conceptions of several mathematical concepts—proportion, speed,
area and perimeter. In the last section, I will address the mathematical competence growth

of students from low social economic status (SES) families.

Non-Routine Mathematics Word Problems

First, let me briefly clarify the term, "non-routine problem." Some mathematics
educators and psychologists (Schoenfeld, 1985; Voss, 1989) saw non-routine problems as
true problems, and routine problems as exercises. They argued that routine problems
Cannot be counted as problems, psychologically. They further argued that a problem is
presumed to exist when the steps needed to achieve the goal are not immediately apparent
as when an individual confronts a problem situation. The term, "non-routine problem,"
Used in this study corresponds to this "problem" idea. Second, the term, "mathematics
problems," refers to designed problem situations that require students to bring their
acqllired mathematics knowledge to the problem situations in order to solve the problems.
Mathematics problems are different from problems solved without any school mathematics
knovVledge base. For example, in the famous Tower of Hanoi Problem (Neves, 1977),

Sub.lects need not use any knowledge of mathematics as a basis for thinking about how they
1 6

_~—

 

Should move the 3 different size disks within a one-by-one format and how they can move
the disks from the left peg to the right peg. Third, the term, "word problems," is used
differently in computational problems. When word problems were presented to the students
in the study, each of them faced a specific situation or story embedded by one or several
different quantities. The Students needed to understand the situation or the story and the
quantities in addition to the numerals and related mathematical operations. Numerical
problems require students to use specific mathematical operation(s) to manipulate the
numerals.
In the following paragraphs, 3 different perspectives—philosophical,

psychological, and educational—will be used to outline the characteristics and the nature of

the problems in this study.

Philosophical Perspective

The nature of "problem" has been a subject of debate in different disciplines, but
mainly in philosophy and psychology (Agre, 1982; Voss, 1989). There is no general
agreement among most investigators on this issue. From Dewey to Popper, the concept of
PPOblenrs, has not been well defined philosophically. Philosophers could only try to lay out
the network of concepts that were closely tied to the ideas of the problem in order to
describe the characteristics of the concept. Here, I would like to use the closely related
conStl'uct, consciousness, to outline my philosophical stance on the concept. I need to
Clarify that it is only a theoretical stance, not the truth. And, I believe that there is still a long
road ahead of us to before we will be able to achieve a true statement on the concept,
problems.

Philosophers have argued with different perspectives as to the existence of the
problem. Is the existence of the problem determined by our consciousness? Some, like
DeWCy (1986), preferred to think that problems are created through consciousness of or

u ticulation about situations rather than being situations. Hence, problems cannot exist

17

 

Without prior consciousness. In a stronger tone, those same philosophers seemed to imply
that the existence of a problem is determined by consciousness. However, some
philosophers (Agre, 1982) maintained that there may be some fault in arguing that the
existence of problems is determined by the human consciousness. There are many counter-
examples revealing that this argument is not true. For example, a grade 7 or 8 student
learning about the mathematical concept, infinity, is probably unaware (in her/his
consciousness) that the concept is embedded in another mathematics problem, continuity.
However, his unawareness does not make the problem, continuity, disappear.

Borrowing words from Merleau-Ponty (1962), "the world is always already out
there before reflection begins," I philosophically admit that problems are not determined by
our consciousness with a phenomenological stance. However, the focus of this study is
educational and psychological, how students psychologically tackle school mathematics
problems. Hence, it would be meaningless for me to study problems that students are
unaware of from a psychological and educational sense. In other words, this study is
focusing on a special kind of problem that humans are psychologically aware of, although I

a(il'nit that the existence of such problems is not determined by our consciousness.

Psychological Perspective

At least since the cognitive revolution (Gardner, 1985), a group of educational
psychologists and psychologists (Dunker, 1945; Hayes, 1989; Holyoak, 1995; Lester,
1983; Mayer, 1985; Mayer & Wittrock, 1996; Newell & Simon, 1972; Thorndike, 1950;
V038. 1989) have implicitly or explicitly suggested a similar psychological model for
problems. Psychologically, a problem comes to one's mind when one has a goal to
ac:hieve, but the current psychological state does not have a very clear route or a ready
solution for helping her/him to achieve the goal state. Although problems can generally be

defined with this psychological sense, there are different kinds of problems under this

18

 

 

Umbrella definition. I would like to specify the characteristics of the problems in this study

in the following paragraphs.

V' i i '

Tower of Hanoi (Neves, 1977) is one of the most frequently cited, classic
psychological experimental problems. Let us take a look at what are the psychological
characteristics of the problem, then I will describe the main difference between the
experimental problems and the problems in the study, or most of the word-problems in
school mathematics lessons. Psychologically, a problem appears in an individual's mind
when there is an obstacle between the initial state and its goal state. Psychologists try to
build a simple and easily defined experimental situation in which individuals can have such
psychological experiences and they easily detect the possible psychological activities in an
individual's mind. The Tower of Hanoi problem creates a kind of situation in which an
individual will come up with such an experience. In the experiment, individuals will be

shown the following situation (Figure 3.1)

 

 

 

 

 

Initial State

IIII A

F’ Goal State
'g‘n‘e 3.1 Tower of Hanoi Problem

and be given two rules: 1) they can only move one disk at a time, and 2) a large disk must

never be placed on a small disk. Individuals who did not work on the puzzle before do not

19

 

 

know immediately what series of actions must be performed to solve the puzzle.
Psychologists consider the individual experience a problem, psychologically.
Comparing the Tower of Hanoi puzzle with the problems given to students in this
study, I can say that the main obvious difference is that students in this study did not have a
clear image about the goal state. In other words, students have a problem situation (initial
state), the problem was written on the paper, but they did not have the final answer (goal
state) as a criterion for evaluating their solution, though they could have the expectation that
the answer should be a number. So, although the students in the study experienced a
psychological obstacle between the initial state and the goal state when they read the
problems, they did not have a sure answer to the problem (the goal state). This difference
influenced how individuals in the experiment and students in the study solved the
problems. The individuals in the experiment used the mean-end analysis to search for the
solution. The students in the study did not use that strategy because they were better
eCIUipped with a deep understanding of some of the relevant mathematical concepts
embedded in the problems. I want to clarify what "efficiency" means here. To me,
"efficiency" is not only related to speed, and how much time students need to solve a
PI'Oblem; it is related to the generalizability of the knowledge. If students have knowledge
that encompasses a wider generalizability, they can then apply their own theories to

Situations and they will be more efficient at solving the related problems.

W

In this section, I will discuss the internal representation, the problem space. Newell
and Simon (1972) argued that there should be a place in which individuals can
psychologically process problem-solving activities. They call the place, the problem space.
In the problem space, individuals encode the problem components—defining goals, rules,
and other aspects of the problem situation—in a space that represents the initial situation

p“merited to them. The situations include the desired goal situation, various intermediate

20

 

states (imagined or experienced), as well as any concepts they use to describe these
situations to themselves. In the Tower of Hanoi puzzle, individuals visualize the very clear
initial situation and the goal state of the problem. They also have two very clear
constraints—large disk can never be placed on the small disk, move one disk at a
time—which can be considered as the concepts they use to describe and understand the
situation. Individuals will probably come up with various imagined, intermediate states,
putting the 3 disks into to different pegs, while they are thinking through the problem and
before they come up with the solution.
If we use this idea to describe the possible problem spaces of students in the study,
We will find that there are differences in the content of the problem space between the
Stuclents in the study and the subjects in the Tower of Hanoi study. First, students in the
study cannot visualize the goal state, as mentioned in the last section. Second, students
bring different concepts to describe and understand the problem situation in different
PFOblems. I did not give any rules or constraints to the students. So, there were a lot of
individual differences among the students as to how they perceived and understood the
PFOblems. Third, these students had more complicated intermediate states than the states
generated by the subjects in the Tower of Hanoi. Some of the intermediate states may have
beBn more difficult for the researchers to detect because they were not as visual and were
more abstract. Fourth, students sometimes experienced difficulties in reporting their

pSYChological phenomena clearly to the researchers.

11‘ fn l N n- he Pro 1 ms
Some readers may ask why I use the term non-routine problem instead of ill-
cl‘efilred problem ifI consider this study to be a psychological study. Some psychologists
(Holyoak, 1995; Reitman, 1964; Simon, 1973; VanLehn, 1989) use the terms ill-defined
and well-defined problems to categorize problems. I think that the categories, ill-defined

and well-defined, were not well-suited for this study. The ill-well defined dimension is

21

better suited for focusing on an individual's psychological problem space. If one knows the
initial state, the goal state, and the constraints of the problem, then it is a well-defined
problem, and vice versa. For instance, the Tower of Hanoi is typical of a well-defined
problem. However, the problems in this study are concerned with students' past
experiences. All the problems, no matter how routine or non-routine, are ill-defined
because students do not know the goal state in advance and they often need to do some
checking to ensure the correctness of their answers. In the coming section, I will talk more

about the relationship between practice and problems within an educational context.

Educational Perspective

In the real world, there are a lot of unsolvable problems, but on the general
education level (primary and secondary school educations), most of the problems for
students are solvable. All problems in this study are solvable. Of course, "solvability" is
181 ated to the problem solver's competence, psychologically. In other words, while one
Pmblem is solvable for an individual, it may not be applicable to another person at the same
time. Or, a problem that was not solvable in the past is solvable now, for the same
individual. According to this relative and psychological sense, we know that the solvability
of a problem must be highly related to an individual's knowledge background and
Competence. Problems designed for this study were selected based on the prior learning
experience students had in school mathematics. So, I claim that all the problems were
s()Ianle to most ordinary students, although some were easily solved and some may have
been solved with more difficulty. My argument follows. There are reasons for problems to
be Solvable at the elementary and secondary educational levels. First, problems designed in
these educational settings should have a common goal of helping students develop a deeper
understanding of the important concepts in the subjects. A problem situation offers a
cljail‘nce for students to think and reflect their understanding about the concepts. In what

8 ‘ . . . . . . . .
ltLlatrons does a concept become applicable? What is its limitation? Or, Wthh concepts

22

have more generalizability power? All these questions may encourage students to think
about their learned concepts when they face a solvable problem. Unsolvable problems
cannot offer such a learning opportunity because students may not have the background
knowledge to think and reflect about the concepts they have learned. Second, the goals of
solvable problems offer the criteria for teachers to assess students' progress. If problems
are unsolvable, it is hard for teachers to set up the assessment criteria. If the result is
opened-ended and no one knows the resolution, then how can teachers know whether their
students are headed in the right direction. Third, idealistically, one of the goals at the
general education level is to offer students a chance to learn important ideas/theories/laws
that have been developed systematically by scholars past and current. And, we believe that
this knowledge is the basis for them to go further when they are ready. On the affective
level, we hope students can value and appreciate similar ideas/theories/laws through their
own problem-solving experiences.

In addition to using a dichotomy, solvable and unsolvable, we can use Polya's
classification (1981) of mathematics problems with a pedagogical perspective for
classifying the problems on a more continuous dimension, degree of difficulty. According

to Polya, mathematics problems can be classified into the following types:

1. One rule under your nose - the type of problem to be
solved by mechanical application of a rule that has just
been presented or discussed.

2. Application with some choice - a problem that can be
solved by application of a rule or procedure given
earlier in class so that the solver has to use some
judgment.

3. Choice of a combination - a problem that requires the
solver to combine two or more rules or examples given
in class.

4. Approaching research level - a problem that also
requires a novel combination of rules or examples but
that has many ramifications and requires a high degree
of independence and the use of plausible reasoning.

(Kilpatrick, 1985, pp. 4)

23

The term, " non-routine problems," used in this study corresponded to a hybrid of Polya's
Type 2 (application with some choice) and Polya's Type 3 (choice of combination). And,
the term, "routine problems" corresponded to a hybrid of Type 1 (one rule under your
nose) and Type 2. Polya used the "difficulty" dimension to locate different kinds of
mathematics problems in schools and the academic world. It has its own advantage,
helping us to see a clearer relationship between individual efforts and the complexity of
problems. More effort is needed when the complexity is increased. Polya seemed to
suggest that the complexity of the problem nature, more rules are needed, makes the
problem more difficult. To me, it is one of the sources for making a problem more difficult.
Another source is familiarity. If students have a lot of practice with some problem
situations, these experiences will help them evaluate the problem as an easy one (or a
routine one). So, although some problems may need students to apply several rules to
achieve the answer, they may find that they can easily solve the problems if their prior
knowledge and understanding about those problems are quite well-established. In other
words, the degree of difficulty has a combined effect on the problem structure (the

complexity of a problem) and one's personal experiences.

To summarize what non-routine problems mean in this study, I will argue that
problems are consciously detected by students in a philosophical and a psychological
sense. Psychologically, students in this study did not have a clear idea about the goal states
in different non-routine problems, and they had some very complicated problem spaces
while they were solving the non-routine problems. Educationally, all problems are
solvable, although with different degrees of difficulty. And, the degree of difficulty is
determined by the complexity of the problem structure and the individual's prior
experiences. For a problem to be qualified as a routine or non-routine problem is highly
related to a student's prior experience and practice. For a problem to become routine is

mostly attributed to practice. The process of routinalization is related to several

24

psychological constructs, problem schemes and problem recognition (Hinsley, Hayes, &
Simon, 1977). In the next section, understanding, I will talk more about, problem scheme,

in detail.

Understanding

The term, understanding, is a crucial but slippery word in many disciplines:
education, philosophy, psychology (Greeno, 1983; Greeno & Riley, 1987 ; Locke, 1706;
Mandler, 1998; Resnick & Omanson, 1987; Thagard, 1992; T oulmin, 1972). Each
discipline is concerned with its own interpretation. In education, educators are concerned
about how teachers help their students achieve understanding; in philosophy, philosophers
are concerned what understanding means; and in psychology, psychologists are concerned
about how humans achieve this psychological state. This study is more concerned with the
psychological state and the processes of understanding in an educational context,
especially, mathematics understanding in elementary schools. In this work, the terms,
"conceptualization" and "understanding," are used to refer to a similar psychological
processes. More psychologists like to use the term, conceptualization, and more educators
like to use the term, understanding. To me, they are using two different terms to describe
how children and students develop or learn to use a more abstract entity to think and reason
about their surroundings.

Recently, there have been more and more psychologists (Greeno & MSMTAP
Group, 1998; Thelen & Smith, 1994) and educators (Cobb, Yackel, & Wood, 1992;
Perkins, 1998) who have suggested that we need not have a representational mind to help
us towards understanding how we think. In other words, those psychologists suggested
that humans who conceptualize their outside world need not have an intermediate
psychological entity, a mental representation of the outside world. Situations themselves
offer enough power to build up conceptualization. However, I will only talk about the

representational understanding in this paper. There are several reasons for me to do this.

25

First, the way in which I collected this data is not suitable for doing a non-representational
analysis. In this study, I did not do any historical or cultural analysis. Second, I agree with
Mandler (1998) that most of the non-representational models (Greeno, et al., 1998; Thelen
& Smith, 1994) are based on perceptual theories and perceptual studies (Gibson, 1977;
Thelen & Smith, 1994). They are more suitable for interpreting perception studies or
categorization studies, than conceptual studies. Third, the non-representational theories
(Perkins, 1998) are still developing, and they do not offer a clear explanation as to how
students achieve understanding. For instance, Perkins' performance view on

understanding, Perkins (1994, 1998) offered,

"In summary, understanding is being able to carry out a
variety of 'performances' that show one 's understanding
of a topic and, at the same time, advance it. We call such

I"

performances, 'understanding performances.
(Educational Leadership, pp. 6, I994)

"..., understanding is the ability to think and act flexibly
with what one knows. " (Teaching for Understanding, pp.
40, 1998)

Those definitions are very similar to the vague definition of intelligence: IQ scores or
students' performances on the intelligence tests is intelligence. Philosophically or
psychologically, these kinds of definitions do not help us understand what understanding is
so I tried not to expend effort on these theories in this review section.

In the coming section, I will discuss two important concepts, conception and
perception, by introducing non-symbolic representational kinds of understanding,
perceptual understanding, and its relationship to conceptual understanding. Then, I will talk
about two hybrids of mental representations (a mix of analogical and symbolic
representations) that are used by many psychologists to describe understanding. They are
the mental models (Johnson-Laird, 1983; Greeno, 1983), and the schema (Hinsley, Hayes,
& Simon, 1977; Cheng & Holyoak, 1985). Finally, I will talk about another mental

26

representation related to understanding, the principle-based understanding (Gelman &

Gallistel, 1978; Resnick & Omanson, 1987).

Perceptual Understanding

From the history of philosophy and psychology we found that there is a great deal
of debate about the differences between "conception" and "perception." Beginning with
Pythagoreans, philosophers have always considered there to be a fundamental separation
between conceptual abstraction and perceptual experience (Goldstone & Barsalou, 1998).
Nowadays, psychologists (Chi, Feltovich, & Glaser, 1981; Murphy & Medin, 1985;
Medin, 1989; Murphy, & Spalding, 1995) have suggested a similar version of distinction
for the psychological theories of perception and conception. They have considered that
concepts or abstract concepts are not organized around clusters of perceptual properties, but
rather around organized systems of knowledge. All of these psychologists and other
developmental psychologists (Carey, 1985; Gelman & Markman, 1986; Gelman, 1988;
Keil, 1989; Mandler, 1998) tried to reveal that there are two different and separate kinds of
processing and mechanisms for understanding perception and conception. And, conception
is not based on perception. However, some psychologists (Goldstone & Barsalou, 1998;
Medin & Ortony, 1989; Prinz & Barsalou, 1997) argued that perceptual properties are often
good indicators of important, concept-defining properties and our perceptual systems have
evolved so as to establish useful concepts.

My theoretical stance is not as radical as the eliminative view on human cognition
(Barsalou, 1993; Barsalou & Prinz, 1997; Prinz & Barsalou, 1997). The eliminative view
of perception argued that perceptual representations constitute all knowledge, and human
knowledge contains no non-perceptual representation. I prefer to think that symbolic
representation, the non-perceptual representation (like the rule-based cognition), and
perceptual representation work hand-in-hand when we think. On the one hand, the object

features (related to perception) sometimes do influence and are helpful and efficient when

27

we conceptualize a situation and a problem because the perceptual and analogical
representations preserve aspects of the represented objects in a relatively direct way. For
example, it is always quite difficult for our elementary and junior high school students to
conceptualize and understand the concept, function, with the algorithms and the equations.
However, using a visual metaphor of selling machines or a pop machine to introduce the
concept, it will be easier for students to capture some ideas of the concept, function. I will
continue this argument with more concrete evidence in the discussion section when we
discuss why students are occupied by the perceptions of the area of square when they
confront an area and perimeter problem in the quiz. On the other hand, occasionally rule-
based cognition does offer us efficiency. Some concepts are difficult to replace with an
analogical representation (internal or external). For instance, students often use the speed
formula to solve speed problems when they do their school work and solve the problem in
an examination because the rule is efficient. Although we do not want to see our students
simply memorize rules without any understanding about why the rules work, using rules to
solve mathematical problems is easily observed in our classrooms. And, it has its part in
understanding, although we may not be satisfied with rule—roting based understanding

sometimes.

Mental Models

Many psychologists (Greeno, 1983; Johnson-Laird, 1983; Gentner & Gentner,
1983; Mayer, 1989; Norman, 1983) have investigated the properties of mental models. In
this section I use ideas developed by Johnson-Laird (1983), Mayer (1989), and Hegarty,
Mayer and Monk (1995) to talk about mental models and conceptual models, and
characterize their characteristics. Before we go into detail, I would like to clarify first, that
the mental models I will discuss in the following section are based on Johnson-Laird's
ideas, not the ones suggested by Norman (1983), and Glaser, Lesgold and Lajoie (1985).

The ideas suggested by Norman and Glaser, et al., are explained by how individuals

28

construct mental representations of physical devices, and how the mental representations
influence an individual's thinking. In contrast, the mental models I will discuss in the
following section are not solely limited to mental representations of physical devices.
Mental models can be analogical models and metaphorical models of abstract concepts. I
will give a very concrete example, a pop-machine, as a mental and analogical model to
mathematical function. A mental model helped me understand the concept, function, in the

Johnson-Laird's mental model section.

Johnson-Lg‘rd‘s Mental Mflels
Johnson-Laird (1983) offered the most detailed psychological and historical

information about mental models although his book, Mental Models, was concerned more
about human reasoning. Theoretically, Johnson-Laird told us why we need another mental
construct, mental models, to describe and explain our cognitive activities. The appearance
of mental models aimed at solving the theoretical issues of mental logic. This is a big debate
in psychology, whether we need to have mental logic in our psychological life. To lay out
the debate in this thesis is impossible and unnecessary, however, it is important to know
about the developmental history of mental models. It is especially important when we try to
study students' mathematical learning. Because, several important figures, Inhelder and
Piaget (1958), and Fischbein, Deri, Nello and Marino (1985), explicitly or implicitly
suggested there is a mental logic in our mind.

Johnson-Laird (1983) mentioned several characteristics of mental models, like

I) ”There are no complete mental models for any
empirical phenomena..., one does not necessarily
increase the usefulness of a model by adding
information to it beyond a certain level. "

2) "[Mental models ] have a content and form that fits
them to their purpose, whether it be to explain, to
predict, or to control. Their structure corresponds to the
perceived or conceived structure of the world, and is
accordingly more constrained than their contents. "

29

3) "Mental models contain some elements, A', that
correspond to physical elements, A, in the world. The
elements constitute an ontology of the physical
contents of the world. Certain abstract relations,
however, hold between these entities ultimately in
virtue of the contents of mental models. That is to say,
people act towards the entities in the world in
particular ways, and make judgments about them,
because of certain relations that exist only in their
models of the world. "

(Johnson-Laird, 1983, pp. 4, 410, 418)

Although Johnson-Laird mentioned a lot of other characteristics, like the
computational, economical, and representational features of mental models, I find the
characteristics shared above to be important in explaining how and why students may be
using mental models to help them understand their surroundings and their school work.
The elements of mental models are based on daily concrete experiences and correspond to
world elements. There are many cases mentioned by psychologists of how students use
those concrete daily experiences as models to comprehend and understand some scientific
concepts. The most famous one is the flowing water model for electricity flow (Gentner &
Gentner, 1983). There were not many empirical studies revealing how students use mental
models and analogies to understand some mathematical concepts although the term,
analogy, was originally a mathematical term (Pimm, 1981). One personal experience
impressed me. I remember learning the concept, function, in 8th grade. My teacher tried to
help us think of "function" as a straight line and its equation on the Cartesian plane. I
struggled for 2 years trying to capture the concept with this line model. Maybe the line
model was still abstract and quite a distance from me. In high school, one of my classmates
told me about another model for drinking about function. He told me functions were like
vending-machines dispensing soft-drinks. He told me that when you put a 2 dollar coin
into the machine and press a flavor button, then the machine will go through a series of
procedures, and eventually you will get a can of your favorite soft drink. So, a function is

just a set of procedures turning something, like a coin, into another thing, a can of soft-

drink. And, you will get different flavors of soft-drinks when you press different buttons,

30

although you put the same 2 dollar coin into the machine. The reason you get a different
kind of soft-drink is that the machine performs different functions when you press different
buttons. After that day, listening to his model explanation, I felt I really understood what a
function was. The line and equation models are mental models, too. However, for young
students, the line and equation model are not easily absorbed. The pop-machine model
really helped me to capture the abstract mathematics concept, function. Although the model
may not be true to any function situation, it helped me understand the concept at the high

school mathematics level.

M d'C a s'on once M 1

There are different kinds of mental or conceptual models suggested by educators
and psychologists that will help students learn better at school. First, consider the
conceptual models suggested by Mayer (1989). Although not all the psychologists consider
conceptual models as instructional models and teachable, I still use the conceptual models ~
suggested by Mayer, Dyck, and Cook (1984) as mental models because students mentally
carry these models and try to use them to understand new situations at the assessment
stage. Mayer and his colleagues (Mayer, 1983; Mayer, Dyck, & Cook, 1984) tried to
reveal how important good illustrations (pictures and diagrams) were for understanding.
Mayer, et al., (1984) suggested that we use a complete and concise illustration for helping
students capture and understand some scientific concepts, Ohm's Law, density, and
nitrogen cycle. Mayer, et al., (Mayer, 1983; Mayer, Dyck, & Cook, 1984) found that
concise and complete illustrations improved students' conceptual recall and problem-
solving transfer on several scientific concepts. We often assume that conceptual retention
and transfer are two criteria needed to assess students' understanding. Logically, we think
that retention should be longer when students learn a concept with a coherent model. And,
we consider that students who can transfer their knowledge to different situations likely

understand the concepts better than students who cannot. The conceptual models suggested

31

by Mayer (1989) are based on a psychological theory, structure-mapping (Gentner &
Gentner, 1983). Structure-mapping theory tries to explain how we make use of analogical
models to think and reason. The models convey the like-relational systems held within two
different domains. The structure-mapping analogies hold the identical operations and
relationships among non-identical things. For example, the pop-machine-function concept
analogy, or solar system-atom analogy are those analogical models with structure-mapping.
We started with a known base domain (pop-machine, solar system) to try to understand the
target domain (function, atom). Structure-mapping is one of the characteristics that holds
many mental models. That is, we solve problems with mental models by mapping the
elements and relationship in the problem onto the corresponding elements and relationship

in the model.

st He M M nk

Hegarty, Mayer and Monk (1995) suggested that successful mathematics word
problem solvers were inclined to build object-based mental models for the problems on
which they were working. They found that successful problem solvers might build
number-line models for helping themselves understand and conceptualize problem
situations. And, the unsuccessful problem solvers relied more on key word(s) searching
and applying relative algorithms. This further reveals the evidence that students are using
non-symbolic rules or principles to conceptualize a mathematical problem. There are two
reasons that make this study special. First, it revealed how students worked on a very
simple mathematics operation, arithmetic operation. Most of us think that students should
rely more on numerical algorithms than models. In fact, those successful students might
still prefer to construct object-based models when they confronted a simple mathematics
word problem. Second, all these subjects were college students who should be skillful
arithmetic problem solvers. Most of us logically think that skillful problem solvers should

solve the simple problem with totally principle-based or symbolic-based rules. However,

32

skillful problem solvers still relied on constructing object-based mental models when they
confronted a simple arithmetic problem. This revealed how powerful the object-based
model is to skillful problem solvers, and our intuition about skillful problem solvers who
might rely more on skill/procedural knowledge than conceptualization when they face
familiar problems.

The more interesting phenomenon for further investigation on the mental or
conceptual models is why these illustrations or object-based models are so useful and
powerful when students try to conceptualize the concept and problem situations which are
not so easy to conceptualize with a propositional or symbolic format. A lot of people
believe this phenomenon is true in a very common sense argument, a picture is worth a
thousand words. In fact, we know very little about why this common sense argument
appears to be true in some situations and not others. Larkin and Simon (1987) only
concluded that pictures or diagrams may offer a good index of information that can support
extremely useful and efficient computational processes. But, psychologists are still trying

to explore how this index function and its mechanism looks.

Schema

Schema is a very broad concept in psychology. From Bartlett (1932) and Piaget
(1952) to Anderson and Pearson (1984) and Rumelhart (1977), psychologists have been
interested in using this psychological construct to explain how we understand our
surroundings and descriptions in text. I see scheme as an experience-based psychological
entity that helps us understand objects and events in our surroundings. Scheme itself is a
complex psychological structure; it collects a lot of information and properties related to an
object or a concept. The scheme is developed through the abstraction of many instances and
experience. For example, Thompson (1994) identified a "quantity scheme," which is
composed of object/concept, the features of the object, and related concepts about the

object. In this section, I cannot conduct a complete review on this broad concept in detail,

33

instead I will only focus on two specific instances, problem schema (Hinsley, Hayes, &

Simon, 1977 ) and pragmatic reasoning schema (Cheng & Holyoak, 1985).

ble - ma

Many studies in the 70's and 80's (Bhaskar & Simon, 1977; Chi, Feltovich &
Glaser, 1981; Hinsley, Hayes, & Simon, 1977; Larkin, 1983; Paige & Simon, 1966;
Silver, 1981; Simon & Simon, 1978) explored the role of problem schema in students'
learning of mathematics and science. This construct has had an influence on how we think
about problem solving and understanding. Second, the concept, problem schema, has its
interpretative power on the data in this study, although you may not agree that these
students understood the problems and their relative concepts. Third, I think it offers a

H N

chance for us to rethink the relationship among "problem-solving, searching" and
"understanding." '

Consider the study done by Hinsley, et a1. (1977) to reveal what problem schema
is. The study (Hinsley, et al., 1977) was theoretically based on the text comprehension
studies done by Minsky (1975). In Minsky's view, schema approach is one in which
individuals rely on their knowledge of specific semantic context in comprehending
passages, and they make the main decisions on how to comprehend the passage very early
on and make relatively minor decisions later. In a series of experiments to test these
conjectures, they found that individuals did categorize problem types at the very beginning
stage, and in some cases categorized the problem type in the initial noun phrase. The
inference of problem type did lead individuals to formulate problem solutions. However,
such problem solving strategies only happened when individuals recognized the problem
type. Individuals went back to line-by-line comprehension strategy when they were unable
to categorize the problems.

This study revealed a number of important points. First, routine or familiar

problems are possibly solved by schema activation (recognizing the problem type at the

34

beginning, using the schema relevant knowledge to solve the problem). Second, different
kinds of problems lead to different kinds of problem solving strategies. Individuals cannot,
by definition, possess problem schema for unfamiliar or non-routine problems because
they have no relevant experience. Finally, familiar and routine problems cannot help us or
teachers assess whether students understand the problem situations and the relevant
conceptual knowledge we expect they need to develop. Students can only do the
categorization and apply relevant procedures when they solve a familiar problem. There is a
possibility that they do not conceptualize the problem situation and do not understand why

the relevant procedures are useful and meaningful in that problem situation.

Maugham

Do we develop our logical reasoning through our daily experiences? Or, do we
have some in-born mental logic helping us see the world logically? Many psychologists
(Evans, 1982; Griggs, 1983; Johnson-Laird, Legrenzi, & Legrenzi, 1972; Wason, 1983;
Wason & Shapiro, 1971) have been very interested in investigating these issues. They
want to know more about how individuals reason and how daily experiences influence their
reasoning. The pragmatic reasoning schema (Cheng & Holyoak, 1985) is related to logical
reasoning, especially the deductive reasoning. Why do I put this psychological construct
here? The first reason is that pragmatic schema studies based on Wason's study (1966)
highlighted the relationship between daily experiences and logical reasoning. Second,
mathematical reasoning is a kind of logical reasoning and does relate to students' daily
experiences. Students quite often use their daily experiences to learn the abstract and formal
mathematical principles in school.

The study done by Cheng, et al. (1985) was simple. The experimenter presented
four envelopes on the table: one envelope had a 20 cent stamp on it, one envelope had a 10
cent stamp on it, one envelope labeled "back of sealed envelope," and the last one labeled

"back on unsealed envelope." The subjects were told that s/he was a postal clerk working

35

in some foreign country. Part of her/his job was to go through letters to check the postage.
The country‘s postal regulations required that if a letter was sealed, it must carry a 20 cent
stamp. In order to verify that regulation was followed, which of the above four envelopes
would s/he turn over? Turn over only those that s/he needed to check to be sure. Subjects
were college students in Hong Kong and Michigan. Hong Kong has similar postage
regulations, so Hong Kong subjects outperformed the Michigan subjects on this item.
Nearly 90% of the Hong Kong subjects got the correct reasoning, and about 50% of the
Michigan subjects got the correct reasoning.

In another logical problem situation, subjects were presented four forms. One
carried the word "TRANSIT," one canied the word "ENTERING," one listed cholera,
typhoid, and hepatitis, the last one listed typhoid and hepatitis. Subjects were told that they
were immigration officers at the international airport in Manila, capital of the Philippines.
Among the documents s/he had to check was a sheet called Form H. One side of this form
indicated whether the passenger was entering the country or in transit, while the other side
of the form listed the names of tropical diseases. S/he had to make sure that if the form said
"ENTERING" on one side, then the other side included cholera among its list of diseases.
Which of the above forms would s/he have had to turn over to check? Indicate only those
that s/he needed to check to be sure. In this case, both subjects from Hong Kong and
Michigan were not familiar with the situation. Their correct responses were nearly the
same, about 50% of the subjects from both countries reasoned correctly. According to
Cheng, et a1. (1985), the subjects successfully solved the logic problem using pragmatic
reasoning schema, but not formal logic rules (If P then Q; If ~P then ~Q). The pragmatic
reasoning schema was based on subjects' daily experiences. The Hong Kong subjects
performed very differently in the two cases in which subjects could solve the problem with
the same formal logic rules. So, we can infer that subjects did not use the formal logic

rules.

36

I did not provide any detail relative to what formal logic reasoning is and how it is
learned because it is not highly related to this study. The pragmatic reasoning schema is
based on individual daily experiences, it may appear in different abstract thinking contexts.
In learning abstract mathematical concepts or principles, students may not acquire abstract
and formal principles, but they may develop schema based on their daily experiences in
several problem situations. For instance, students may develop a very intuitive function
schema for the concept, proportion. They may not acquire the cross-multiplicative principle
taught by their teachers, but they may develop a "turn P to ~P and turn Q to ~Q" schema
through their daily experiences with vending machines, turning coins and dollars into
tickets, opening-the-gates, food, and drinks. I will reveal such schema in the discussion

section in detail.

Principle-based Understanding

Another line of psychological studies (Gelman & Gallistel, 1978; Greeno, Riley &
Gelman, 1984; Resnick & Omanson, 1989) have argued that understanding is based on the
acquisition of underlying principles in mathematics. Beginning with counting (Gelman, et
al., 1978) to subtraction (Resnick, et al., 1989), these studies are based on the same
theoretical background. I will give a brief review on these two studies in the following
section. The reason I have chosen to have a section on principle-based understanding is that
this line of studies reveal a very important characteristic of mathematical thinking, no matter
how simple or procedural a mathematical task (counting, adding, or subtracting) is, it is

constrained by some abstract principles in the background.

Counting hinciples
Like many other cognitive studies, research on counting principles is related to a

historical debate about perception and conception. Gelman (1984) wanted to show that the

perceptual stance is false. Perceptual stance argued that young children who are about 3

37

years old initially develop a direct perceptual apprehension of direct quantities, at least for
small numbers. Just like other animals (for example, birds), young children cannot abstract
numerosity and have limited reasoning abilities, so they can only form a perception of
quantity. The most obvious evidence is that young children have difficulty judging a set
size with quantity more than 5. Gelman and other developmental psychologists
(Beckmann, 1924; Gleman and Tucker, 1975) suggested another theoretical stance.
Gelman thought that young children count to quantify a given small number of objects
before taking advantage of a subitizing or perceptual grouping strategy. Gelman and
Gallistel (1978) also found that children's counting performances were constrained by five
counting principles. These principles constrain how children count.

1. The one-one principle: The use of this principle involves ticking off items in an
array with distinct ticks (tags, numerons, numerlogs) in such a way that one and only one
tick is used for each item in the array.

2. The stable-order principle: The tags children use to correspond to items in an
array must be arranged or chosen in a stable and repeatable order. In other words, children
have a stable tag-pattem for assigning tags to the items.

3. The cardinal principle: This principle says that the final tag in the series has a
special significance. This tag, unlike any of the preceding tags, represents a property of the
set as a whole.

4. The abstraction principle: This principle states that the preceding principles can
be applied to any array or collection of entities.

5. The order-irrelevance principle: This principle states that the order of
enumeration is irrelevant; the order in which the items are tagged and hence which item
receives which tag, is irrelevant. In other words, the children know that it does not matter
how and in order they count the items.

Furthermore, there is a developmental relationship among these principles. Gelman

and Gallistel (1978) believe that children develop the one-one and stable-order principle

38

first. Then, they develop the cardinal principle. When we say children have the competence
to count, they should develop these principles in their mind, although the principles may be
implicit to them. According to Gelman and Gallistel's conclusion (1978), these principles
govern and constitute a scheme in that they both guide and motivate the development of

proficiency at counting.

S l . E . . 1
With the theoretical base built by Gelman and Gallistel (1978) on counting

principles, and the subtraction buggy algorithms found by Brown and VanLehn (1980,
1982), Resnick and Omanson (1989) tried to reveal how the number and quantity
principles constrain students' work on subtraction problems. The main focus of the study
was to test the effect of different kinds of instructional designs, mapping instruction and
prohibition instruction. They tried to verify whether the conceptual understanding and
learning of subtraction concepts can be explained by analogical mapping as suggested by
Anderson, Greeno, Kline and Neves (1981). I am more concerned about the principles
behind procedural and numerical manipulations in subtraction multi-digit. The subtraction
principles identified by Resnick and Omanson (1989) were:

1. Additive composition of quantities: This is the principle that states that all
quantities are compositions of other quantities. For example, 9 is not only a cardinal that
describes a set of given size, it is also a composition of 3 and 6, of 2 and 7, and so forth.

2. Convention of decimal place value notation: In the decimal system, each position
in a multi-digit number represents a successively higher power of ten. For example, digits
in the right-most column have a unit value—that is, the digit is multiplied by 1 to find its
value. Digits in the next column to the left have a tens value, digits in the next column after
that have a hundreds value, and so on indefinitely.

3. Calculation through partitioning: This principle means recomposing the problem

into a set of convenient sub-problems and cumulating partial results. For example, the

39

partition principle allows an individual to convert the problem 38 - 12 to perhaps a more
easily soluble problem, (30 -10) + (8 - 2).

4. Recomposition and conservation of the minuend quantity: It is easier to illustrate
this principle with an example:

832 - 267 = (800 - 200) + (30 - 60) + (2 - 7)

There are several within-columns yielding negative numbers. To avoid negative partial
results, the minuend can be recomposed to:

832-267=(700+130+2)-(200+60+7)

=(700+ 120+ 12)—(200+60+7)

= (700 - 200) + (120 - 60) +(12 - 7)

This recomposing is permitted by the additive composition principle, subject to the
important constraint that the total quantity in the minuend be conserved. The basic idea of
these principles is that they underlie and constrain the steps in students' procedures for
subtraction. Resnick, et al. (1989) thought that students will use the subtraction buggy
algorithms (Brown & Burton, 197 8), smaller-from-larger, borrow-from-zero, etc. .., when
they face challenging subtraction problems if students do not develop the subtraction
principles.

Unlike Gelman and Gallistel (1978), Resnick and Omanson (1989) did not offer a
clear picture of how students develop these principles. They only revealed that their
designed instruction sessions that emphasized these principles did not bring an impressive
learning effect for correcting the buggy algorithms. So, they argued that the conceptual
understanding was not attributed to the analogical mapping. They were more inclined to
think that the connection between block, writing, and quantities, seemed to be more
important for helping students develop understanding. They argued that understanding was
related to the semantic understanding of quantities more so than the syntactic rules of

subtraction. Students' errors in the post-test were related to the gap between semantic

40

understanding of subtraction in quantities and their automatic buggy algorithms in syntactic
subtraction.

Studies done by Gelman and Gallistel (1978) and Resnick and Omanson (1989)
were concerned with the mathematical competence of young children, but the principles
they found could not be applied to the data in this study. However, the studies highlighted
several important characteristics of children's conceptual thinking in the mathematics
domain. First, two studies revealed that no matter how simple and procedural a task is,
there are some abstract principles or structures constraining how children perform.
Although we may think counting and subtraction are simple and procedural manipulations
or tasks, children need to develop some principles before they can perform the
manipulation or task proficiently. Second, conceptual understanding may be easily

developed through "quantities" understanding than automatic "syntactic" procedures.

To summarize the understanding section, there is evidence that perceptions, object-
based mental models, and abstract principles (structures) have an impact on how children
conceptualize and understand their surroundings and mathematics problems. Although I
cannot offer a final definition of "understanding" based on the reviewed studies, I think the
results helped me to outline its characteristics. First, all of these mental entities are
representations. How I view representations is quite different from the commonly used
definition of representation. The most commonly used definition sees representation as a
symbol standing for another referent. I prefer to view representation as a complex concept,
the first sense of representation brought up by Mandler (1983). This conception involves
factual and experiential knowledge and how this knowledge is structured and organized. It
is also related and similar to a Piaget's term. Piaget referred to this kind of representation as
"conceptual representation" (Piaget, 1951). From the more sub-conscious level of mental
activity, perception, to the higher level mental activity, principle-based thinking, different

kinds of mental representations constrain how we think and conceptualize problem

41

situations. Using a Gestalt's figure-and—ground idea, students' perception of a square will
have an influential effect on how they solve the area and perimeter problem. It is possible
that a student perceives the borders of the square as the figure and the surface as the
ground. He may think that the sum of the length of the four borders is the area of a square.
And, Gelman and Gallistel (197 8) already showed us how the counting principles constrain
how young children count items proficiently. In other words, I suggest that children should
have two psychological assets, factual and experiential knowledge, and psychological
structures, to help them organize this knowledge when we think children conceptualize or
understand a situation or a problem. Second, nearly all these mental entities are experience-
based All of them are gradually built through students' daily or school experiences. But,
they are not a copy of all the actual experiences. They are newly constructed or abstracted
through a lot of experiences. Finally, J ohnson-Laird (1983) suggested a very important
characteristic for understanding; different individuals can achieve different levels of
understanding. For example, when we think that an individual understands computers
really well on the computer programmer level, we can assume that the individual
understands computers well if s/he can use one or two computer programming languages
skillfully. On the system expert level, not only can an individual program with one or two
programming languages, s/he should also know how to heck the kernel of the system and
build hardware drives for the system. I think it is the same when we think about students in
primary or secondary schools. We should expect students to achieve different levels of
understanding of mathematics according to their schooling level. In the school mathematics
context, we consider that students in elementary schools understand the concept,
proportion, with their own intuitive conceptual model—like an intuitive functional model

for proportion that we will explore a sample in the results chapter.

42

 

Multiplicative Reasoning

In order to talk about multiplicative reasoning, I would like to talk about an
important concept, quantity. In this study, quantity does not refer to the numbers given in
the problems alone, it is a measurable aspect of some object in a problem situation. It is
also a good criterion for teachers to use when checking to see whether their students are
trying to understand the problem situation. If students do not try to understand the problem
situation and instead use the key-word matching strategy to solve word problems they often
use a wrong quantity to solve the problem. In some situations, quantities also embed
mathematical concepts. The geometrical measurement units are the most common examples
in elementary school mathematics. The quantity for length and quantity for surface are
related to linear measurement and 2-dimensional measurement. Students can use an additive
model to conceptualize the linear measurement, but they need a multiplicative model to
conceptualize the 2-dimensional measurement.

Like Vergnaud (1994), I prefer to think that students conceptualize multiplication
with many mathematical concepts and experiences: multiplication, division, linear and
bilinear function, ratio, proportion, dimensional analysis, and linear mapping. So, I put
proportion, speed, and area problems under the overall umbrella idea in this section.

In addition to these related concepts, students' multiplicative reasoning has a
developmental relationship with addition. A number of studies and reports (Nesher, 1988;
Peled & Nesher, 1988; Resnick, 1989) found that children often first understand
multiplication with a kind of repeated addition model. For example, students will think
about 4 x 5 as 4 + 4 + 4 + 4 + 4 = 20. This additive model is helpful when the problems
can be read as a complicated additive problems. However, this model can also be an
obstacle for children learning more advanced mathematical concepts like, ratio, fraction,
and area. In the following paragraphs, I will briefly talk about recent studies about

students' work on proportion, speed, and area and perimeter.

43

mm

Proportional reasoning is a form of mathematical reasoning that involves a sense of
co-variation and multiple comparison. Many educators and psychologists (Karplus, Pulos
& Stage, 1983b; Lesh, Post, & Behr, 1988; Piaget & Inhelder, 1975; Schwartz & Moore
1998; Stigler, 1976) have tried to understand how students develop and learn to think
proportionally. There are two main reasons why educators and psychologists are so
concerned about students' proportional reasoning competence. First, proportional
reasoning is the "conceptual watershed" (Lesh, et al., 1988). That is, proportional
reasoning is both (1) one of the most elementary higher order understandings, and (2) one
of the highest level elementary understandings. The same point can be made from a
developmental perspective: proportional reasoning can be recognized as a capability that
ushers in a significant concept shift from concrete operational levels of thought to formal
operational levels of thought (Piaget & Beth, 1966). The second reason is related to the
essential character of proportional reasoning, abstracting similarities among different
systems—the relationship between two elements in each system that have the same
variation pattern. The competence to abstract similarities among different systems has its
important role in scientific discovery and creativity in different domains of knowledge
(V osniadou & Ortony, 1989).

Different models have proven how students reason proportionally in primary and
secondary schools. According to Piaget and Inhelder (1975), the essential characteristic of
proportional reasoning is that it must involve a relationships between two relationships.
That is, rather than a simple relationship between two concrete objects, a second-order
relationship is involved. And, Piagetians have also argued that an early phase in children's
proportional reasoning often involves "additive reasoning" (Lesh, Post, & Behr, 1988). In
other words, young children may see proportional relationships among the elements in a
form like, A - B = C - D, instead of A/B = C/D. Another model suggested by Karplus,

Pulos and Stage (1983a, 1983b) is that proportional reasoning must involve a linear

44

 

relationship between two variables. Their main argument was an objection to proportional
reasoning seen as a relationship between numbers. Instead, they argued that students must
think about the relationship between two variables when considering students' development
of proportional reasoning competence. In other words, students might think about the
functional relationship among the elements in a proportional problem. The thinking model
may look like, Y= fix) or Y = Mx, when students think about proportion. The last model is
the most commonly observed model used by students in school settings. It is the algorithm
for cross multiplication, N3 = x/D, x is the value for which students are looking for.
Some educators (Lesh, Post, & Behr, 1988; Hart, 1984) argued that this is not a real
proportional reasoning model if students do not understand why the algorithm works and
do not naturally develop the algorithm by themselves. They also warned that using the rote
cross multiplication algorithm without sense will impede students from thinking

proportionally.

Steed
Speed is a kind of proportional reasoning. Like other kinds of proportional

reasoning, it involves a sense of co-variation between units. However, the units in speed
have different qualities, distance measurement and time measurement. Children develop the
concept of speed through the experience of movement and motion (Thompson, 1994).
Piaget (1970) argued that the conception of speed is constructed as a proportional
correspondence between distance moved and time of the movement—the elaboration first
of concrete, and later of formal measurement operations. And, Thompson (1994) tried to
elaborate this idea in a more empirical format. He revealed how a student developed the
conception of speed, from a distance-centered conception (more concrete) to a distance-time
co-varying relationship conception (more formal). In her early conception of speed, the
student relied more on the distance quantity than time quantity to conceptualize the concept,

speed. She saw time implicitly in how many "speed-lengths" were required to make a

45

distance. So, her conception of speed was evaluated by partitioning the traveled distance by

the rate-length (Figure 3.2).

Distance to be travelled is 100 ft.
' How much time is need to
travel the distance, when speed is

 

 

 

 

fit- 30 ft per second.

1 sec
' 30 ft '

1 sec
} 1 Time exists only
30 ft 30 ft implicitly, as a

1 sec 1 sec number of

durations - one

p30 ft 30 ft 30 ft : duration to travel

lsec

1 sec lsec

one rate-length.

 

fio ft

lsec

3on 30ft i

lsec lsec partofa

second

Figure 3.2 Distance-centered Conception of Speed

After several teaching experimental sessions, she began to conceive the co-varying

accumulation of the quantities of distance and time (Figure 3.3).

46

. 100 ft. in 3.5 second, what is the
speed?

 

‘-

 

I Each second corresponds to
one rate-length.

 

 

 

 

 

Id I

7‘. ft T? —?a T's-‘ift
Figure 3.3 Distance-Time Conception of Speed

In the later stage, her conception of speed involved two dimensions, distance and
time, and they were coordinated. In this new model, a time quantity was not an attachment
to the distance quantity. Time and distance were separated, but had a co-varying
relationship. Increasing a time quantity would cause an increase in distance quantity. The
most obvious differences between the early and later stage of the conception are (1) the
number of truly independent quantities in the student's model, and (2) the co-varying
relationship between the quantities. In the early stage, the student only relied on one
dimension (distance) to conceptualize speed. In the later stage, she integrated the two

dimensions with the co-varying relationships to conceptualize the concept of speed.

WM

Area and perimeter are two additional important topics in elementary mathematics
that involves additive and multiplicative reasoning. These concepts are essential in various
measurement activities in everyday life. The concepts are also related to other mathematics
topics, such as multiplication, fraction, and algebra (Freudenthal, 1983; Schultz, 1991).

Although the concepts are used in daily life, students have difficulty conceptualizing the

47

 

concepts, especially the concept "area" (Bell, Hughes, & Rogers, 1975; Hirstein, Lamb, &
Osborne, 1978). There are several interpretations of students' difficulty in learning these
geometrical concepts. A number of researchers (Simon & Blume, 1994; Tierney, Boyd, &
Davis, 1990) attributed students' poor performance on the area problems to a tendency to
learn the area formula by rote, instead of developing a conceptual basis for the formula. For
instance, Carpenter, Lindquist, Brown, Kouba, Silver and Swafford (1988) reported the
results of a large-scale assessment study in the United States. About 50% of the Grade 7
students could calculate the area of a rectangle when they were given both dimensions; only
13% of the students applied their knowledge of the area formula to a square when they
were given the length of one side. In addition to the learning habit, students seemed to have
some conceptual difficulties in learning the concept. 2

Some researchers (Dickson, 1989; Hart & Sinkinson, 1988) found that the
instructional activities in which students experienced using concrete materials, like unit
tiles, to cover the rectangular figures, were not very effective for helping students
conceptualize the area concept. Mostly, students would find how many unit squares
covered the rectangular figures, but they were unable to state the area of the figures. These
results seem to suggest that students' difficulties might be related to failure to distinguish
between additive knowledge from multiplicative knowledge and read their relationships. To
summarize the two lines of studies, some researchers interpreted that the difficulties were
related to the transfer issue; students learned about area by using manipulatives with an
additive model, adding the unit squares. Those students may have experienced difficulties
in transferring this additive model to a multiplicative model to understand the concept, area.
Some other researchers thought that the conceptual obstacle was related to students'
learning habits, students tried to rote-learn the area formula instead of building a conceptual
base for the concept.

In addition to the line of studies focusing on instruction and learning that

mathematics educators and educational psychologists are interested in, another line of

48

studies done by developmental psychologists (Case, Marra, Bleiker, & Okamoto, 1996;
Piaget & Inhelder, 1956) suggested another psychological reason for explaining why
students have difficulties with area. According to Case (1998), by the age of 4, most
children who live in developed countries and grow up in middle class families have
developed a general schema for representing familiar 3-dimensional objects on a 2-
dimensional surface. However, most children need to reach age 10 to reference a whole
field of objects to two discrete mental reference axes that are orthogonal to each other. In
other words, they can really represent objects in 2—dimensional models. Maybe this
developmental evidence explains why students in elementary schools have so much
difficulty understanding the 2-dimensional concept, area. Children need to reference two
axes in a 2-dimensional model to really understand "area."

Although students' misconceptions about area are very common among students in
elementary schools, there are only a few studies investigating why students have
difficulties learning the concept. The above explanations are mostly cited. However,
students' difficulties are more diversified than what researchers would suggest. For
example, students may be confused by the concept, square root. Some subjects in this
study tried to divide the area by 2. And, one of the scorers in this study suggested that this
mistake can be explained. She thought students might confuse "square root" with dividing
by 2. This common mistake cannot be interpreted by the above suggested explanations.

And, I think that we should put more effort into this issue in the future.

Students from Low SES Families
Change is always taking place, but we sometimes neglect the changing. It is the
case in Hong Kong. Hong Kong educators do not think that social economic status (SES)
is an important factor influencing students' achievements in schools because they neglect
the changes in the population of our students. It is a very serious issue in education,

because the changing population of students is an important dimension in schooling

49

(Pallas, Natriello, & McDill, 1995). It is difficult to find any study related to students'
achievement and 8138 in Hong Kong. To me, it might be related to an historical reason.

Fifty years ago, most of the population in Hong Kong was comprised of the new
immigrants from Mainland China because the Chinese Communist Party took over the
Mainland at that time. Most of new immigrants were not able to bring their assets or money
to Hong Kong. So, most of them possessed lower SES status and corresponding resources
upon settling in Hong Kong. That is also the reason why there was relatively no social
class issues in the last generation; the SES structure was relatively "flat." However, there
have been a number of changes in the last 50 years. At least, a new social class was and is
emerging and growing. And, this new middle class also brings issues in education.
Students from different social classes may need different styles of teaching and services at
schools. For instance, more social workers are needed for the students from the low SES
families. And, different kinds of instructional methods given by teachers may have
different learning effects on students from different social classes.

Many studies and reports done in the United States (Duncan, Featherman &
Dumcan, 1972; Entwisle, Alexander, & Olson, 1997; Ginsburg, Klein, & Starkey, 1998;
Griffin, Case & Capodilupo, 1995; Natriello, McDill, & Pallas, 1990; Siegler, 1993;
Zucker, 1995) revealed that social class does have an effect on students' achievements in
mathematics in schools. Students from low SES families do not perform as well as
students from those of upper and middle class families. Although not many studies focus
on the SES issue at the elementary school level (Entwisle, Alexander, & Olson, 1997),
some studies revealed that the SES effect on students‘ academic performance was an
accumulative One that started in elementary schools or earlier (Broman, Nichols, &
Kennedy, 1975; Entwisle, Alexander, & Olson, 1997). Recently, more and more studies
(Entwisle & Hayduk, 1988; Ensminger & Slusarcick, 1992; Kerckhoff, 1993; Alexander,
Entwisle, & Dauber, 1994) have revealed that elementary grades (Grades 1 to 6) have a

great influence on children's future successes in and out of schools.

50

 

There is no doubt that social economic status is strongly correlated to children's
schooling in different ways, from achievement and understanding to social and affective
development. In the elementary school level, students' achievement is possibly related to a
number of variables. The mostly cited, one may be related to students' starting competence.
A number of studies (Huston, 1994; Saxe, Guberman & Gearhart, 1987; Siegler, 1993;
Smith, 1972) revealed that students from low SES families arrive at the first grade with
poorly developed verbal and mathematical skills compared to that of their counter-parts. In
addition to this frequently cited family-related reason, Entwistle and Alexander (1992;
1994), Heyns (1978; 1987), and Mumane (1975) found that students from poor families
gained little or no mathematics and reading competence over the summer. Entwistle,
Alexander and Olson (1997) argued that poverty limited cognitive growth mainly by
denying young children the resources they need to grow outside of school. Summer
vacations had such an effect on growth. Entwistle, et a1. (1997) also found that the
resource effect only mattered in the summer, not over the winter holiday or during school
days.

Students live in two worlds when they are attending school. One is with their
families, another is in their schools. Students' learning and development is highly related to
these two contexts. Some studies put more foci on the family side, and others try to explore
the school effect. Oakes (1990) found that teachers who taught in schools serving large
numbers of children from low SES families were more likely to focus on "computational
basics" in their mathematics lessons. Zucker (1995) found that many teachers in those
schools thought that their students appeared to have more deficiencies in their grasp of
basic mathematical skills so they needed to put their effort in building the missing
foundation. I had the same feeling when I talked to the head-teacher of mathematics at my
low SES school during data collecting. The teacher told me that his students were not

smart, and he needed to focus on enhancing the students' basic skills.

51

According to the previous studies, there were many interactive effects on
"computational skills without understanding" and "summer vacation." Logically, when
students learn a number of computational skills without understanding, they easily forget
what they learned after a period of time. Entwistle, et al. (1997) found that when students
from low SES families did not experience any growth in mathematical competence, it may
have been related to how the students learned mathematics in schools. "Summer vacation
effect" may be an auxiliary cause making the phenomenon easily observed, but the main
cause may be related to "learning without understanding," so students do not experience

retention of the knowledge they learned in their schools.

52

Chapter 4

METHOD

This study is a small-scale study. This study is not an experimental nor quasi-
experimental study. Subjects and sites in this study were not randomly selected or
assigned. The main goal of this study was to explore how Hong Kong students worked on
non—routine mathematics word problems, and whether students from different social
backgrounds had different kinds or levels of performance on routine and non-routine

problems.

Subjects

Of the 123 subjects who voluntarily participated the first phase of this study, 27 of
them voluntarily participated the second phase of the study as well. All of the subjects
participated in the study with the permission of their parents. All of them were grade 5
students studying in two different schools in Hong Kong, although their chronographical
ages were not the same. Most immigrant students were 1 to 3 years older than the local
students.

The two schools were selected because of the socioeconomic status (SES) of the
community. And the study were completed with the permission of the school principals.
One of the schools was located in a low SES community, the other was in a middle-class
community. In each school, there were two 5th grade classes that participated in the first
phase. One was a high-achiever class and the other was a mid-to-low-achiever class. In the
first phase, participants were asked to finish a 10 problem quiz within an hour in class. Of
the 123 students who participated in the first phase of the study, 27 students were selected
to participate in the second phase. Based on the school's achievement and social
background (different SES background and whether they are new immigrants), students

were selected to participate in the second section. Students were then categorized into 3

53

different groups, based on their performances on two recent tests given in mathematics.
Subjects who had average scores equal to or higher than 90 (full mark was 100) were
categorized as the high-achiever group; subjects who had average scores between 60 and
89 were categorized as the middle-achiever group; and subjects who had average scores
less than 60 were categorized as the low-achiever group. In the second phase, selection
criteria for the subjects were based on their social background, school achievement level,
and their performance on the first phase. Some subjects were selected because their
performance was quite impressive. Each participant was asked to sort some problem cards
and solve one to three non-routine word problems within a one hour interview.

Tables 4.1 and 4.2 shows the frequency of student achievement level by school for
the quiz and interview sections.

Table 4.1 Students Distribution in guiz Section

 

School / Class High-achiever Class Mid-to-low Achiever Class
Middle Class School 28 27
Low SES School 34 34

 

Table 4.2 Students Distribution in Interview Section

Social
Background I High Achievers Middle Achievers Low Achievers
Achievement

Hong Kong middle
class 3 2 4
Hong Kong low
SES

Immigrants from
China 3 4 3

Tools
Problem Development
All the problems used in the quiz and interview were developed from two sources:

1) the textbooks and the exercise books used by the students. All the routine problems and

54

 

the Balance problem were based on those books. 2) External resources included, "Ideal
Problem Solver (2nd ed. )," Bransford and Stein (1993); Connected Mathematics Project
(1998); "Problem Solving: A Handbook for Elementary School Teachers" Krulik and
Rudnick (1988); the TIMSS report; and "A Study on the Mathematics Problem-solving
Processes of Grade 5 and Grade 6 Students with Different Achievement Levels," Yang
(1994). For example, the idea of the Glass House problem is based on a Connected
Mathematics Project module, Covering and Surrounding, in the curriculum developed by
Lappan, Fey, Fitzgerald, Friel, and Phillips (1998). The idea for the Lake problem comes
from the book, Ideal Problem Solver, written by Bransford and Stein (1993). The reason
for using external resources is that most of the students had no prior experience with these

problems.

9mm

Two forms (FORM A and FORM B) of a 10 problem quiz were developed to
collect data. The routine problems in the two forms were parallel. However, the non-
routine problems were not. The reason for having different non-routine problems in the
two forms was to provide data on a variety of non-routine problems, which yields more
information about how students tried to solve the problems. In each form, there were 4
routine problems and 6 non-routine problems (all the problems in the two forms can be
found in Appendix A). The following table (Table 4.3) lists the order and names of

problems on the quiz.

55

 

Table 4.3 Problem Distribution on Two Forms

 

 

 

 

Form & Order Problem Name Expected Problem Type
Form A l Multiplication Routine
Form A 2 Unknown Digit Routine
Form A 3 Balance Non-routine
Form A 4 Glass House Non-routine
Form A 5 Coins Non-routine
Form A 6 Lake N on-routine
Form A 7 Photo-Robber (Ratio) N on-routine
Form A 8 Extended Square N on-routine
Form A 9 Saving Routine
Form A 10 Rectangle Width Routine
Form B 1 Multiplication Routine
Form B 2 Unknown Digit Routine
Form B 3 Number Pattern Non-routine
Form B 4 Lake Non-routine
Form B 5 Cakes Non-routine
Form B 6 Photo—You (Ratio) Non-routine
Form B 7 Balance Non-routine
Form B 8 Magazine Routine
Form B 9 Extended Square Non-routine
Form B 10 Rectangle Length Routine
lowest/Mon

Problem-Cards: There are 15 problems—4 routine problems and 11 non-routine
problems—printed on individual cards (all problems for the interview can be found in

Appendix B). These problems were used in a problem-sorting task.

56

Table 4.4 Problems for the Interview

Problem Name

Expected Problem Type

4 Squares Non—routine
Balance Non-routine
Buying Magazine Routine

Classroom Board N on-routine
Fraction Division Routine

Lake N on-routine
Moon Non-routine
Number-triangle Routine

Paper Recycle Non-routine
Poster Designer Non-routine
Rope & Rectangle Routine

Table-tennis N on-routine
Tiles Non-routine
Train Non-routine
Wallpaper Non-routine

 

Design and Procedures

Quiz Section

Students were given one hour to take the quiz individually in class. They were
asked to try their best to solve the problems on the quiz and told that they would have an
hour to finish the problems. The testing was done on 3 different days, one day in
December 1997 in the low SES school, and two days in January 1998 in the middle class
school. The days were arranged by the two principals in the two different schools. The
arrangement was made so that no major subject lessons (language and mathematics) in their
schools were disturbed. All the quizzes were given in the students' physical education or
art class time. In the briefing, I spoke with the students about why I came to their school
and what the study was investigating. Finally, I asked them to try their best to finish the

quiz, although it is not a test.

57

Interview Section

Each student worked for an hour with me. There were two activities in this section.
First, students were asked to read and sort the problem cards into three categories. Second,
they were asked to solve one to three problems individually. Because each interview was
restricted to one hour, the number of problem attempted differed. As students finished the
first, I gave them another problem if time permitted.

Problem Sort: Students were asked to read the 15 problems aloud, one by one, and
sort them into 3 categories depending on the level of their confidence in being able to solve
the problem. The categories were "Not Know," "Not Sure," and "Know." This activity
was used to assess whether students had difficulty reading the problem. It is possible that
reading could be the source of a student's difficulty in solving word problems. Also, the
activity was used to distinguish which problems were judged routine (the problems
categorized as "Known" were routine) and non-routine (the problems categorized as "Not
Sure" or "Not Known" were non-routine) by each student as routine. The instruction given
was,

"Now, I want you to read aloud all these problems. After
reading each problem, you need to categorize the problem
into one of the 3 categories. The first one is "Not Know":
After the reading, you think that they do not know how to
solve the problem. The second one is "N ot Sure": After the
reading, you are not very sure whether they can solve the
problem. The Third one is "Know": After the reading, you
are sure that they know how to solve the problem. Do you

have any question on what we need to do on those
problems?"

Problem Solving: Students were asked to solve one to three problems from the
following problem set, "Train" problem, "4 Square" problem, "Board" problem, and
"Wallpaper" problem (All the problem cards are shown in Appendix B). All the students
were given the "Train" problem first. If students performed well on the "Train" problem,"

they were then assigned the "Board" problem. If time allowed them to do one more

58

 

problem, students were asked to try the "Wallpaper" problem. Students were assigned the
"4 Square" problem, if they had difficulty solving the "Train" problem. Each student was
given the following instruction:

"Later, I will pick some problems for you to solve. These

problems are a little bit different with what you usually do. I

would like you tell me what you are thinking while you are

solving the problem. In other words, I would like you to

report everything that comes up to your mind when you

solve the problem. If I do not hear any report from you, I

will ask, "What are you thinking?" But, I want you to know

that I am not pushing you to finish the problem, I just want

to remind you to tell me what you are thinking. Do you have
any questions?"

There were two reasons for assigning the "Train" problem as the first trial problem.
It is a speed problem, one of the focus problem types in the study. Second, I hoped that by
using a story problem with a simple structure like the "Train" problem, students would be
able to solve at least one problem in the interview. The "Board" and "4 Square" problems
are area and perimeter problems. If the students had trouble solving the "Train" problem, I
chose a simpler problem, the "4—Square" problem on which for them to work. If the
students did not have trouble solving the "Train" problem, I chose the more complicated
problem, the "Board" problem, for them to solve. I was more interested in collecting data
on students' conceptualization of the concepts, area and perimeter, rather than text
comprehension and unit transformation. Although all of these data (text comprehension and
unit transformation) are important, I needed to make a priority decision in completing this
study.

I picked a problem card from the stack and students were asked if they wanted to
re-read the problem to refresh their memory. After the re-reading, students were asked
which element(s) they found important in solving the problem. By asking students to report
the important elements in the problems, I expected to gain a general sense of their

competence in comprehending the text, and how they conceptualized a problem.

59

 

Scoring

Quiz Section

Two different rounds were used in scoring the quizes. The graders for the first
round of scoring were operating under a time constraint and did not finish scoring all the
problems. The outcome of this scoring trial did not result in a good inter-rater reliability. I
will explain this observation in the following section. As a consequence, a second round of
scoring was arranged and the results of the second round of scoring became the focus of
this study. Although there were two different trials for scoring, the scoring schemes used
were the same in each trial. The only differences were the instructions for the scorers and
the time spent in scoring. In the second round, I intentionally reminded the scorer of the
possibility of more than one conceptual or computational error being embedded in the
students' work, and that answers were to be categorized into three groups (correct,
numerically correct, or incorrect), with numerically correct not counting as correct. In the
first trial, scorers had one night, about 8 hours for scoring. In the second trial, the scorers
had a week for scoring the students' work. There were 3 scorers, including myself, in the
first trial. The three scorers were graduate students majoring in educational psychology and
who had completed their elementary and secondary schooling in Hong Kong. All of them
were fluent in reading Chinese. There were 2 scorers, including myself, in the second trial.
Both of them were graduate students majoring in educational psychology. The new scorer

in the trial was a graduate student from Taiwan who was fluent in reading Chinese.

Scoring Scheme

The scoring scheme was developed using several scoring pilots and an examination
of the students' work on the quiz and in interviews. First, I decided which problems
should be scored for conceptual understanding. In this part of the scoring, I asked scorers
to identify what sorts of conceptual errors the students made. Based on the written work

given by the students and the importance of the embedded mathematical concepts (ratio,

60

area and perimeter) in the upper grades of primary school, I decided which problems
should be the main focus. Although all the problems were related to multiplicative
reasoning, based on their written work, students seemed to be most confused by ratio, and
area and perimeter. "Glass House" and "Extended Square" both focused on the
mathematical concepts of area and perimeter. And, work done by students revealed their
difficulty in solving these problems. However, there was more written work offered by the
students on the "Extended Square" problem, thus giving me more information with which
to investigate their conceptual understanding. So, the "Extended Square" problem was
chosen as the area problem for conceptual scoring. In the two forms (Form A and B), a
conceptual analysis was done on 5 problems—a pair of "Ratio" problems, a pair of
"Extended Square" problems, and the "Cake" problem. For the other problems, scorers
assessed whether the students' answers were—correct, numerically correct, incorrect, no
answer, or no attempt. Next, I tried to categorize the error types. I found many mistakes
related to students' neglect of the quantity given in problem situations. The issue of unit
confusion seems to be common across all problems. Another frequently used code for
conceptual error was the "Other" category. Errors made by students that I could not
understand were categorized as "Other." All the other error types were limited to the
specific problems. For example, in the ratio problem, a conceptual error is the
rrrisconceptualization of proportion as an additive relationship between entities. In the
"Extended Square" problem, there were three other types of conceptual errors. One was
related to the confusion between area and length. Students thought that they could get
length when they divided the area by 4. The second type was related to confusion about the
relationship between area and perimeter. Students gave the same number for both area and
perimeter. The last type was related to adding a small square to the original square. (A
detailed example can be found in the Appendix A)

Scorers were given procedures to use in assessing students' work. Figure 4.1

shows the flowchart for assessing students' work. In the half-hour scorers' training

61

session, I read through the flowchart and explained every step of the assessment to the
scorers with an example. The scoring sheet (Appendix C) offered the option for scorers to
check each step. Scorers made an evaluation decision at each step, and then circled his/her
decision on the scoring sheet. For instance, in the first step, scorers decided whether the
student attempted the problem based on the student's written work on the quiz. The first
column of the scoring sheet was the variable, Attempt. If a scorer decided that the student
did not attempt the problem, s/he circled the sign, NA. If a scorer determined that the
student did try to solve the problem, then s/he went to the 2nd step and examined whether
the student provided procedures that led to the given answer. The scoring format used was
a problem—by-problem format. In other words, scorers assessed the same problem done by
all students, then moved on to the next problem.

For the problems that required conceptual analysis, scorers followed the scoring
key (Appendix A) for making the scoring decision. Before scorers proceeded to the

scoring, I did an introduction to the scoring key on each conceptual analysis problem.

62

 
 

D . s the subject
write anything
bout the proble

 

yes

 
    
 

 

Check whether the
procedures led to
the answer.

 

Check there is evidence
they misread the
problem

   

 
  

 

Score: NA

NA for no attempt

 

 

 

 

 

 

 

Check there is a
given answer.

 
  

.LScore: Y or N

   

 

 

Score: MR

 

  

Y for with written procedures
N for no written procedures

(Procedures have direct relationship
with the answer)

MR for mis-reading

 

 

 

 
 

 

 

Check the
correctness of the
given answer

Check how many
conceptual
mistake(s).

 

 

Score: Y or N

  
 

Y for having answer
N for no answer

 

 

I
J

 

 

b Score:CorI

   

C for correct
I for incorrect
NC for numerical correct

 

  

 

 
 

 

 

 

 

check the error type
on problems: ratio
and extended square

Score: 0, 1, 2

    
 

0 for no mistake
I for one mistake
2 for more than 1 mistake

 

 

 

 

 

Check how many
computational
mistake(s).

 

  

   

 
 

A for "Type A" error
B for "Type B" error
C for "Type C" error
D for "Type D" error
E for "Type E" error

 

 

 

Figure 4.1 Scoring Procedure

>1 Score: 0, 1, 2

63

0 for no mistake
I for one mistake
2 for more than 1 mistake

 

Inter-rater Reliability

While some parts of the quiz data (for example, attempt or not) were not relatively
trivial to score, other parts were. To ensure the objectivity of the scoring, other scorers
were trained to use the scoring scheme to score the students' performances on the quiz in
two rounds of scoring.

Kappa coefficient and Spearrnan correlation were used to measure the inter-rater
reliability. One reason for using these non—parametric statistics, Kappa coefficient and
Spearman correlation, was that the data I collected may not match with the population
distribution (Siegel and Castellan, 1988). In fact, I do not know the exact nature of
population distribution. In addition, all of the scoring categories were ordinal and
categorical, and non-parametric tests are more suitable for analyzing these kinds of
measurement. Because the error type is a categorical variable, the Kappa coefficient was
used for assessing the reliability among raters on scores for the conceptual mistakes. For
example, there were 3 different types of conceptual mistakes, 1) unit confusion; 2) additive
relationship between variables; 3) others, to categorize students' conceptual mistakes in the
"Ratio" problem (the Photo-Robber and the Photo-You problem). Because the correctness
is an ordinal variable, the Speannan correlation was used for assessing the reliability
among raters on scores of the correctness. For the correctness variable, students'
performances were categorized into 3 ordinal categories, 1) Correct; 2) Numerical Correct;

3) Incorrect.

First Round Inter-rater Reliability

In the first round, 3 raters, including myself, scored the papers. Scorers did not
discuss their scoring after each session. I underestimated how much time was needed for
scorers who were new to the students' work. The two additional scorers only evaluated 3

problems on the quiz in one night. Under the original plan, I estimated one night would be

64

enough for scoring all the of problems. The time arrangement pressed scorers and they
made more mistakes when they scored the third problem, the "Cake" problem. As a result
of these issues, the inter-rater reliability coefficients were not as robust. Also, I did not
emphasize the possibility that students might make double or triple errors in a problem. The
result was that the two new scorers easily neglected the second and third errors that some
students made. Based on the unsatisfactory results of this trial, I decide to have another go

at the scoring.

Table 4.6 First Round Inter-rater
Reliability in Form B (N = 65)

Table 4.5 First Round Inter-rater
Reliability in Form A (N = 58)

 

 

 

 

Problem Problem Conceptual Problem Problem Conceptual
Mistake Mistake
Name Type (k=inter-rater Name Type (k=inter-rater
reliability) reliability)
Photo- N0": Type A Error: k = .75* Cake (A) N0": Type A Error: k = .02*
Robb“ ROW“ Type B Error: k = .73* Romm" Type B Error: 1: = .64*
Type C Error: k = .55,“
Extended N°“'_ Type A Error: k = .46* Cake (B) None Type A Error: k = .05*
(Sg)uare Routine Type B Error: k = .74: Routine Type B Error: k = .60*
Type C Error: k = .53
Type D Error: k = .71”:
Type E Error: k = .77*
Extended Non: Type A Error: k = .58* Photo- Non-I Type A Error: k = .76*
:8?“ Row“ Type B Error: k = .75: Yo“ Romme Type B Error: 1: = .77:

Type C Error: k = .57
Type D Error: k = .69,“
Type E Error: k = .70"‘

Type C Error: k = .75

* Key: k = Kappa Coefficient

Extended Non: Type A Error: k = 155*
(Siffre Rout” Type B Error: k = .76:
Type C Error: k = .62
Type D Error: k = .73*
Type E Error: k = .68.“
Extended Non: Type A Error: k = .76*
(Sgrare Routine Type B Error: k = .77:
Type C Error: k = .75
Type D Error: k = .82*

Type E Error: k = .74*

Second Round Inter-rater Reliability

In the second trial, I had two raters for scoring, including myself. Two raters
discussed their differences on scoring after they finished the scoring on all of the students'
performances. The scorers took sufficient time to score the papers—one week. The scorers
scored 2 problems per day. Also, the additional scorer was reminded of the possibility that
students might have double or triple errors in their work. There were 45 instances of
disagreement between the two scorers, but the scorers reached an agreement on all the
scores after discussion—except for two scores that will be described and reported in the

coming paragraphs.

66

Table 4.7 Second Round Inter-rater Reliability in Form A (N = 58)

 

 

 

 

Problem Answer Correctness Conceptual Mistake Computing Error
NamteElLLCMABQDEQlZ.
MUlthl i ' rs=l rs=l
cation
[R]
Unknown rs=l rs=l rs=l
Digit
[R]
Balance .95 .95 .95 .95 .95
[N] <rs< <rs< <rs< <rs< <rs<
.99 .99 .99 .99 .99
Glass rs=l rs=1 rs=l
House (A)
[N]
61355 rs=l rs=1 {5:1 rs=l rs=l
House (B)
[M
Coins rS=l rs=l rs=l
[N]
Lake rs=l rs=l rs=l rs=1 rs=l
[N]
PhOtO- {5:1 [5:1 {5:1 rS=l rs=l k=l k=l k=l rs=1 rs=l rs=l
Robber
[N]
EXW rs=l rs=1 rs=l rs=l rs=1 k=l k=l k=l k=l k=l rs=1 [5:1 T's-=1
Square (B)
[N]
EXW 1' 5:1 [8:1 rs=l rs=l k=l k=l k=l k=l k=1 r5=l rs=l TS=1
Square (C)
[N]
Saving rs=l rs=l rs=1 rs=l
[R]
Rectangle rs=l {5:1 [8:1 rs=l
Width
[R]
Keys: Q: Correct k: Kappa coefficient
K: Numerically Correct rs: Spearman correlation
I: Incorrect [R]: Routine Problem
HQ: Unclassified [N]: Non-routine Problem
11A: Unattempted

67

Table 4.8 Second Round Inter-rater Reliability in Form B (N = 65)

 

 

 

 

Problem Answer Correctness Conceptual Mistake Computing Error
NameQMILLCIi/AAEQDEQJ.
Multipli- rs=1 rs=l
cation
[R]
Unknown rs=l rs=l rS=l
Digit
[R]
Numw {5:1 rszl rs=1
Pattern (B)
[N]
Number rs=l rs=l rs=l
Pattern (C)
[N]
NUHIbCI' 1'5: 1 T5: 1 T521
Pattern (D)
[N]
Lake rs=l rs: rs=l rs=l
[N]
Cake (A) rs=1 rs=l rs=l =1 k=l rs=l rS=l rs=l
[N]
Cake (B) T's-'31 1.5:] rs: kzl k=l rs=l rs=l T's-:1
[N]
Photo-You rs=l rs=l r5=1 rs=l rS=l k=l k=l k=l rs=1 rs=l rs=l
[N]
Balance T's-=1 rs: rs=l {5:1 r3=l
[N]
Maguinc X's-=1 rszl rszl rszl {5:1
[R]
EXW rs=l rs=l rszl rs=l rs=l k=l k=1 k=l k=l k=.97 rs=.98 rS=.98 rs=.98
Square
(B)[N]
Em r521 rs=1 rszl rs=l k=l k=l =1 k=l k=.97 rs=.98 rs=.98 rs=.98
Square (C)
[N]
Rectangle rs=l rs=l rs=l
Width
1R]
Keys: Q: Correct k: Kappa coefficient
11g: Numerically Correct rs: Spearman correlation
1: Incorrect [R]: Routine Problem
UQ: Unclassified [N]: Non-routine Problem
LIA: Unattempted

Observations with Disagreement
Although there were discussions between the two scorers after scoring in the
second scoring trial, there were still two unresolved scores. The two instances appeared on

different problems in two different forms. One was the "Balance" problem in Form A and

68

another was the "Extended Square" problem in Form B. These two results made the
Spearman correlation and Kappa coefficient lower than the other scores. In the following
paragraphs, I will discuss the disagreements between the two scorers on these two

problems.

Insmce 1

The "Balance" problem was one in which the two scorers did not agree. One of the
scorers considered the answers, 9 and 5 written on the balance stand as incorrect. The other

scorer judged that the student tried the problem, but no answer was given.

Quiz Form A

r armrraerr enrprarerarrr?
iidu"§§?z““

 

Figure 4.2 A Student's Work on the Balance Problem

69

Instance
On the "Extended Square" problem, the scorers also disagreed. The students did

not provide the complete procedures showing how s/he figured out that the new area was
169 and the new perimeter was 52. One scorer maintained that the student committed a
Type E error. The Type E error was the "Others" type that the scoring scheme did not
classify. Another scorer preferred to judge that the student made a computational error; the
student tried to find the square root of 121 cm2 and added this length to 3 em, but the

student made a computing error in adding two number and got the sum, 13.

Quiz Form B

9' 5“}.73'913 ’ fifiilflflifiAg} , if: if. ~ .. .
”51‘15fié9fiflfi o A HE fififi-‘i’ fiiéfifd’kfi‘

m tea—makeup? ~4~4~lmwe¢.-
m arrearzr? ’ 1%.
1 j; .1 V ”1 g ' 77
i M 3 ~
c>rarari;r? 15

’1

Figure 4.3 A Student's Work on the Extended Square Problem

I‘VXU .;.\" 7.,

70

Chapter 5

RESULT

In this chapter, results from two kinds of data are reported: 1) an analysis of all 123
student performances on the paper-and-pencil quiz, and 2) a summary of the performances
of 27 students from two different schools who worked on one to three word problems in
individual interviews. The main foci of the quiz analysis were the answers' correctness, the
conceptual mistakes made by students, and their computing errors. In the second part, the

main focus was the conceptual analysis of their performance.

Quiz Result
The quiz scoring scheme was designed to assess 3 different dimensions of
students' performances. The first dimension was the correctness of the answer. Raters
judged the answers given by the students; they did not try to assess their thinking
processes. In other words, we classified students' performances into 5 different categories,

1) correct: the answer is correct,

2) numerically correct: the number is correct, but a wrong or no unit was given. For
example, the correct answer should be 196 square centimeters, but a student
wrote 196 cm or 196 without any unit as her/his answer. As a result, the answer
will be judged as numerically correct,

3) incorrect: the answer is numerically incorrect,

4) unclassified: students tried to work on the problem, but no answer was given;

5) unattempted: there is no evidence that the students tried the problem was given.

The second dimension was a scheme to assess possible conceptual mistakes. From

the students' performances on the problems, several conceptual error categories were
developed. All the conceptual error categories were based on common mistakes made by

students in which the error(s) could be explained and were related to one or more

71

mathematical concepts. All uncommon errors and unexplainable errors were categorized as

"Other." There were 3 types of errors —"unit error, additive error," and "other"—in the

"Ratio-Photo" problem. There were 5 types of errors—"unit error," "area & length

confusion, area & perimeter confusion, adding a new area," and "other"—in the
"Extended Square" problem. And, there were 2 types of errors—"unit error" and
"other"—in the "Cake" problem.

The third dimension assessed students' computational errors. All computational
mistakes in using the wrong numbers for computation were categorized as an error in
computation. Scorers also counted how many computational mistake(s) were made by
students in each problem after they assessed the students' performances. So, there were
three numerical codes for computational mistakes in this category, 1) no mistake, 2) one
computational mistake, 3) more than one computational mistake.

To summarize the results in the second round of scoring, I will present the general
result first. In both schools, most of the students performed quite well on the routine
problems—more than 65% of the students got the correct or numerically correct answer on
those problems. In Form A, students generally performed quite well on the 3 routine
problems—"Multiplication," "Unknown Digit," and "Rectangle Width"—more than 74%
of the students gave correct responses. Only the "Saving" problem was an outlier, only
43% of the students completed this problem with the correct answer. Many did not give the
unit in their answers. In Form B, students performed quite well on the 3 routine
problems—"Multiplication," "Unknown Digit," and " Rectangle Width"—more than 69%
of the students were able to finish the problems with correct responses. The students did
not perform as well on the routine problem, the "Magazine". Only 55% of the students
finished the problem with correct responses, and 12% of those students gave a wrong unit
or missed the unit in their answer. The students' performances on the two forms were quite

consistent. Their correctness on the routine problems were consistent on the four parallel

problems. "Saving" and "Magazine" were two parallel problems for which students needed

72

similar mathematical competence—understanding the text and relationship (addition,
subtraction, multiplication, and division) between the quantities in the problems. Two
problems, "Saving" and "Magazine," had similar story structures, but different story
contents. And, students seemed to have more difficulty solving these problems in contrast
to the other 3 routine problems.

On the non-routine problems, students' performances were not as good as those on
the routine problems. Except in the "Cake" problem, more than 63% of the students solved
this non-routine problems correctly. There was one interesting phenomenon on the two
parallel "Ratio" problems; students ,who worked on Form B out-performed students who
worked on Form A. The percent of correctness was 44.6% and 19% respectively.

There were several general characteristics of students' performances on the quiz.
First, the simpler the language and the fewer the computations required in the problem, the
higher the correctness of performance was on the routine problems. For example,
"Multiplication," "Rectangle Length," and the "Rectangle Width" were the problems with
simplest language content and fewest computational requirements. Their percentages of
correctness were the highest, with more than 80% of the students getting the correct or
numerically correct answers. Second, except for the "Cake" and "Number Pattern"
problems, students did not perform as well on non—routine problems. Less than 45 percent
of the students finished the non-routine problems with the correct answers. Third, the non-
routine problems, the "Cake" and "Glass House" problems, were problems with similar
story structure and with complicated stories that needed to be comprehended. The problems
were unfamiliar to students. However, the students' performances on these two problems
were quite different. Less than 4% of the students were able to finish the "Glass House"
problem, while more than 63% of the students could solve the "Cake" problem. Fourth,
among the non-routine problems, students' performances on the two "Ratio" problems in
the two different Forms (A/B) were quite surprising. The two problems were very similar

in structure and language content. The only difference was the story situation. Less than

73

7:7

 

 

20% of the students who worked on Form A could solve the problem. However, more
than 44% of the students who worked on Form B could solve the problem. Finally, the
conceptual analysis among the three non-routine problems ("Cake," "Ratio," and
"Extended Square") revealed that unit confusion was a major difficulty for students.

Comparing the two schools from two communities, students' performances in the
low SES school were weaker than that of the students' performances in the middle class
school. Except in the cases of the "Multiplication," "Cake" (Part A), and "Balance"
problems, students from the middle class school out-performed the students from the low
SES school. On 14 out of 24 problem-items, there was more than a 10% difference
between the two schools, favoring the middle class school. The problems that magnified
the differences (more than 20% on both forms) between the two groups were the "Ratio"
and the "Extended Square" problems. The conceptual analysis of these problems revealed
that students in the low SES community experienced more difficulty in understanding the
context of the problems, and experienced greater confusion with the units given in these
problems. More than 25% of the students in the low SES schools confused the units on the
"Ratio" (Photo-Robber and Photo-You) and "Extended Square" problems. In addition to
their achievement level, students in the low SES school were quite different than their
counterparts in their level of motivation. Students in the low SES school seemed to give up
more easily on difficult problems. For example, more than 75% of the students did not
attempt the "Glass House, part b" problem; and more than 25% of the students did not
attempt the "Extended Square, part c" problem. Only 52% of the students from the middle
class school did not attempt the "Glass House, part b" problem, and 3% of them did not
attempted the "Extended Square, part c" problem.

Over the different achievement levels, low achievers performed poorly, in general,
but quite well on the "Multiplication" and "Cake" problems. Their performances on the
"Cake" problem were surprising as most of them did not have conceptual difficulty with the

problem. The differences between the high achievers and the mid achievers were generally

74

not very great. The only exception was the "Coins" problem, for which more than 55% of
the high achievers solved the problem. Only about 5% of the middle achievers gave correct
solutions. On a conceptual level, low achievers had more difficulty on the "Extended
Square" problem. Many of their errors were classified as "Other" type, which means I
could not interpret their errors. Between 30 to 53% of the student responses reflected an
"Other" type error on the "Extended Square" problem in both forms. However, less than
16% of the other two groups (high and middle achievers) showed this type of error. Low
achievers showed less motivation and seemed to give up easily on difficult problems. For
instance, more than 92% of the low achievers did not attempted the "Glass House"
problem, compared to other groups whose highest unattempted percent was 63% .

In the following tables, students' performances on both forms of the quiz will be
revealed. The first set of tables present the general performance of the students from both
schools. The second set of tables present the performance data based on students from
different schools. These tables let us examine the differences in performance between the
two groups. The last set of tables present the performance data based on students'
achievement levels. These tables allow us to examine whether students with different

achievement levels performed differently on two different kinds of problems.

Table Description

These tables have a unique format; they reveal the students' performances on three
dimensions—correctness, conceptual, and computational. Under the correctness
dimension, students' performances were categorized into 5 types:

1) "C" for a correct answer was given,

2) "NC" for a numerically correct answer was given,

3) "I" for an incorrect answer was given,

75

4) "UC" for an unclassified trial was given, which means that students tried to
solve the problem, but did not given an answer, based on what they wrote on
the quiz sheet,

5) "UA" for unattempt, which means no written information was provided on the
quiz sheet.

Under the computational dimension, students' performances were categorized into

3 types: 1) 0 for no computational mistake, 2) l for one computational mistake, 3) 2 for
more than one computational mistake.

The conceptual dimension was more complicated. Three different problems had
their own scoring schemes for conceptual mistakes. The following paragraphs briefly
describe the mistake categories on each problem:

Cake Problem

W is a unit confusion mistake. For example, a student wrote, $963 -

11 X $45 = 468 pieces. This algorithm can only offer the dollar unit, not pieces.

Because, 11 cake multiplied by 45 dollars equals the total price of 11 cakes.

W is the "Other" type of mistake. This means that I could not

understand or interpret the written work given by the students. For example, a

student wrote the following algorithms and answer, 12 x 6 = 72 => 62 x 6 = 372

=> 372 - 72 = $300. This students seemed to put all the given numbers in the

problem and manipulated them with arithmetic operations. And, all the arithmetic
operations did not make sense within the context of the problem.
Ratio Problem (the Photo-Robber and the Photo-You problem)

W is a unit confusion mistake. For example, a student wrote 10 cm x

4 cm = 40 cm (it should be 40 cm2). Although there is a multiplicative relationship

between the variables in the problem, the relationship should not consist of picking

up 2 numbers and multiplying them together. And, some students wrote, 1.4 m x 4

cm = 5.6 cm.

76

1ny B Mistake is an additive model mistake. For example, a student wrote 1.4 + 6
= 2 m (6 is equal to 10 cm - 4 cm). the student thought that ratio was an additive
relationship between the variables. In addition to the additive model mistake, this
student violated another conceptual mistake (unit confusion) and a computational
error in this case.
Type C Mistake is the "Other" type of mistake. For example, a student wrote on the
quiz, 1.4 / 2 + 1.4 / 2 = 1.4. I could not interpret this algorithm. I really did not
know what and how the student thought about ratio and the problem.

Extended Square Problem
Type A Mistake is a unit confusion mistake. There were many types of confusion in
this unit. The most common one was adding cm2 with cm. For example, a student
wrote, (121 + 3)(121), on the quiz. The number 121 was an area, its unit was cmz.
The number 3 was a length and its unit was cm.
1m B Mistake is an area-length confusion mistake. In other words, the students
confused the two different measurements for two different dimensions (area and
length). For example, many students thought that dividing the area of a square by 4
would give them the length of a side.

C Mistake is an area-perimeter confusion mistake. Few students could not
distinguish between area and perimeter, they gave the same answer to two different
questions in this problem set.
1ny D Mistake is an additive type mistake. A small square was added to the
original square. For example, a student wrote, 121 cm2 + 3 cm x 3 cm = 130 cm2,
on the quiz sheet.
flfype E Mistake is the "Other" type of mistake. This type of error was more diverse
and most of the mistakes could not be interpreted within the problem context and

mathematically. For example, a student wrote 6.25 x 4 = 3200 as the answer for

77

perimeter. Or, students just tried to use all the given numbers with different

mathematical operations, but without any contextual and mathematical sense.

Table 5.1 Performance from Both Schools on Form A (N = 58)

 

 

 

. ml n._- L‘- 'v m

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQIUQUAABQDEQ l 2
Multipli- 1 84.5 15.5
cation (49) (9)
[R0]
Rectangle 2 79.3 10.3 8.6 1.7
Width (46) (6) (5) ( 1)
[R0]
Unknown 3 74.1 24.1 1.7
Digit (43) (14) (1)
[R0]
Saving 4 43.1 22.4 32.8 1.7
[RC] (25) (13) (19) (1)
Balance 11 3.4 1.7 77.6 13.8 3.4
[NR] (2) (1) (45) (8) (2)
Glass 12 77.6 1.7 20.7
House (45) (1) (12)
(AHNRI
Glass 10 3.4 3.4 25.9 1.7 65.5
House (2) (2) (15) (1) (38)
(B) [NR]
Coins 7 12.1 77.6 10.3
[NR] (7) (45) (6)
Lake 9 10.3 1.7 67.2 12.1 8.6
[NR] (6) (1) (39) (7) (5)
Photo- 5 19.0 5.2 63.8 5.2 6.9 39.7 36.2 13.8 60.3 17.2 0.0
Robber (11) (3) (38) (2) (4) (23) (21) (8) (35) (10) (0)
[NR]
Extended 8 10.3 5.2 72.4 6.9 5.2 27.6 24.1 5.2 6.9 19.0 56.9 20.7 0.0
Square (6) (3) (42) (4) (3) (16) (14) (3) (4) (11) (33) (12) (0)
(13) [NR]
Extended 6 19.0 69.0 1.7 10.3 27.6 27.6 6.9 0.0 20.7 74.1 5.2 0.0
Square (11) (40) (l) (6) (16) (16) (4) (0) (12) (42) (3) (0)
(C) [NR]
Keys: Q: Correct ILA; Unattempted
HQ: Numerically Correct R: Rank of the Correctness
I: Incorrect R0: Routine Problem
E: Unclassified NR: Non-Routine Problem

78

Table 5.2 Performance from Two Schools on Form B (N = 65)

 

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] ENCILCUAAEQDE
Multipli- 1 81.5 18.5
cation (53) (12)
[R0]
Rectangle 3 74.4 15.4 9.2
Width (49) (10) (6)
[R01
Unknown 4 69.2 26.2 4.6
Digit (45) (17) (3)
[R01
Magazine 6 55.4 12.3 27.7 3.1 1.5
[R0] (36) (8) (18) (2) (1)
Number 7 55.4 43.1 1.5
Pattern (36) (28) (1)
(13)le
Number 9 41.5 47.7 10.8
Pattern (27) (31) (7)
(C) [NR]
Number 12 23.1 72.3 4.6
Pattern (15) (47) (3)
(D)[NRl
Lake 13 13.8 75.4 1.5 9.2
[NR] (9) (49) (1) (6)
Cake(A) 2 76.9 1.5 20.0 1.5 10.8 6.2 93.8 4.6 0.0
[NR] (50) (1) (13) (1) (7) (4) (61) (3) (0)
Cake(B) 5 63.162 30.8 169123 81.5 15.4 3.1
[NR] (41) (4) (20) (11) (8) (53) (10) (2)
Photo-You 8 44.6 3.1 44.6 3.1 4.6 24.6 10.8 21.5 73.8 6.2 0.0
[NR] (29) (2) (29) (2) (3) (16) (7) (14) (48) (4) (0)
Balm 14 1.5 1.5 83.1 9.2 4.6
[NR] (1) (l) (54) (6) (3)
Extended 11 23.1 1.5 67.7 4.6 3.1 26215.4 3.1 3.1 21.5 69.2 12.3 4.6
Square (15) (1) (44) (3) (2) (17) (10) (2) (2) (14) (45) (8) (3)
(B)[NR]
Extended 10 23.1 3.1 58.5 15.4 20.0 15.4 1.5 0.0 15.4 70.8 9.2 0.0
Square (15) (2) (38) (10) (13) (10) (1) (0) (10) (46) (6) (0)
(C)INR1
Keys: Q: Correct m: Unattempted
NC: Numerically Correct R: Rank of the Correctness
I: Incorrect R0: Routine Problem
QC: Unclassified NR: Non-Routine Problem

On the correctness dimension, students in both schools performed well on the
routine problems on both forms. On Form A, the routine problems received the highest

percentage of correct responses. On Form B, the three routine problems ("Multiplication,"

79

LA“. ..ur'— —..————-l-

 

"Rectangle Width," and "Unknown Digit") received the highest percentage of correct
responses. The "Cake" problem was the only outlier in this study. It was a non-routine
problem, but its percentage of correct responses was higher than that of the routine
problem, "Saving".

On the conceptual dimension, the major mistake in the 5 problems (4 of them were
2 sets of parallel problems) was related to unit confusion. On both forms, unit confusion
had the highest percentage of conceptual errors on each item. This result revealed that
students had more difficulty conceptualizing or understanding the units, or did not pay
attention to the units when they tried to solve the problems.

On the computational dimension, using the 5 analyzed problems to examine the
students' computational errors, the results revealed that the students in both schools did not
experience many difficulties doing the computations. The highest percentage of error was
20%. On both forms, students seemed to make more computational errors on the

"Extended Square" problem.

Table 5.3 Students with Different Achievement Levels Distribution

 

 

High Achievers Mid Achievers Low Achievers
4 (Form A) 22 (Form A) 6 (Form A)
Low SES School* 3 (Form B) 23 (Form B) 8 (Form B)
Total = 7 Total = 45 Total = 14
. 5 (Form A) 13 (Form A) 7 (Form A)
Middle Class School 9 (Form B) 16 (Form B) 5 (Form B)
Total = 14 Total = 29 Total = 12

 

 

 

 

 

* Two students in the low SES school were new to the school; their performance records were not available
when I collected the data, so both were not in these achievement level categories. Because of these two
missing students, the total number of students were 121 in this table.

80

 

Table 5.4 Performance from Low SES School on Form A (N = 33)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QMIUQQAABQDEQLZ
Multipli- 1 87.9 12.1
cation (29) (4)
[R0]
Rectangle 2 75.8 12.1 9.1 3.0
Width (25) (4) (3) (1)
[RC]
Unknown 3 66.7 30.3 3.0
Digit (22) (10) (1)
[RC]
Saving 4 36.4 30.3 30.3 3.0
[RC] (12) (10) (10) (1)
Balance 8 3.0 72.7 18.2 6.1
[NR] (1) (24) (6) (2)
Glass 9 66.7 3.0 30.3
House (22) (1) (10)
(AHNRI
Glass 8 3.0 18.2 3.0 75.8
House (1) (6) (1) (25)
(B) [NR]
Coins 6 9.1 78.8 12.1
[NR] (3) (26) (4)
Lake 5 9.1 3.0 69.7 9.1 9.1
[NR] (3) (1) (24) (3) (3)
Photo- 5 9.1 3.0 72.7 6.1 9.1 42.4 39.4152 54.5 15.2 0.0
Robber (3) (1) (23) (2) (3) (14) (13) (5) (l8) (5) (0)
[NR]
Extended 7 6.1 3.0 69712191 36.4 30.3 6.1 9.1 21.2 48.5 21.2 0.0
Square (2) (l) (23) (4) (3) (12) (10) (2) (3) (7) (l6) (7) (0)
(B)[NR]
Extended 6 9.1 72.7 3.0 15.2 36.4 36.4 9.1 0.0 24.2 72.7 0.0 0.0
Square (3) (24) (1) (5) (12) (12) (3) (0) (8) (24) (0) (0)
LC) [NR]
Keys: Q: Correct U_A_: Unattempted
NC: Numerically Correct R: Rank of the Correctness
1: Incorrect RO: Routine Problem
I_J_C_: Unclassified NR: Non-Routine Problem

81

’ :7 1-". ."

 

Table 5.5 Performance from Low SES School on Form B (N = 35)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QM!IEUAABQDEQL-
Multipli- 1 82.9 17.1
cation (29) (6)
[RC]
Rectangle 3 68.6 20.0 11.4
Width (24) (7) (4)
[RC]
Unknown 5 57.1 37.1 5.7
Digit (20) (13) (2)
[R0]
Magazine 6 51.4 20.0( 22.9 5.7
[R0] (18) 7) (8) (2)
Number 7 48.6 48.6 2.9
Pattern (17) (17) (1)
(13)le
Number 8 31.4 48.6 20.0
Pattern (ll) (17) (7)
(C) [NR]
Number 10 14.3 80.0 5.7
Pattern (5) (20) (2)
(D) [NR]
Lake 13 8.6 80.0 11.4
[NR] (3) (28) (4)
Cake(A) 2 77.1 2.9 20.0 11.4 2.9 91.4 8.6
[NR] (27) (1) (7) (4) (1) (32) (3)
Cake(B) 4 60.0 2.9 37.1 229171 80.0 17.1 2.9
[NR] (21) (1) (13) (8) (6) (28) (6) (1)
Photo-You 9 28.6 2.9 57.1 5.7 5.7 37.1 11.4 31.4 71.4 8.6 0.0
[NR] (10) (1) (20) (2) (2) (13) (4) (11) (25) (3) (0)
Balance 14 82.9 11.4 5.7
[NR] (29) (4) (2)
Extended 12 8.6 80.0 5.7 5.7 40.0 22.9 5.7 5.7 28.6 51.4 20.0 5.7
Square (3) (28) (2) (2) (14) (8) (2) (2) (10) (18) (7) (2)
(B)[NRl
Extended 11 8.6 2.9 62.9 25.7 28.6 22.9 2.9 0.0 20.0 48.6 17.1 0.0
Square (3) (1) (22) (9) (10) (8) (1) (0) (7) (l7) (6) (0)
(C) [NR]
Keys: Q: Correct 1g: Unattempted
NC: Numerically Correct R: Rank of the Correctness
1: Incorrect R0: Routine Problem
U_Q Unclassified NR: Non-Routine Problem

82

 

Table 5.6 Performance from Middle Class School on Form A (N = 25)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QEIQQLLAABQDE
Rectangle 1 84.0 8.0 8.0
Width (21) (2) (2)
[R0]
Unknown 2 84.0 16.0
Digit (21) (4)
[RC]
Multipli- 3 80.0 20.0
cation (20) (5)
[R0]
Saving 4 52.0 12.0 36.0
[R0] (13) (3) (9)
Balance 11 8.0 84.0 8.0
[NR] (2) (21) (2)
Glass 12 92.0 8.0
House (23) (2)
(A) [NR]
Glass 10 8.0 4.0 36.0 52.0
House (2) (1) (9) (13)
(3)le
Coins 8 16.0 76.0 8.0
[NR] (4) (19) (2)
Lake 9 12.0 64.0 16.0 8.0
[NR] (3) (16) (4) (2)
Photo- 5 32.0 8.0 56.0 4.0 36.0 32.0 12.0 68.0 20.0 0.0
Robber (8) (2) (14) (1) (9) (8) (3) (17) (5) (0)
[NR]
Extended 7 16.0 8.0 76.0 16.0 16.0 4.0 4.0 16.0 68.0 20.0 0.0
Square (4) (2) (19) (4) (4) (1) (1) (4) (20) (5) (0)
(B) [NR]
Extended 6 32.0 64.0 4.0 16016.0 4.0 0.0 16.0 72.0 12.0 0.0
Square (8) (16) (1) (4) (4) (1) (0) (4) (l8) (3) (0)
QINR]
Keys: Q: Correct LIA: Unattempted
NC: Numerically Correct R: Rank of the Correctness
I: Incorrect R0: Routine Problem
U0: Unclassified NR: Non-Routine Problem

83

7

 

Table 5.7 Performance from Middle Class School on Form B (N = 30)

 

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQIUQUAABQDEQL
Rectangle 1 83.3 10.0 6.7
Width (25) (3) (2)
[RC]
Unknown 2 83.3 13.3 3.3
Digit (25) (4) (I)
[RC]
Multipli- 3 80.0 20.0
cation (24) (6)
[R0]
Magazine 8 60.0 3.3 33.3 3.3 It
[R0] (18) (1) (10) (1) 1
Number 7 63.3 36.7
Pattern (19) (11)
(B)[NRl
Number 9 53.3 46.7
Pattern (16) (14) i
(C) [NR]
Number 11 33.3 63.3 3.3
Pattern (10) (19) (1)
(D) [NR]
Lake 12 20.0 70.0 3.3 6.7
[NR] (6) (21) (1) (2)
Cake(A) 4 76.7 23.3 3.3 10.0 10.0 96.7 0.0 0.0
[NR] (23) (6) (1) (3) (3) (29) (0) (0)
Cake(B) 5 667100 23.3 10.0 6.7 83.3 13.3 3.3
[NR] (20) (3) (7) (3) (2) (25) (4) (1)
Photo—You 6 63.3 3.3 30.0 3.3 10.0 10.0 10.0 76.7 3.3 0.0
[NR] (19)(1) (9) (1) (3) (3) (3) (23) (1) (0)
Balance 13 3.3 3.3 83.3 6.7 3.3
[NR] (1) (1) (25) (2) (1)
Extended 10 40.0 3.3 53.3 3.3 10.0 6.7 0.0 0.0 13.3 90.0 3.3 3.3
Square (12) (1) (16) (1) (3) (2) (0) (0) (4) (27) (1) (l)
(B)[NRl
Extended 10 40.0 3.3 53.3 3.3 10.0 6.7 0.0 0.0 10.0 96.7 0.0 0.0
Square (12) (1) (16) (1) (3) (2) (0) (0) (3) (29) (0) (0)
(C) [NR]
Keys: Q: Correct DA: Unattempted

M2: Numerically Correct R: Rank of the Correctness

I: Incorrect R0: Routine Problem

QC: Unclassified NR: Non-Routine Problem

Comparing students' performances across two different schools, the results

revealed that students from different schools did not perform differently on most routine

problems, except in the case of the "Unknown Di git" problem on Form B. Students in the

84

middle class school outperformed their counterparts. When we consider "Correct" and
"Numerically Correct" as correct answers, there is no obvious difference between the two
groups on the "Saving" problem. If we see the two categories of correctness as different
levels of achievement, students from the middle class school made fewer mistakes (mostly
unit mistakes) on the "Saving" problem. The performance of the two groups on the non-
routine problems revealed another story. Except in the cases of the 4 problems ("Balance,"
"Glass House," "Coins," and "Lake") on Form A and 2 problems ("Cake" and "Balance")
on Form B, students in the middle class school outperformed their counterparts by more
than 10% on the dimension of correctness.

0n the conceptual dimension, students in the low SES school seemed to experience
more difficulty on the "Extended Square" problem on Form A. They made more conceptual
mistakes on unit and area-length. Their error percentage was double that of those made by
their counterparts—Extended Square [B]: unit confusion (36% versus 16%) and area-
length confusion (36% versus 16%); Extended Square [C]: unit confusion (36% versus
16%) and area-length confusion (36% versus 16%). On Form B, students in the low SES
school performed even more poorly. Except in the case of the "Cake" problem Part A, the
conceptual mistakes were related to unit confusion. The other non-routine problems
("Ratio" and "Extended Square") completed by the students in the low SES school were
double that of the mistakes made by their counterparts (Cake (B): 37% versus 10%; Ratio:
37% versus 10%; Extended Square (B): 40% versus 10%; Extended Square (C): 28%
versus 10%). On the "Extended Square" problem, the mistakes made by the students in the
low SES school were 3 times that of their counterparts (22% versus 6%) on area-length
confusion category.

0n the computational dimension, the two groups of students did quite well. The
only obvious differences between the two groups were that students in the low SES school

made more mistakes on the "Extended Square" problem Part (A). Of the students in the low

85

SES school, 20% made one computational mistake on this problem, compared to only 3%

of the students in the middle class school.

Table 5.8 High Achievers Performance on Form A (N = 9)

 

 

 

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQLUQUAABQDEQL
Multipli- 1 100
cation (9)
[R0]
Rectangle 2 88.9 11.1
Width (8) (1)
[RC]
Unknown 3 88.9 11.1
Digit (8) (1)
[R0]
Saving 4 66.7 11.1 22.2
[R0] (6) (1) (2)
Balance 9 22.2 11.1 44.4 22.2
[NR] (2) (1) (4) (2)
Glass 11 77.8 22.2
House (7) (2)
(A) [NR]
Glass 10 11.1 11.1(333 44.4
House (1) 1) (3) (4)
(B) [NR1
Coins 7 55.6 33.3 11.1
[NR] (5) (3) (1)
Lake 8 33.3 22.2 44.4
[NR] (3) (2) (4)
Photo- 8 33.3 55.6 11.1 44.4 44.4 33.3 88.9 0.0 0.0
Robber (3) (5) (1) (4) (4) (3) (8) (0) (0)
[NR]
Extended 6 55.611.133.3 0.0 0.0 0.0 0.0 0.0 77.8 0.0 0.0
Square (5) (1) (3) (0) (0) (0) (0) (0) (7) (0) (O)
(B)[NRl
Extended 5 66.7 33.3 11.1 11.1 11.1 0.0 0.0 100 0.0 0.0
Square (6) (3) (1) (1) (1) (0) (0) (9) (0) (0)
(C) [NR]
Keys: Q: Correct LIA: Unattempted
_N_Q: Numerically Correct R: Rank of the Correctness
1: Incorrect R0: Routine Problem
U_Q: Unclassified NR: Non-Routine Problem

86

Table 5.9 High Achievers Performance on Form B (N = 12)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQIEQABQDEQLZ
Multipli- 1 91.7 8. 3
cation (11) (1)
[R0]
Rectangle 1 91.7 8.3
Width (11) (1)
[RC]
Unknown 2 83.3 16.7
Digit (10) (2)
[RC]
Magazine 5 66.7 33.3
[R0] (8) (4)
Number 7 50.0 50.0
Pattern (6) (6)
(B)[NR1
Number 6 58.3 33.3 8.3
Pattern (7) (4) (1)
(C) [NR]
Number 8 25.0 75.0
Pattern (3) (9)
(D) [NR]
Lake 10 91.7 8.3
[NR] (11) (1)
Cake(A) 2 83.3 16.7 8.3 8.3 100 0.0 0.0
[NR] (10) (2) (1) (1) (12) (O) (0)
Cake(B) 4 66.7 8.3 25.0 16.7 16.7 83.3 8.3 8.3
[NR] (8) (1) (3) (2) (2) (10) (1) (1)
Photo-You 3 75.0 25.0 8.3 0.0 16.7 83.3 8.3 0.0
[NR] (9) (3) (1) (0) (2) (10) (1) (0)
Balance 10 91.7 8.3
[NR] (11) (1)
Extended 9 16.7 83.3 16.7 8.3 0.0 0.0 16.7 83.3 16.7 0.0
Square (2) (10) (2) (1) (0) (0) (2) (10) (2) (0)
(13)le
Extended 9 16.7 83.3 16.7 8.3 0.0 0.0 16.7 0.0 0.0 0.0
Square (2) (10) (2) (1) (0) (0) (2) (0) (0) (0)
JCMNRJ
Keys: Q: Correct LIA: Unattempted

NC: Numerically Correct R: Rank of the Correctness

1: Incorrect R0: Routine Problem

LE: Unclassified NR: Non-Routine Problem

87

Table 5.10 Mid Achievers Performance on Form A (N = 35)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQILLQMABQDEQLZ
Rectangle 1 88.6 5.7 2.9 2.9
Width (31) (2) (1) (1)
[RC]
Multipli- 2 82.9 17.1
cation (29) (6)
[R0]
Unknown 3 77.1 20.0 2.9
Digit (27) (7) (1 )
[R0]
Saving 4 37.1 31.4 28.6 2.9
[R0] (13) (11) (10) (1)
Balance 10 88.6 8.6 2.9
[NR] (31) (3) (1)
Glass 10 82.9 17.1
House (29) (6)
(A) [NR]
Glass 9 2.9 2.9 31.4 62.9
House (1) (1) (11) (22)
(13)le
Coins 7 5.7 85.7 8.6
[NR] (2) (30) (3)
Lake 7 5.7 74.3 8.6 11.4
[NR] (2) (26) (3) (4)
Photo- 5 22.9 8.6 51.4 5.7 11.4 31.4 28.6 5.7 62.9 11.4 0.0
Robber (8) (3) (18) (2) (4) (11) (10) (2) (22) (4) (0)
[NR]
Extended 8 2.9 5.7 74.3 8.6 8.6 28.6 34.3 0.0 5.7 14.3 60.0 20.0 0.0
Square (1) (2) (26) (3) (3) (10) (12) (0) (2) (5) (21) (7) (0)
(13) [NR]
Extended 6 14.3 68.6 2.9 14.3 28.6 34.3 0.0 0.0 11.4 74.3 2.9 0.0
Square (5) (24) (1) (5) (10) (12) (0) (0) (4) (26) (1) (0)
iClfljR]
Keys: Q: Correct ILA: Unattempted
NC: Numerically Correct R: Rank of the Correctness
1: Incorrect R0: Routine Problem
110: Unclassified NR: Non-Routine Problem

88

Table 5.11 Mid Achievers Performance on Form B (N = 39)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QELLCUAABQDEQLZ
Rectangle 1 76.9 17.9 5.1
Width (30) (7) (2)
[R01
Multipli- 2 76.9 23.1
cation (30) (9)
[RC]
Unknown 4 69.2 28.2 2.6
Digit (27) (11) (1)
[R0]
Magazine 7 59.0 10.3 25.6 5.1
[R0] (23) (4) (10) (2)
Number 6 61.5 35.9 2.6
Pattern (24) (14) (1)
(B)[NRl
Number 9 38.5 51.3 10.3
Pattern (15) (20) (4)
(C) [NR]
Number 12 28.2 64.1 7.7
Pattern (11) (25) (3)
(D) [NR] '
Lake 13 23.1 61.5 15.4
[NR] (9) (24) (6)
Cake(A) 3 74.4 23.1 2.6 12.8 5.1 92.3 5.1 0.0
[NR] (29) (9) (1) (5) (2) (36) (2) (0)
Cake(B) 5 66.7 2.6 30.8 20.5 7.7 82.1 15.4 2.6
[NR] (26) (1) (12) (8) (3) (32) (6) (1)
Photo-You 8 43.6 5.1 41.0 5.1 5.1 25.6 12.8 23.1 74.4 7.7 0.0
[NR] (17) (2) (16) (2) (2) (10) (5) (9) (29) (3) (0)
Balance 14 2.6 2.6 79.5 10_3( 5.1
[NR] (1) (1) (31) 4) (2)
Extended 11 28.2 2.6 59.0 5.1 5.1 28.2 20.5 5.1 5.1 12.8 66.7 10.3 7.7
Square (11) (1) (23) (2) (2) (11) (8) (2) (2) (5) (26) (4) (3)
(13)le
Extended 10 28.2 5.1 51.3 15.4 23.1 20.5 0.0 0.0 10.3 69.2 10.3 0.0
Square (11) (2) (20) (6) (9) (8) (0) (0) (4) (27) (4) (0)
(C) [NR]
Keys: Q: Correct U_A: Unattempted

N_C: Numerically Correct R: Rank of the Correctness

1: Incorrect R0: Routine Problem

U_C,: Unclassified NR: Non-Routine Problem

89

Table 5.12 Low Achievers Performance on Form A (N = 13)

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QMIUQUAABQDEQL2
Multipli- 1 84.6 15.4
cation (11) (2)
[R0]
Unknown 2 53.8 46.2
Digit (7) (6)
[R0]
Rectangle 3 46.2 23.1 30.8
Width (6) (3) (4)
[RO]
Saving 4 46.2 53.8
[R0] (6) (7)
Balance 6 69.2 23.1 7.7
[NR] (9) (3) (1)
Glass 6 61.5 7.7 30.8
House (8) (1) (4)
(ANNRI
Glass 6 7.7 92.3
House (1) (12)
(13)le
Coins 6 84.6 15.4
[NR] (11) (2)
Lake 5 7.7 84.6 7.7
[NR] (1) (11) (1)
Photo— 6 100 53.8 46.2 23.1 30.8 46.2 0.0
Robber (13) (7) (6) (3) (4) (6) (0)
[NR]
Extended 6 92.3 7.7 46.2 15.4 23.1 7.7 46.2 23.1 38.5 0.0
Square (12) (1) (6) (2) (3) (1) .(6) (3) (5) (0)
(B)[NR]
Extended 6 92.3 7.7 38.5 15.4 23.1 0.0 53.8 53.8 7.7 0.0
Square (12) (1) (5) (2) (3) (0) (7) (7) (1) (0)
(C) [NR]
Keys Q. Correct MI Unattempted
15E: Numerically Correct R: Rank of the Correctness
I: Incorrect R0: Routine Problem
[LC Unclassified NR: Non-Routine Problem

90

Table 5.13 Low Achievers Performance on Form B (N = 13)

 

 

 

 

Problem R Answer Correctness Conceptual Mistake Computing Error
[Type] QNQILLQUAABQDE
Multipli- 1 84.6 15.4
cation (11) (2)
[RC]
Rectangle 3 53.8 23.1 23.1
Width (7) (3) (3)
[R0]
Unknown 4 53.8 30.8 15.4
Digit (7) (4) (2)
[RC]
Magazine 8 30.8 30.8 30.8 7.7
[R0] (4) (4) (4) (1)
Number 6 46.2 53.8
Pattern (6) (7)
(BHNRI
Number 7 38.5 46.2 15.4
Pattern (5) (6) (2)
(C) [NR]
Number 10 7.7 92.3
Pattern (l) (12)
(D) [NR]
Lake 11 100
[NR] (13)
Cake(A) 2 76.9 7.7 15.4 7.7 7.7 92.3 7.7 0.0
[NR] (10) (1) (2) (1) (1) (12) (1) (0)
Cake(B) 5 46215.4 38.5 7.7 23.1 76.9 23.1 0.0
[NR] (6) (2) (5) (1) (3) (10) (3) (0)
Photo-You 9 15.4 76.9 7.7 38515.4 23.1 61.5 0.0 0.0
[NR] (2) (10) (1) (5) (2) (3) (8) (0) (0)
Balance 11 84.6 7.7 7.7
[NR] (11) (1) (1)
Extended 10 7.7 84.6 7.7 30.8 7.7 0.0 0.0 53.8 61.5 15.4 0.0
Square (1) (11) (1) (4) (1) (0) (0) (7) (8) (2) (0)
(B)[NR]
Extended 10 7.7 61.5 30815.4 7.7 7.7 0.0 30.8 53.8 7.7 0.0
Square (1) (8) (4) (2) (1) (1) (0) (4) (7) (1) (0)
(QINE
Keys: C: Correct U_A: Unattempted
NC: Numerically Correct R: Rank of the Correctness
_I_: Incorrect R0: Routine Problem
QC: Unclassified NR: Non-Routine Problem

0n the dimension of correctness, there were no obvious differences among 3
groups at the general level of pattern. Routine problems received the highest percentage of

correctness among the 3 groups. The "Cake" problem was the only exceptional non-routine

91

 

problem. The students received a high percentage of correctness on this problem. On the
routine problems, the high achiever group received the highest percentages on all of these
problems. The middle achiever group performed at a level between the two extreme
groups, high and low achiever groups. The only exception to this case was the
"Multiplication" problem; the middle achiever group received the lowest percentage of
correctness on both forms.

The results on the non-routine problems were mixed. The high achiever group
received the highest percentages of correctness on most of these problems. However, they
performed poorly on the "Extended Square" problem on Form B. It seemed strange, the
other group of high achievers did quite well on the parallel problem on Form A.

On Form A, the high achiever group outperformed the other two groups on all non-
routine problems. And, the low achiever group did not respond correctly on these
problems.

On Form B, the differences among the 3 groups were not so obvious when it came
to dimension of correctness. The middle achiever group performed better than other two
groups on several non-routine problems ("Lake," "Balance," "Extended Square,"),
although the differences were not as great.

0n the conceptual dimension, unit confusion mistakes were the most common ones
made by the 3 groups across the 5 non-routine problems. In general, high achievers made
fewer mistake on these 5 problems. The only exception was the "Ratio" problem on Form
A. The "Extended Square" problems on both forms distinguished the high achievers from
the other two groups, on the dimension of conception. The total percentage of mistakes
made by the high achievers were far fewer than that of the other two groups.

On the computational dimension, 3 groups did quite well on the 5 non-routine
problems. The only exception was the low achiever group who worked on the From A.
They made a number of computational mistakes on the "Ratio" and "Extended Square (B)"

problems, 46% and 38% respectively.

92

 

A Special Example

In addition to the general summary of the students' performances on the quiz, a
special example will be presented in this section. This example will reveal a student who
had a wrong idea about the problem, but got the correct answer. It was the only case in the
data where an incorrect way of thinking lead to a correct answer. This example may have a
special implication for assessment. Is the final answer the only criterion teachers should use
to evaluate students work and performance? In this case, the researcher considered the
student's answer to be a correct answer. I also considered the fact that the student violated

the "other type" conceptual error.

Dimple

This response is the only example in which a student misconceptualized the
problem, but got the correct answer. The student worked on the "Cake" problem section b.

The student did not correctly conceptualize the problem, but got the correct answer. The
correct algorithm should be ($963 - $45 x 11) + 6. The student thought the algorithm was

($963 -:-11x $45) + 6, and after dividing 963 by 495 (the product of 11 and 45), s/he took
the remainder and divided it by 6, and got 78. The answer is correct, but his/her thinking

did not make sense.

 

W:
A cake-shop owner found that some customers like to buy a slice of cakes and others like to buy whole
cakes. So he cuts some of his cakes into 8 equal slices and sells them for $6 per slice. He sells the whole

cakes for $45.

a) One day, he sold 12 whole cakes and 62 slices. How much money he got that day?

Answer: $ 912

b) Another day, he had $963 in his cash-register at the end of the day. He knew he sold 11 whole cakes.
How many slices did he sell on that day?

 

Answer: 78 pieces

Figure 5.1 The Cake Problem on Form B

 

93

MES—“k ’ £Hfi79637'0 ’ mkfiwf'flle-ifi ' if 81711155; 1111822311
~i$7 firhll'ffli-téil‘ ’ 12* thfil-‘t H?

 

(7QitllMfiJ‘78 .5-
; “65> 1.21 ——i~
: 7111}? .1 __q 1'

. “(‘1 9' I

(iii/(r77.

r 035

- a.“ 618

Figure 5.2 A Student's Response on the Cake Problem

94

Interview Result

Card-Sorting

In the card-sorting activity, the card sorting result matched well with my
characterization of an item as a routine problem or a non-routine problem. My reasoning
was that students would categorize the items into "Not Know" and "Not Sure" when no
problem scheme came to their rrrind instantly. These items should be seen as non-routine
problems. The result revealed that more than 50% of the students categorized the non-
routine problems into "Not Know" or "Not Sure" categories. 0n the two non-routine
problems ("Moon" and "Paper Recycle"), my characterization did not match with the
students' categorization. Although it is a possibility that students over-estimated their own
competence or misread the problems, the results still revealed that the problem
categorization (routine and non-routine) that I did matched quite well with that of the

students' categorization.

Table 5.14 Frequencies from Students' Card-Sorting Results

 

 

 

 

 

 

 

Problem Name Responses
_ Mm NLSure Know
Balance 12 13 2
Board 6 15 6
Division <Routine> 0 0 27
Lake 14 12 1
Magazine <Routine> 2 8 17
Moon 5 3 19*
Number Triangle <Routine> 0 3 24
Poster 4 12 1 1
Paper Recycle 0 7 20*
Rope & Triangle <Routine> 1 3 23
4 Squares <Routine> 3 6 18
Table Tennis 3 18 6
Tiles 5 12 10
Train 2 14 1 1
Wallpaper 1 1 10 6

 

* Problems did not match with my characterization and student's categorization

95

 

Students' Performance on the Interview Problems

This interview section was designed to collect more information about how students
solved non-routine problems and to let students experience less pressure of time to finish
the non-routine problems. Two kinds of problems were selected. One was related to the
concept of speed, the "Train" problem. The other two were related to the concepts, area and
perimeter, the "Classroom Board" and "4 Squares" problems. These problems were
selected because students seemed to have more difficulty conceptualizing the

concepts—rate and area.

The Tran Problem;

Two trains (A and B) are starting out from Mongkok Station in two different directions at the same time
with the same speed, 66 km per hour. 2 minutes and 30 second later, how far apart distance are the trains?

+— >

Mongkok Station I

Figure 5.3 The Train Problem

 

 

 

 

In this problem, we examined the students' performances from two perspectives, a)
conceptual correctness: the train's speed is based on the ratio between time elapsed and
distance covered; b) computational correctness: all the computation between given and
generated numbers are correct. By 'conceptually correct' we mean that the students' work
indicated that he/she understood the essential relationships between the quantities stated in
the problem. For example, in the Train problem, clear evidence was given that the students
grasped the need to convert the speed in km/hr. to km/minute, and that speed were the
result of a multiplicative relationship between distance and time. Many students committed
a conceptual mistake on this problem, thinking it was a time-conversion problem. Other
students made computational mistakes related to forgetting that the problem asked for the
distance between two trains. They sought the distance traveled by a train and thought that

they had found the answer.

96

 

There are 3 separate sections in Table 5.16. Each section represents students with
different social backgrounds. The original plan of this study attempted to include students
from 3 different social backgrounds—middle class, low SES, and new immigrant from
Mainland China. Unfortunately, there were only a few students who belonged to the third
category so I decide not to consider this category when I did the quantitative analysis.
However, I would like to keep this group separately in the interview analysis.

In General, the result revealed that most high achievers did not have any conceptual
or computational difficulty on the speed problem. The middle achievers gave a mixed
result. About half of them experienced conceptual difficulty, although most of them did not
have any computational difficulty. In the low achiever group, most had conceptual

difficulties and did not have any serious difficulties on computation.

97

Fl. - r‘h ‘81 .‘L :11 ‘1; r\. -<.U r” \‘ Cu

Table 5.15: Brief Interview Performance on the Train Problem

ichool /

Middle Achievers Low Achievers

 

Achievement Hi h Achievers
§chool 2 Aaron (2104):

 

 

98

Alan (2207): Angela (2206):
(Middle Class) Conceptual x Conceptual Conceptual x
Native H.K.. Computational ‘1 Computational x Computational
students (Know) (Not Sure) (Not Sure)
Adam (21 10): Alice (21 12): Andy (2203):
Conceptual Conceptual Conceptual x
Computational Computational Computational
(Not Sure) (Know) (Not Sure)
Adrian (2133) Ashley (2220):
Conceptual ‘1 Conceptual
Computational \/ Computational
(Not Sure) (Not Know)
Austin (2212):
Conceptual x
Computational
(KnOW)
§clrool 1 Barry (1118): Blenda (1116): Brandon (1207):
(Low SES) Conceptual \/ Conceptual x Conceptual x
Native H.K. Computational V Computational \/ Computational
students (Know) (Know) (Not Sure)
Ben (1108): Billy (1121): Brian (1231):
Conceptual ‘1 Conceptual x Did not engage in the
Computational ‘1 Computational \l task.
(Know) (Not Sure)
Betsy (1122): Bobby (1217):
Conceptual x Conceptual
Computational \/ Computational
M (Know) (Not Sure)
Keys:
3 Correct
XI Incorrect

Table 5.15 (cont.): Brief Interview Performance on the Train Problem

 

 

 

School /
Achievement High Achievers Middle Achievers Low Achievers
School 1 Carl(llll): Cindy(1205): Crystai(l220):
(Low SES) Conceptual Conceptual x Conceptual x
new immigrant Computational Computational \/ Computational x
students (Not Sure) (Know) (Not Sure)
Danny (1223): Corey (1226) Doug (1202):
Conceptual Conceptual Conceptual x
Computational Computational Computational ‘1
(Know) (Know) (Not Know)
Debby (1229): Dorothy (1225):
Conceptual x Conceptual x
Computational \/ Computational \/
(Not Sure) (Not Sure) ‘
Diane (1228):
Conceptual \1
Computational \/
(Not Sure)
Dion (1230):
Conceptual x
Computational \1
(Not Sure)
Keys:
‘1: correct
x: Incorrect

On this interview problem, the high achievers performed better than the other two

groups. There were two exceptions, Bobby and Ashley. Bobby was a middle achiever, but

his conceptualization on the speed problem was quite different than that of the other

students. He used a more advanced proportional concept to see the relationship between the

quantities in the problem. He wrote down a formula, 66/(60/2.5) + 66/(60/2.5), for the

train problem after he thought about the problem for about 3 minutes. Ashley seemed to be

an under-achiever; she under-estimated her own competence. She lacked self—confidence in

doing mathematics. In the interview, she was asked if she liked her mathematics classes.

She reported that she felt sleepy when she saw mathematics. In fact, her performance

revealed that she understood the problem and the concept of speed.

99

4- m

 

The following figure shows a rectangle consisting of 4 equal squares. If the area of the
rectangle is 196 cm2,

 

 

 

 

 

 

 

a) What is the area of each square?
b) What is the perimeter of each square?

0) What is the perimeter of the rectangle showed in the figure?

(NR)

 

 

 

Figure 5.4 The 4-Square Problem

Only 5 students were given the 4-Square Problem in the interview. All of them
were low achievers at the two schools. I exarrrined the students' performances from two
perspectives, a) conceptual correctness: concepts on the square and perimeter of a square;
b) computational correctness. By "conceptually correct," I mean that students' observed
work indicated that he/she understood the relationship between the area of a square and the
length of a side and the fact that area and length are in two different dimensions. The
mistakes made by students on conceptual dimension were related to the concept of area.
Those students thought that they could determine the length of a side of a square by
dividing the area by 4. However, most of the students did not have any problem with the
resulting computation. Ashley was still a special case. She did not have any difficulty with

the concept of area and its relationship to the length of its side.

100

Table 5.16: Brief Interview-Performance on the 4-Squares Problem

 

 

 

 

School /
Achievement High Achievers Middle Achievers Low Achievers
Level
School 2 Andy (2203):
(Middle Class) Conceptual x
Native H.K. Computational \/
students (Know)
Ashley (2220):
Conceptual ‘/
Computational \/
(Not Know)
School 1 Brandon (1207):
(Low SES) Conceptual x
Native H.K. Computational ‘1
students (Not Sure)
School 1 Crystal (1220):
(Low SES) Conceptual x
new immigrant Computational \/
students (Not Sure)
Doug (1202):
Conceptual x
Computational ‘1
(KDOW)
Keys:
\1: correct
x: Incorrect

Table 5.17 shows that all the low achievers had conceptual difficulties relative to the
concept, area. The only exception was Ashley. However, all low achievers in the interview

did not make any computational errors.

101

l mB ar le

 

You and 3 other classmates are assigned to design your classroom bulletin board this
semester. The board is a rectangular shape, 2.5 meters by 1 meter. First you think about
the background.

1) You choose the background color (light blue), and find there are two sizes of that color
paper. One is 30 cm by 50 cm, and another is 40 cm by 50 cm. You want to buy the size
that wastes the least paper

a) Which will you buy?

b) How many pieces of your chosen size do you need to buy to cover the whole board?

 

(NR)

 

 

Figure 5.5 The Classroom Board Problem

This was the most complicated problem in the interview. Because the problem did
not offer a visual model, students needed to understand the context of the problem. Also,
there were a lot of unit-transformations students needed to handle in solving the problem.
In addition to the concepts of area and perimeter, students needed to track the meaning of
the numbers they generated. The investigators examined the students' performances from
four perspectives, a) area formula, b) unit transformation, c) problem interpretation, (1)
number meaning. By "conceptually correct," I mean that the students' work I observed
indicated that he/she understood the concept of area. For example, they knew how to
transform 2.5 m2 to 25000 cm2, and they could assign a correct unit and meaning to the
number they generated. For example, they knew that the unit of the remainder (1000) could
be generated by dividing the area of the board (25000 cm2) by area of the paper (1500 cm2
or 2000 cmz). None of the students experienced any difficulty giving the area formula of a
rectangle. On unit transformation dimension, the students' rrristakes were related to the
misconception of the relationship between m2 and cmz. Students thought that they could
turn 2.5 m2 to 250 cm2 by multiplying 2.5 by 100. On the problem interpretation
dimension, some students interpreted the word, "waste," in several different unexpected

ways. Some students thought that the more sheets one bought, the more waste would be

102

created. Others thought that bigger paper would create more waste. On the number meaning
dimension, a lot of students did not know what a number represented after they completed
some conversions or operations on the numbers. For instance, students divided the board
area (25000 cmz) by the paper area (1500 cm2 or 2000 cmz), and then were confused by
the remainder (1000 cm2). They did not know what that remainder represented. Some
students thought that the " 1000 cm2" was the left-over paper. They concluded that there
was no difference in "waste" between the two different sizes of paper.

For this problem, none of the student gave a completely correct answer. The table
5.18 showed that all experienced one or more conceptual difficulties with the problem.
However, most of the high achievers showed only one conceptual difficulty with the
problem, and most of the mid and low achievers showed two or more conceptual
difficulties. As on the train problem, Ashley and Bobby showed a similar pattern to that of

the high achievers.

103

Table 5.17: Brief Interview Performance on the Classroom Board Problem

 

 

 

 

School I
Achievement High Achievers Middle Achievers Low Achievers
Level
School 2 Aaron (2104): Alan (2207): Ashley (2220):
(Middle Class) Area formula: \1 Area formula: \/ Area formula: ‘1
Native HK. Unit Transformation: x Unit Transformation: x Unit Transformation: \/
students Interpretation: \/ Interpretation: \1 Interpretation: x
Number Meaning: \1 Number Meaning: x Number Meaning: \/
(Not Sure) (Not Sure) (Not Know)
Adam (2110): Alice (2112):
Area formula: ‘1 Area formula: ‘1
Unit Transformation: ‘1 Unit Transformation: x
Interpretation: ‘1 Interpretation: x
Number Meaning: x Number Meaning: x
(KDOW) (KDOW)
Adrian (2133)
Area formula: ‘1
Unit Transformation: \’
Interpretation: \/
Number Meaning: x
(Not Sure)
School 1 Barry (1118): Blenda (1116):
(Low SES) Area formula: ‘1 Area formula: \I
Native H-K- Unit Transformation: ‘1 Unit Transformation: x
5111951115 Interpretation: x Interpretation: x
Number Meaning: )1 Number Meaning: ‘1
(Not Know) (Know)
Ben (1108): Bobby (1217):
Area formula: \/ Area formula: \1
Unit Transformation: \/ Unit Transformation: ‘1
Interpretation: \1 Interpretation: \1
Number Meaning: x Number Meaning: x
(Not Sure) (Not Sure)
Betsy (1 122):
Area formula: ‘1
Unit Transformation: x
Interpretation: x
Number Meaning: x
(Not Sure)
Keys:
4: correct
x: Incorrect

104

Table 5.17 (cont.): Brief Interview Performance
on the Classroom Board Problem

 

 

 

School /
Achievement High Achievers Middle Achievers Low Achievers
Level
School 1 Carl (1111): Corey (1226) Dorothy (1225):
(Low SES) Area formula: ‘1 Area formula: ‘1 Area formula:
new immigrant Unit Transformation: x Unit Transformation: x Unit Transformation: x
students Interpretation: x Interpretation: x Interpretation: x
Number Meaning: x Number Meaning: ‘1 Number Meaning: x
(Not Know) (Not Sure) (Not Sure)
Danny (1223): Debby (1229):
Area formula: \1 Area formula: \1
Unit Transformation: x Unit Transformation: x
Interpretation: ‘1 Interpretation: x
Number Meaning: x Number Meaning: x
(Not Sure) (Not Sure)
Diane (1228):
Area formula: ‘1
Unit Transformation: \/
Interpretation: x
Number Meaning: x
(Not Sure)
Keys:
\1: correct
x: Incorrect

Generally, the performance of high achievers in the conceptual domain was better
than that of the two other groups except for the two special cases, Bobby and Ashley. 0n
the computational domain, students from the three groups performed quite well; only 3
students on the "Train" problem made computational mistakes. So, students' difficulties on
the non-routine mathematics word problems seemed to be related to their conceptual
difficulties. Students relied quite heavily on mathematics formulas they had learned when
they solved the problems. There were no examples of a student trying to draw a model for
the "Classroom Board" problem. Based on the data for the "4 Square" problem and the
"Classroom Board Problem," students had difficulty in understanding the concept of
"area." The mistakes they made on unit transformation revealed their difficulty with the

concept.

105

Chapter 6

NATURE OF CONCEPTUAL MISTAKES AND THEIR IMPACT
ON PERFORMANCE

The results presented in Chapter 5 revealed that students had more difficulty solving
non-routine problems than routine problems. However, there are many elements of non-
routine problems (story structure, embedded mathematics concepts, answer format,
number of sentences and words, different kinds of numbers—integers and decimals,
transformation of measurement units, etc...) that could contribute to their difficulty. Some
elements, like the embedded mathematics concepts, ratio, and the meaning of different
measurement units, make non-routine problems more conceptually challenging. Some
elements, like the length of the problem text, contribute to the difficulty without increasing
the conceptual or mathematical challenge. For example, the "Cake" problem revealed that a
story structure does not always make non-routine problems more difficult conceptually.
Although this problem had a very complicated and unfamiliar story structure, most students
performed quite well on this problem. More than 60% of the students were able to solve the
problem correctly. A non-routine problem can be conceptually difficult when the
mathematical concept(s) embedded in the problem is not well understood by the students.
For example, the "Ratio" and "Extended Square" problems revealed such difficulty in
conceptual understanding. Students did not understand the concepts, proportion and area,
very well. They performed poorly on these two problems and committed several sorts of
conceptual errors. For instance, many students used an additive model to think about
proportion in the "Ratio" problem, and thought that dividing the area of a square by 4 could
give them the length of a side in the "Extended Square" problem. If the problem structure is
unfamiliar and complicated and the embedded concept is not well understood by students,
the problem may be extremely difficult for students. For example, no student was able to

solve the "Glass House" problem with two correct answers in two sections. In this chapter,

106

I will give a deeper conceptual analysis of students' performances on non-routine
problems. In particular, I will analyze the kinds of errors students made on some non-
routine problems. In the following sections, the "Ratio" and the "Extended Square"
problems will be the main focus. In each problem, I will show the students' conceptual

difficulties with ratio and area and the kinds of errors they made.

Two Focus Problems

There were 12 non-routine problems distributed between two different forms in the
quiz. Each form contained 6 non-routine problems. Among these problems, there were 4
pairs of parallel problems (Balance, Extended Square, Lake, Ratio), and 4 other non-
parallel problems (Cake, Coin, Glass House, Number Pattern). Most parallel problems
contained the same story-content, but the given numbers on each problem were different on
the two different forms. The only exception was the "Ratio" problem, in which the two
parallel problems had two different story-contents. However, the mathematical concept,
proportional reasoning, and the story structure was the same. The two problems used the
ratios between an object on a photograph and the actual object as the main theme, and asked
students to figure the actual height of one object when they knew its represented height on
the photograph.

My reasons for choosing the "Extended Square" and "Ratio" problems were three-
fold. First, there were more written responses from students on these problems. If the
conceptual analysis was to be the main focus, more students' written protocols were
needed in order for an inference about their thinking. Second, the concepts embedded in
these two problems introduced two important mathematical concepts (proportion and area)
that were taught in grades 4 and 5. Proportion and area were two special multiplicative
concepts and marked a new level of multiplicative understanding in the curriculum. Before
these new multiplicative concepts were introduced, students treated multiplication as

another arithmetic computation. Proportion and area required that they think about and see

107

 

 

multiplication differently. The additive model is not a helpful or efficient one for students to
use to think about multiplication in the context of proportion and area. Third, "Extended
Square" and "Ratio" are two pairs of parallel problems so they offered us a more complete

picture of the performances of all students.

The "Extended Square" Problem

Two versions of this problem were presented in two different forms (Form A and
Form B). The two problems had the same story content; the only difference was the given
numbers. In Forrrr A, the given area was 196 cm2 and the dimension of the square was
lengthened by 5 cm. In Form B, the given area was 121 cm2 and the extended length was 3

C11].

 

Ex 11 Pr 1 in F A:
The area of a square is 196 cm2, now you lengthen one dimension by 5 cm to form a new shape.

a. Draw the new shape.
Answer: a rectangle
b. What is its area?

Answer: 14 cm x 19 em = 266 cm2
c. What is its perimeter?

Answer: (14+19)cm x 2 = 66 cm

Figure 6.1 The Extended Square Problem in Form A

 

 

W:

The area of a square is 121 cm2, now you lengthen one dimension by 3 cm to form a new shape.

a. Draw the new shape.
Answer: a rectangle
b. What is its area?

Answer: 11 cm x 14 cm = 154 cm2
c. What is its perimeter?

Answer: (ll+l4)cm x 2 = 50 cm

Figure 6.2 The Extended Square Problem in Form B

 

The main focus of this problem was to investigate how well students conceptualized
the concepts, area and perimeter. Students were given the area of a square; they needed to

extend the two opposite sides of this given square to build a rectangle. They were asked to

108

 

 

figure out the area and the perimeter of the new rectangle. Based on the collected written
protocols, five types of conceptual errors were identified. The first three types were related
to students' confusion between l-dimensional and 2-dimensiona1 measurements. The
fourth type may have been related to difficulty in conceptualizing the problem situation. All
the different kinds of errors that I could not logically and contextually interpret were placed

into an "Other" class.
In the second round of scoring, one of the scorers discovered a sixth type of
conceptual error not found in the original scoring scheme. There were 4 students who

thought that dividing the area by 2 would give them the length of a side (see Figure 6.3).

W
9' t‘zte-aatrzlraeet it. -
art-nanny o (#1! 91113? flirt-filth}?

A) ‘fik’fifi‘fl'fi ?
, mfg? ("1.731; t

l

.m—m V ' .'

‘
"D

m filfiifitfiy?
[34 '1 HST-{i

-‘\ . or.
- .." 1".1 . o
_.‘\;.} "I

e - “‘"v.T

.4 Ii" f

C 1 filfllfil} 11* ?
tbiS'rM'lfr‘z

'- "‘12:";

Figure 6.3 The Square-root Confusion
This thinking might be related to the confusion between dividing by 2 and taking the square
root. In this case, the student tried to get the length of a side by dividing the area by 2, in

one case, 121 cm2 -:- 2, which is 60.5 cm. S/he kept the original area 121 cm2 and added it

109

to a new area, 60.5 x 3 cm2. These four cases were originally scored as examples of
"Other" errors.

In the next section, the first five different types of conceptual mistakes that appeared
on the "Extended Square" problem will be described and illustrated. The student work
revealed several conceptual mistakes in this one problem. This is important because these
mistakes were related to the core concept, measurement in 1 and 2 dimensions. In other
words, they confused the length and area measurements. They used the length
measurement to think about area and operated on area quantity, for example, l2lcm2, as if

they were lengths.

Wis a unit confusion mistake. Of these students, 26.8% (33 students) committed this
mistake. In Form A, there were 36.4% ( 12 students) in the low SES school, and 16% (4
students) of those in the middle class school made this mistake. In Form B, there were
40% (14 students) in the low SES school, and 10% (3 students) in the middle class school
who committed this mistake. In this problem, the most common unit confusion was adding
a quantity in cm2 to a quantity in cm. For example, in the following case (Figure 6.4), the
student added 5 cm to 196 cm2. However, there were several students who multiplied 196
cm2 by 5 cm. And, the second most common case of unit confusion was done by dividing
the area by 4 and adding the quotient to 5 cm or 3 cm. In fact, dividing the area by 4 did not
change the unit of the area. The division just partitioned the area into four equal sectors. All
of these cases were categorized as a unit confusion rrristake. These mistakes revealed that
the students could not distinguish between the two different kinds of measurement, linear
measurement (cm) and area measurement (cmz). Students might not understand the
difference between two different kinds of measurement, one for length and one for
surfaces. Or, their knowledge of measuring length still had an influence on how they

thought about area. Some students might still think about the additive function in the

110

context of surface. This conceptual difficulty is quite common among grade 4 and 5

students, because they often learn the area concept by adding small tiles.

 

 

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Figure 6.4 Type A Conceptual Mistake on the Extended Square Problem
The illustrated case (Figure 6.4) included multiple conceptual and computational mistakes.
In addition to the unit confusion, the student might not comprehend the problem correctly.
First, s/he might be confused with length and area. S/he appeared to see 196 cm2 as the
length of a side; s/he added 5 cm to 196 cm2 revealing length and area confusion. Her/his
drawing revealed that s/he did not comprehend the problem correctly, the correct drawing
should be a rectangle. The product of 196 and 2 is 392, and not 292. Her/iris answer
indicated that a computational mistake had been made. Finally, perimeter cannot be the

product of the area (196 cm2) and 2. This was another conceptual mistake that s/he made.

111

IyLBis also a confusion of area and length, but in a slightly different way. Of these
students, 21% (26 students) made this conceptual mistake. In Form A, there were 36.4%
(12 students) in the low SES school and 16% (4 students) in the middle class school who
committed this mistake. In Form B, there were 22.9% (8 students) in the low SES school
and 6.7% (2 students) in the middle class school who committed this mistake. They
thought that dividing the area by 4 would give them the length of a side. A lot of students
who made this mistake were also confused about the units for different kinds of
measurement. For instance, the student work in Figure 6.5 showed the same unit, cm, for

area and perimeter.
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Figure 6.5 Type B Conceptual Mistake on the Extended Square Problem
This student thought that dividing 196 cm2 by 4 would give her/him the length of a side.

And, it was interesting that s/he thought that the product of sides (54 cm x 54 cm) would

112

give her/him the new area. The student did not realize that s/he measured area in two totally
different methods and procedures. One was done by multiplying the length of one side by 4
(dividing the area by 4 was revealed in her/his thinking about how s/he thought she could
find the area). Another method was to multiply one side with its adjacent side (54 cm x 54
cm). The methods and procedures were not reversible. It revealed that the student had some
idea about area, but her/his concepts about area was not firm. S/he might remember the
formula for rectangular shapes, but s/he did not understand how and why the formula
worked. There are several possible reasons for students making this conceptual mistake,
dividing area by 4 to get the side. Students could be misled by perception. They saw the
perimeter as the figure and the area as the ground. So, they did not realize that the focus
should be on the surface and not the sides. Students may also have been confused by the
concepts of area and perimeter. They may see area as the total length of four sides. That is,

they might not clearly be able to distinguish between area and perimeter.

M is a complete confusion of area and perimeter. Of these students, only 0.5% (6
students) made this mistake. In Form A, there were 9.1% (3 students) in the low SES
school and 4% (1 student) in the middle class school who made this mistake. In Form B,
there were 5.7% (2 students) in the low SES school and 0% (no students) in the middle
class school who made this mistake. So, only a few students could not distinguish between
area and perimeter. The students who made this mistake gave the same answer to both
questions. The student's work in Figure 6.6 revealed that s/he could not distinguish
between area and perimeter. S/he might think that the sum of the length of 4 sides was area

and it was also the perimeter.

113

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Figure 6.6 Type C Conceptual Mistake on the Extended Square Problem

hpe_12 is the mistake of adding a small square to the original square. In this case, the
students struggled more with their understanding of the problem statement than with the
concept of area and perimeter. There were only 0.5% (6 students) who made this mistake.
In Form A, there were 9.1% (3 students) in the low SES school and 4% (1 student) in the
middle class school who made the mistake. In Form B, there were 5.7% (2 students) in the
low SES school and 0% (no students) in the middle class school who made the mistake. In
the illustrated case (Figure 6.7), a student wrote, 121 cm2 + 3 cm x 3 cm = 130 cm2, for

the new area.

114

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Figure 6.7 Type D Conceptual Mistake on the Extended Square Problem

In this case, the student seemed to experience a conceptual difficulty in addition to the
additive mistake. Her/his work on section C revealed that s/he had a Type B conceptual
mistake as well. S/he thought that dividing the area by 4 would give her/him the length of a
side. The additive mistake seemed to be related to her/his difficulty in comprehending the

context of the problem.

Type—E is "Other" conceptual mistakes. This type of error was more diverse and these
mistakes could not be easily interpreted relative to the context of the problem. In Form A,
there were 7 students (21.2%) from the low SES school who committed this type of
mistake on Part B, and 8 students (24.2%) on Part C. In the middle class school, 4
students (16%) committed this mistake on Part B, and 4 students (16%) committed this
mistake on Part C. In Form B, 10 students (28.6%) from the low SES school committed
this mistake in Part B, and 7 students (20%) committed this mistake in Part C. In the

middle class school, 4 students (13.3%) committed this mistake in Part B, and 3 students

115

(10%) committed this mistake in Part C. For example, a student wrote, 6.25 x 4 = 3200,
as the answer for perimeter. I could not interpret the source of the number, 6.25—how s/he
figured out this number. Or, some students tried to use all the given numbers with different
mathematical operations, but without any contextual and mathematical sense. Two different
illustrative examples will be presented; they will suggest how diverse this type of

conceptual mistake was.

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Figure 6.8 Type E Conceptual Mistake 1 on the Extended Square Problem
In the case 1 (Figure 6.8), the student tried to use all the given numbers in the problem to
solve the problem. In the area section, s/he divided 123 cm2 (I presumed s/he misread the
given area 123 cm2 for 121 cm2) by 3, and got 41. Then, s/he multiplied 41 by 41 and got
1681 cm. In the perimeter section, s/he multiplied 41 by 4 and got 164 cm. Obviously, this
student did know about area and perimeter. At least, s/he remembered the formula for the
area of a rectangular square (length times width) and the perimeter (one length of a side
times 4). However, s/he seemed to have difficulty transforming or integrating this

knowledge into this problem situation. And, s/he manipulated all the given numbers

116

without a full understanding of the context of the problem. For example, she divided the
misread area (123 cm2) by 3, which did not make any contextual sense.

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Figure 6.9 Type E Conceptual Mistake 2 on the Extended Square Problem
In case 2 (Figure 6.9), the student drew a very strange diagram. The diagram seemed to
indicate that s/he did not comprehend the problem well. In the area section, s/he wrote 121
+ 3 x 4 = 133. S/he did not give any unit to this number, and her/his written response did
not make any contextual and mathematical sense in this problem situation. I could not
explain how and why s/he came up with this resolution— adding the original area, 121
cm2, to 3 multiplied by 4. If we examined this solution relative to her/his drawing, it would
still be difficult to interpret the writing, 3 x 4. In the perimeter section, the written response
revealed that the student had some idea about perimeter. S/he knew the formula for the
perimeter of a square (one side length times 4). However, s/he made a Type B
mistake—length-area confusion. S/he thought that dividing the area by 4 would give

her/him the length of a side.

117

Summary

If we take a closer look at the five types of mistakes, these mistakes can be
categorized into two higher level categories—measurement confusion, and comprehension
and knowledge integration difficulty. The first three types, A, B, and C, and the square
root confusion are related to measurement confusion. Students were confused by two
different kinds of measurement, length and surface area. The confusion may be related to
three sources. One source may be related to prior knowledge about linear measurement
(length measurement). The students still used linear measurement to think about surface
measurement. The unit confusion (Type A) might be related to a careless mistake, using a
wrong unit, but it might also be related to confusion between two different types of
measurement. Type B might be related to understanding of the concepts. Students did not
really understand the differences between these measurements, and why and how the
formula for area and perimeter worked. However, most of them knew the formula for area
and perimeter. The case presented in Figure 6.5 showed that a student knew the formula,
but did not understand that s/he used two incompatible algorithms in reasoning about area.
In Part B, s/he divided the area by 4 to find the length of a side. In Part C, s/he used the
"correct" method of multiplying the adjacent side lengths to find the area. The Type C error
might have been related to perception. Some students may not see surface as the figure, but
as the ground. They might see the perimeter as the figure as perceptually salient because the
surface is blank, but the perimeter lines are obvious. So, they might easily perceive the
sides and concentrate on how to measure them first. Finally, students' confusion could
quite possibly be related to several sources. Memorizing the formula without deep
understanding, and interacting this partial knowledge with the perceptual effect might easily
trap students in confusion about measurement.

The second higher level category was comprehension and knowledge integration
difficulties. Type D and E mistakes were related to this category. Students might not be

able to comprehend the problem context, so they established a wrong model for the

118

situation. For instance, students who added another small square beside the big square
showed a miscomprehension of the problem. Students who tried to use all the given
numbers to solve the problem without any contextual and mathematical sense showed that
they might have difficulty comprehending the problem situation and in integrating prior
knowledge they learned in school. Case 1 in the Type E mistake (Figure 6.8) showed this
kind of phenomenon.

In general, students' work showed that their mistakes were related to their level of
understanding of the concept of area. Most students had some sense of area and perimeter,
but their ideas were strongly tied to the formula and less so to a conceptual model. Many
knew that the product of adjacent sides gave the area, and 4 times the length of the side
gave the perimeter of a square. However, they did not seem to clearly know why and how
these formulas become the measurements of area and perimeter.

After studying these different types of mistakes, I examined students from different
social classes according to the type of conceptual mistakes they made on this problem. In
the following tables (Table 6.1, 6.2, 6.4, and 6.5), The abbreviations represent the
students' performances on three dimensions—correctness, conceptual, and computational.
Under the correctness dimension, students' performances are categorized into 5 types: 1) C
for a correct answer was given; 2) NC for a numerically correct answer was given; 3) I for
an incorrect answer was given; 4) UC for an unclassified trial was given, that means
students tried to solve the problem, but no answer was given, based on what they wrote on
the quiz sheet; 5) UA for unattempted, that means no written information was provided on
the quiz sheet. Under the conceptual dimension, there were 5 different types of mistakes
that were described in the last section. Under the computational dimension, students'
performances were categorized into 3 types: 1) O for no computational mistake, 2) l for one
computational mistake, 3) 2 for more than one computational mistake. And, "R" stands for

the rank of correctness; "t" stands for the total rank of the problems on different forms.

119

Table 6.1 Performance on the Extended Square Problem on Form A
School Problem Answer Correctness Conceptual Mistake Computing Error

QDLCIUQUAABQDEQIZ

Low Extended 6.1 3.0 69.7 12.1 9.1 36.4 30.3 6.1 9.1- 21.2 48.5 21.2 0.0
SES Square (2) (1) (23) (4) (3) (12) (10) (2) (3) (7) (16) (7) (0)
School (PartB)

 

(N=33)
Extended 9.1 72.7 3.0 15.2 36.4 36.4 9.1 0.0 24.2 72.7 0.0 0.0
Square (3) (24) (1) (5) (12) (12) (3) (0) (8) (24) (0) (0)
(PartC)
Middle Extended 16.0 8.0 76.0 16.0 16.0 4.0 4.0 16.0 68.0 20.0 0.0
Class Square (4) (2) (19) (4) (4) (1) (1) (4) (20) (5) (0)
School (PartB)
(N=25)
Extended 32.0 64.0 4.0 16.0 16.0 4.0 0.0 16.0 72.0 12.0 0.0
Square (8) (l6) (1) (4) (4) (1) (0) (4) (18) (3) (0)
(PartC)

 

Based on the results in the Table 6.1, students in the low SES school had more
difficulty with measurement confusion (Type A and Type B) in Form A. The students from
the low SES school made, on the average, nearly twice as many mistakes across all types
of mistakes, compared to their counterparts. However, the patterns of the mistakes were
the same. Students who made these mistakes revealed that they could not distinguish
between the two different kinds of units of measurement, and were conceptually confused
with the relationship between length and area. The conceptual confusion of the relationship
may have been caused by students who experienced confusion about units. Nearly one
third of the students (30%) in the school thought that dividing the area by 4 would give
them the length of a side. Yet, only 16% of the students in the middle class school showed

this confusion.

120

Table 6.2 Performance on the Extended Square Problem on Form B

 

 

 

School Problem Answer Correctness Conceptual Mistake Computing Error
C M; .1. HQ U_A. A. E C D B.
Low Extended 8.6 80.0 5.7 5.7 40.0 22.9 5.7 5.7 28.6 51.4 20.0 5.7
SES Square (3) (28) (2) (2) (14) (8) (2) (2) (10) (18) (7) (2)
School (PartB)
(N=3S)
Extended 8.6 2.9 62.9 25.7 28.6 22.9 2.9 0.0 20.0 48.6 17.1 0.0
Square (3) (1) (22) (9) (10) (8) (1) (0) (7) (l7) (6) (0)
(PartC)
Middle Extended 40.0 3.3 53.3 3.3 10.0 6.7 0.0 0.0 13.3 90.0 3.3 3.3
Class Square (12) (1) (l6) (1) (3) (2) (0) (0) (4) (27) (1) (1)
School (PartB)
(N=30)
Extended 40.0 3.3 53.3 3.3 10.0 6.7 0.0 0.0 10.0 96.7 0.0 0.0
Square (12) (1) (16) (I) (3) (2) (0) (0) (3) (29) (0) (0)
(PartC)

 

Based on the results in Table 6.2, students in the low SES school had more
difficulty with the measurement confusion (Type A and Type B) and conceptualization of
the problem situation (Type E) in Form B. In these three types of mistakes, the percentage
of students making such mistakes in the low SES school were more than double that of
their counterparts in the middle class school. In addition to measurement confusion,
students in the low SES school had difficulty with comprehension in the problem situation,
or integrating their learned mathematical knowledge into the problem situation. However,

the students in both schools showed the same patterns of mistake.

121

Table 6.3 Incorrect Responses Frequency on the Extended Square Problem

 

 

 

 

 

Minimum Incorrect
Incorrect Response
Frequency, Response Frequency
School Form Problem Incorrect Frequency related to
(N =) Response related to Computational
Conceptual Errors
_ Mistakes
Low SES Extended
School Square 23 16 7
(33) (Part B)
Extended
Square 24 24 0
(Part C)
Middle Class A Extended
School Square 19 14 5
(25) (Part B)
Extended
Square 16 13 3
(Part C)
Low SES Extended
School Square 28 19 9
(3 5) (Part B)
Extended
Square 22 16 6
t (Part C)
Middle Class B Extended
School Square 16 14 2
(30) (Part B)
Extended
Square 16 16 0
(Part C)

 

Based on the results in Table 6.3, students in both schools who responded to the

problems incorrectly, seemed to have more difficulty conceptually than computationally. In

FOI‘m A, 23 out of 33 students in the low SES school responded incorrectly in Part B, and

24 students in Part C. Among the 23 students who responded incorrectly in Part B, there

W(Ere only 7 students who committed one computational error. This means that at least 16

Stlldents committed one or more conceptual mistakes in this part. Among the 24 students

who responded incorrectly in Part C, all of them committed at least one conceptual mistake.

Students who responded incorrectly in the middle class school also showed the same

pattem—most of them had more difficulty conceptually than computationally. In Form A,

122

19 out of 25 students in the middle class school responded incorrectly in Part B, and 16
students in Part C. Among the 19 students who responded incorrectly in Part B, only 5
committed one computational error. This means that at least 14 students committed one or
more conceptual mistakes. Among the 16 students who responded incorrectly in Part C,
only 3 of them committed one computational error. This means that at least 13 students
committed one or more conceptual mistakes. In Form B, 28 out of 35 students in the low
SES school responded incorrectly in Part B, and 22 students in Part C. Among the 28
students who responded incorrectly in Part B, only 9 students committed one or two
computational errors. This means that at least 19 students committed one or more
conceptual mistakes. Among the 22 students who responded incorrectly in Part C, only 6
students committed one computational mistake. This means that at least 16 students
committed one or more conceptual mistakes. In Form B, 16 out 30 students in the middle
class school responded incorrectly in Part B, and 16 students in Part C. Among the 16
students in Part B, only 2 of them committed one or two computational mistakes. This
means that at least 14 students committed one or more conceptual mistakes. Among the 16
students in Part C who responded incorrectly, all of them committed at least one conceptual
mistake.

Based on the data collected in this study, I cannot explain why there was such a big
difference in the numbers of conceptual mistakes made between students from the two
schools. There are many possible reasons that could be used in interpreting the results.
First, the middle class school had a screening system that picked the smarter students and
rejected the students with learning difficulties. But, the school in the low SES community
did not have such a screening system and they accepted any student who applied for a seat.
In fact, there were a lot of new immigrant students in the low SES school. This
phenomenon implies that this particular school did not have a screening system for picking
their students. In Hong Kong, most good schools that have a lot of applicants will establish

a screening system for picking their students. The schools that have a lot of new immigrant

123

students are the ones without a screening system. So, the middle class school may have
smarter students from middle class families. Second, students from the middle class may
have several learning support resources—parents who know how to educate their children,
and private tutors may be employed. On the other hand, students from the low SES
families do not have access to resources for helping themselves learn. Third, the middle
class schools may have good teachers who motivate their students to work hard. These
teachers may also offer their students better classroom practice. Of course, the differences
may result in an interactive effect between these reasons. Clearly, no matter what the
reasons are for the differences, we need to worry about students from low SES families.
They seemed to began experiencing difficulty with learning important upper grade

mathematical concepts at the primary school level.

The "Ratio" Problems

There were two versions of the "Ratio" problems on the quiz (Photo-Robber on
Form A and Photo-You on Form B). They were not perfectly parallel. The two problems
were intentionally designed to ask students to use the same mathematical knowledge
(proportion and ratio) to solve the problems, but they had their own story content in
addition to different given numbers. Both problems used a photograph as the medium for
asking students to think about the proportional relationship between actual objects and the
represented objects on the photographs. However, the two photographs were taken in two
different situations creating two different story contexts. One photograph was taken of a
bank-robbery, the other in a leisure situation. The given numbers in these problems may
have affected students' performances. The Photo—Robber problem had one more decimal
than the Photo-You problem. The percentage of correctness for the Photo—Robber problem

was 19%, whereas 44% of the students solved the Photo-You problem correctly.

124

 

W:

A hidden camera took the picture below during a robbery. In the picture, you can see the robber and a
service desk. The height of the desk is 3 cm, and the height of the robber is 4.4 cm. The real height of the
desk is 1.2 meter. Can you figure out the height of the robber? How?

 

 

[service desk ] T
TI 1

30m

 

 

 

 

 

 

 

 

Answer: 1.76 m or 176 cm

 

 

Figure 6.10 The Photo-Robber Problem

 

PW:

Last Sunday, you stood beside a sculpture and had your picture taken. Your height on the photo is 4 cm,
and the height of the sculpture is 10 cm. You know your real height is 1.4 meters. Can you figure out the
real height of the sculpture? How?

 

 

 

 

 

 

 

 

Answer: 3.5 mm 350 cm

 

 

Figure 6.11 The Photo-You Problem

The main focus of the ratio problems was to investigate how students conceptualize
and reason about the concept, ratio. Based on students' responses, 3 types of conceptual
mistakes were categorized. They were: 1) mistakes related to the confusion of measuring
the units, meter and centimeter (Type A), 2) mistakes related to inappropriate additive
reasoning about ratio (Type B), and 3) mistakes which could not be interpreted in the
problem context (Type C). The Type B and Type C mistakes interacted with the Type A
mistake. Most students who made Type B and Type C mistakes also made Type A
mistakes. In Form A, 39.7% (23 cases) of the students committed Type A and Type B or C

125

mistakes, and only 5.2% (3 cases) of the students committed a single mistake. In Form B,
12% (15 cases) of the students committed Type A and Type B or C mistakes, and only
6.2% (4 cases) of the students committed a single mistake.

In the following paragraphs, three different types of conceptual mistakes will be
described with one or more illustrated examples.
Wis a unit confusion mistake. For example, a student wrote 10 cm x 4 cm = 40 cm (it
should be 40 cm2). Or, when a student wrote 3 — 1.2 = 1.8, it indicated that the student
confused or neglected the units for 3 and 1.2. In the problem, 3 is for 3 centimeters and 1.2
is for 1.2 meters. This was the most common mistake made by the students; one third of
the students made this mistake (33% or 41 students). The students experienced two types

of unit confusion. They are illustrated in Figure 6.12 and Figure 6.13.
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Figure 6.12 Type A Mistake Case 1 on the Photo-Robber Problem
In Figure 6.12, students subtracted two different units of measure, centimeter and meter,
without noticing the difference. In the illustrated case, the student subtracted 1.2 m from 3

cm. The difference could not be equal to 1.8 m, but the student did not notice that the

126

subtraction did not make sense within the problem context. The mistake that the student
made involved confusion with unit measurement, and also included incorrect additive

reasoning.
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Figure 6.13 Type A Mistake Case 2 on the Photo-You Problem

In Figure 6.13, the student multiplied quantities given in two different units of
measurement (6 cm and 1.4 m), without noticing their differences. S/he first subtracted 4
cm from 10 cm and got 6 cm. Then, s/he multiplied the difference times 1.4 m. The
product should not be equal to 8.4 (centimeters or meters). The unit should be in "square"

units. S/he did not know that the product of 6 cm and 1.4 m should yield square units.

[m B is an additive reasoning mistake. In the last section, Figure 6.12 showed that a
student thought that 3 cm minus 1.2 m corresponded to 4.4 cm minus the difference
between 3 cm and 1.2 m. The student seemed to think about ratio as an additive or

subtractive relationship between two related numbers. The most common additive mistake

127

made by students was to (1) interpret the relationship between quantities in the photo
additively (Q1 - Q2) and then (2) apply that difference to increase the size of the missing

real object.

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Figure 6.14 Type B Conceptual Mistake on the Photo-Robber Problem

A more logical additive mistake will be presented in Figure 6.14. In this case, the student
thought about how s/he could turn 3 cm into 1.2 m in an additive relationship. S/he
reasoned that 3 minus 2.8 equaled to 1.2 (s/he made a computational mistake: 3 minus 2.8
actually equals 0.2). S/he then carried out the same subtraction pattern on 4.4 cm (4.4 -
2.8) to give her/him the robber's height. This student seemed to have an additive model of
ratio in her/his mind. Compared to the case illustrated in Figure 6.12, this student had a

functional idea about ratio, but her/her functional idea about ratio was an incorrect additive.

TM mistakes could not be interpreted in relationship to the context of the problem.
Students might not have comprehended the problem situation or been unable to integrate
their ideas about ratio and proportion into the problem situation. So, many just tried to use

all the given numbers without any contextual sense.

128

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Figure 6.15 Type C Mistake Case 1 on the Photo-Robber Problem

In Figure 6.15, the student seemed to use all the numbers with additive and subtractive
operations to solve the problem. S/he first subtracted 1.2 m from 3 cm, and got 1.8 m, then
she added 1.8 m with 4.4 cm. S/he thought the answer should be 6.2 m. Based on her/his
responses, s/he made an additive reasoning mistake on the concept of ratio, and also
neglected the unit of measure among the given quantities in the quiz. However, it was very

difficult to explain why s/he used the operations among quantities in this problem context.

129

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Figure 6.16 Type C Mistake Case 2 on the Photo-Robber Problem

In Figure 6.16, the student seemed to use another arithmetic operation, division, without a
deep contextual sense. S/he used the given number "3" to divide another given number
"4.4", and thought the answer was 1.5. Based on her/his responses, it was revealed that
the student was confused by the unit measurement, but it is difficult to interpret why s/he

did the division operation on those quantities.

m

From the examples given above, we can see that the unit confusion mistake often
co-occurred with the other two types of mistakes. These two types of mistakes (B and C)
represented two different levels of understanding, although both of them were categorized
as conceptual mistakes in the scoring scheme. The "Additive" mistake (Type B) is related to
misconceptions or false models for thinking about ratio. Students who made this mistake
seemed to have been confused by the concept, ratio, as an additive relationship between the
quantities, and not as a multiplicative relationship. The "Other" mistake was related to

comprehension or recognition of the problem situation. Students who made this type of

130

mistake may have been unable to comprehend the problem or relate the problem situation to
the learned concept, ratio.

In this section, I would like to examine the differences between students at the two
schools through the lens of conceptual mistakes. On Form A (see Table 6.4), there were no
obvious differences between the two groups. They executed the same patterns of mistake,
with unit confusion (Type A). It was the most common mistake made among all of the
students. Although students from the middle class school seemed to perform better, the
differences between the two groups on the 3 types of mistake were less than 8%. In
contrast to the "Extended Square" problem, the frequency of these errors between the two

schools was much closer.

Table 6.4 Performance on the Photo-Robber Problem at Two Schools

 

 

 

 

School Problem Answer Correctness Conceptual Mistake Computing Error
LLC. 1 UC DA _A_ B. Q Q 1.3. .Q I 2

Low Photo- 9.1 3.0 69.7 9.1 9.1 42.4 39.4152 54.5 15.2 0.0

SES Robber (3) (1) (23) (3) (3) (14) (13) (5) (18) (5) (0)

School

(N=33)

Middle Photo- 32.0 8.0 56.0 4.0 36.0 32.0 12.0 68.0 20.0 0.0

Class Robber (8) (2) (14) (1) (9) (8) (3) (17) (5) (0)

School

IN=25L

 

On Form B (see Table 6.5), students from the middle class school outperformed
their counterparts. Only one tenth of the students at the middle class school made this
mistake on each type. In general, these two groups of students made these mistakes in
different patterns. In the low SES schools, more students made unit confusion mistakes
(Type A) than other mistakes (Type C). More than one third of the students in this group
made mistakes of these two types. This seems to imply that many students in this group did
not understand the problem situation or were unable to recognize the problem as related to
ratio and proportion. About the same number of students in the two schools reasoned

additively about ratio (Type B mistake).

131

Table 6.5 Performance on the Photo-You Problem at Two Schools

 

 

 

School Problem Answer Correctness Conceptual Mistake Computing Error
9. N_C_ l .U_C % A. B Q .12 E _ _

Low Photo- 28.6 2.9 57.15.7 5.7 37.1 11.4 31.4 71.4 8.6 0.0

SES You (10) (1) (20) (2) (2) (13)(4) (11) (25) (3) (0)

School

(N=35)

Middle Photo- 63.3 3.3 30.0 3.3 10.0 10.0 10.0 76.7 3.3 0.0

Class You (19) (1) (9) (l) (3) (3) (3) (23) (1) (0)

School

(N=30)

 

Based on the results in Table 6.6, we can conclude that the incorrect responses

were related to conceptual difficulties that students had, rather than the computational errors

 

 

 

they made.

Table 6.6 Incorrect Responses Frequency on lthfe Two Ratio Problems
Minimum Incorrect
Incorrect Response

Frequency, Response Frequency

School Form Problem Incorrect Frequency related to

(N:) Response related to Computational
Conceptual Errors

i Mistakes

Low SES A Photo-

School Robber 24 19 5

(33)

Middle Class A Photo-

School Robber 14 9 5

(25)

Low SES B Photo-You

School 20 17 3

(35)

Middle Class B Photo-You

School 9 8 l

(30)

 

Among the 24 out of 33 students who responded to the "Photo-Robber" problem
incorrectly from the low SES school, at least 19 of them committed one or more conceptual
mistakes. Only 5 of them responded incorrectly, and their incorrect responses could
possibly be attributed to computational errors. And, among the 14 out of 25 students who
responded to this problem incorrectly from the middle class school, at least 9 of them
committed one or more conceptual mistakes. Only 5 of them responded incorrectly and

their incorrect responses may have been attributable to computational errors. On the

132

"Photo—You" problem, 20 out of 35 students in the low SES school responded incorrectly.
Among the 20 students, 17 of them committed one or more conceptual mistakes. Only 3 of
them responded incorrectly, and their responses may have been caused by computational
errors. Also, 9 out of 30’students from the middle class school responded to the problem
incorrectly. Among the 9 students, 8 of them committed one or more conceptual mistakes.
Only 1 of them responded incorrectly, and her/his response might have been caused by a

computational error.

General Summary

In general, the students who responded to these two problem incorrectly seemed to
experience more conceptual difficulty, rather than lack of computational skills or careless
computational errors. Based on the mistakes made by students in the two problems, we can
find several common characteristics. First, the most obvious one is that unit confusion was
the most common mistake made by students across the two problems. And, this mistake
seemed to be related to other conceptual or problem comprehension mistakes. Students
who did not understand the concepts well made a unit confusion mistake. Most of these
students were also confused by the concepts of area and ratio. They may have acquired an
additive rather than multiplicative model for these concepts. Students who did not
comprehend the problem situation also made the unit mistake. It would be interesting to
learn why students who had conceptual and comprehension difficulties would neglect the
quantities given. I do not have any answer, but I think it would be a good question for
further investigation. Second, if we take out the comprehension problem that some students
had and just focus on the conceptual mistakes, then we will find that the conceptual
mistakes made by the students were related to the students' prior experiences. The mistake
does not emerge suddenly without history. The common conceptual mistake in the
"Extended Square" problem was related to confusion between length measurement and area

measurement. Students used their length measurement knowledge in the area situation. In

133

the "Photo-Ratio" problem, students used their knowledge of addition to think about
multiplication. In this chapter, we could see that many conceptual mistakes were related to
additive thinking. In the next chapter, we will see that several students made correct use of

their additive knowledge to solve these ratio problems.

Other Non-Routine Problems

There were 6 more non-routine problems in the quiz and they were arranged
between the two different forms. Among these problems, most of the answers given on 4
problems (Glass House, Coins, Lake, and Balance) by the students were incorrect, and
most students did not outline the procedures that revealed their thinking. 80, it is difficult to
make any inference about their reasoning. Based on the students' responses, two problems
(Number Pattern and Cake) were quite different from the four we mentioned. In the
"Number Pattern" problem, students did very well on Parts A, B, and C, however, their
answers for Part D were mostly incorrect. Neither students wrote their procedures in Part
D. So, I cannot make any inference about their reasoning.

The "Cake" problem is the only one on which we can make some inference about
students' reasoning. In general, students performed very well on this non-routine problem.
But, the students who responded incorrectly seemed to have more conceptual difficulty,

rather than making computational errors. Based on the results on Table 6.7,

134

Table 6.7 Incorrect Responses Frequency on the Cake Problem

 

 

 

Minimum Incorrect
Incorrect Response
Frequency, Response Frequency
School Form Problem Incorrect Frequency related to
(N =) Response related to Computational
Conceptual Errors
Mistakes
Low SES B Cake
School Part A 7 4 3
(35)
Cake
Part B 13 8 7
Middle class B Cake
School Part A 6 6 O
(30)
B Cake
Part B 7 3 5

 

we found that students committed more conceptual mistakes than computational errors. The

only exception was the middle class school students' performance. Among the 7 students

who gave the incorrect responses in Part B, only 3 of them committed conceptual mistakes;

and the other 4 only made computational errors.

135

Chapter 7

STUDENTS' CONCEPTUAL STRENGTHS AS EVIDENCED IN NOVEL
SOLUTIONS

This chapter will describe some unexpected student strengths in understanding and
solving mathematical problems. Before I go into a brief introduction to this chapter, let me
explain why I decided to include a chapter about these unexpected solutions. This chapter is
written for two audiences—teachers and educational psychologists. In particular, I would
like to reach primary school teachers. Sometimes, these novel student solutions are difficult
for the teachers to detect and discover in their students' reasoning. To present my
inferences about student reasoning to primary school teachers is the goal of this chapter. I
anticipate that the teachers would like to know more about the students' actual thinking,
rather than the statistics I presented in the previous chapters. Hopefully, teachers will find
that this information is useful in their classrooms. The second audience I would like to
address is educational psychologists. The data that will be presented in this chapter will
reveal several characteristics of student learning in which educators and psychologists are
interested. The students in this study showed the power of prior knowledge and intuition in
learning mathematics. And, one student, Bobby, showed the potential that students have
for learning the concept, proportion.

Several students in this study offered some unexpected solutions to quiz and
interview problems including "Photo-Robber," "Photo-You," Lake," and "Train." Their
solutions were closely related to students' intuitive ideas about several mathematical
concepts, rather than ideas they learned in school. The basis of this inference is (1) that
there were only a few students who offered such unexpected solutions, (2) they learned the
mathematical topics from different mathematics teachers at two different schools, and most
importantly, (3) these solutions did not easily flow into the solutions that the researcher

expected. The solutions were less likely to be taught in schools.

136

There are two main sections in this chapter. In the first section, a common solution
to the "Ratio" problem will be presented, then several unexpected solution to student work
on the two "Ratio" problems will be presented. In the second section, I will focus on a
special case, Bobby. All of his work on the quiz and in the interview will be analyzed in
detail. At the end of the chapter, I will summarize the unexpected solutions by students and
discuss their implications in the learning and teaching of mathematics in elementary
schools. I would argue that such unexpected solutions offer us a chance to think about how
students construct their mathematical knowledge in schools and consider what the possible
conditions are for helping them learn mathematics with meaning. And, finally, I will focus

on how teachers can build such learning environments in their classes.

Students' Typical Solutions on the Two Ratio Problems

Case 1

 

WA):

A hidden camera took the picture below during a robbery. In the picture, you can see the robber and a
service desk. The height of the desk is 3 cm, and the height of the robber is 4.4 cm. The real height of the
desk is 1.2 meter. Can you figure out the height of the robber? How?

 

 

[service deskJ T

 

 

 

 

 

 

 

 

 

 

Of the students who gave a correct answer, 35 produced the following solutions on
the two "Ratio" problems on the two different forms. On Form A, most students with a
correct answer wrote the formula (see Figure 7.2), 120 cm / 3 cm x 4.4 or 1.2 / 3 x 4.4.

Given only written formulas and correct answers given by the students, I cannot infer how

137

they thought about the concept of ratio, but this was the most typical written protocol
collected in this study. It was impossible for me to know whether they might be solving the
problems by using a cross-multiplication strategy, or some other strategy. These students
might be using the formula, Ratio x = Ratio y. Using the "Photo-Robber" problem as an
example, these students may have had the formula, 120 cm / 3 cm = y / 4.4 cm, on their

mind when they worked on the "Photo-Robber" problem.
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138

 

Case 2

 

MI

Last Sunday, you stood beside a sculpture and had your picture taken. Your height on the photo is 4 cm,
and the height of the sculpture is 10 cm. You know your real height is 1.4 meter. Can you figure out the
real height of the sculpture? How?

 

 

 

 

 

 

 

 

Figure 7.3 The Photo-You Problem on Form B

On Form B, the most typical written protocol collected was the following scanned
example (Figure 7.4). Most students wrote a similar formula, 1.4 / 4 x 10. As with Photo—
Robber, this protocol did not offer sufficient evidence to infer how these students thought
about the concept of ratio. But, these responses seemed to suggest that these students might
be solving the problem using a cross-multiplication strategy because this is the strategy

most students learn at school.

139

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Students' Unexpected Solutions on the Two Ratio Problems
Five out of the 123 students gave two kinds of unexpected solutions on the two
ratio problems. Thirty-five students solved these ratio problems using the learned
algorithms mentioned in the last section. Among the five students, two of them used an
intuitive functional model to think about ratio, and the other three used an additive model to

think about the concept.

Intuitive Function Strategy

This answer is of special interest because the students used a model that was not
directly related to the "cross—multiplication" algorithm that they probably learned in school
to think about a proportional problem. There were two students who used this idea (I have

chosen to call it, intuitive functional thinking) to think about proportion. I infer that the

140

students thought about a rule to turn 3 centimeters into 1.2 meters. They may intuitively
have used an implicit formula similar to the following:

1.2 m = fix) 3 cm
They found that the functional rule, fix), is equal to dividing 3 by 5 and multiplying times
2, (3 + 5 x 2 = 1.2), which equals 1.2. Then, they applied the function rule to 4.4 cm and
got 1.76 m, (4.4 + 5 x 2 = 1.76). Figure 7.5 shows the written response given by one
student. Both students did similar work on their quiz sheet. Both students gave the correct
unit, meter, in their answers. To me, the search for a rule was a primitive and intuitive
functional idea—searching for an algorithm that helps them turn "a" into "b." The two

students were from two different schools so it did not appear to be an instructional effect of

their classes or schools.

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Figure 7.5 An Example of Functional Thinking on Proportion

141

Karplus, Pulos, and Stage (1983a, 1983b) suggested students should develop a similar
model, as that given by these two students, when they learn about the concept, proportion.
Karplus et al., (1983a, 1983b) suggested that students should read "proportion" as a
relationship between two variables, and not as a relationship between two numbers.

Students should see "proportion" as a functional relationship between two variables.

Additive Strategy

Three students used sort of an additive, "building it up," thinking (see Figure 7.6)
to solve the ratio problem on Form B. Students reasoned that 10 cm is equal to 4 cm + 4
cm + 2 cm, and inferred that 1.4 m + 1.4 m + 1.4 m + 2 = 3.5 m, because each 4 cm was
corresponding to 1.4 m. Figure 7.6 shows that the written responses given by one student;
were also duplicated by three students who gave similar written work on their quiz sheets.
All three students who used this reasoning worked on the "Ratio-Photo" problem on Form

B. No student used this reasoning in Form A.

142

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Figure 7.6 An Example of Additive Thinking on Proportion

 

 

 

 

 

 

This "building up" model is very common among students when they develop the concept,
proportion, in their early phase (Lesh, Post, & Behr, 1988, Nesher, 1988; Peled and
Nesher, 1988; Resnick, 1989). It is quite common for students to carry their additive
knowledge to a multiplicative context. Proportional reasoning is a multiplicative
relationship between two variables, and students try to use their additive knowledge to

understand the proportional relationship between two variables.

Summary and Conclusion

There are two interesting characteristics of these unexpected solutions. First, the
intuitive functional model only happened on the ratio problem on Form A, and the additive
model only appeared on the ratio problem on Form B. So, these solutions seem to be tied

in some way to the problem context. The relationship among given numbers in the

143

situations seemed to have influenced how students constructed model(s) to conceptualize
the problem situation and its related mathematical concepts. In the "Photo-Robber"
problem, the functional model, 1.2 m =f(x) 3 cm, may be more suitable in this context.
The relationship between the represented quantities, 3 cm and 4.4 cm, makes it more
difficult for students to apply an additive model. Unlike the "Photo-You" problem, the
relationship between the two represented quantities, 4 cm and 10 cm, seemed to make it
easier for several students to construct an additive model to solve the problem because 10
cm is equal to 4 cm plus 4 cm plus 2 cm. The relationship between these two quantities is
2.5 times, 4(2.5) = 10. Second, it is interesting that all of these unexpected solutions were
given by the middle achievers in two different schools.

The main difference between these five cases and the other students who finished
the problem using a cross-multiplication algorithm is that these five special cases revealed
their understanding of the concept of proportion. Although some of their understandings of
the concept was formed in an early phase of development (additive and building-up
models), they still captured the co—variant relationship between the two variables when they
conceptualize the proportional relationship. Many mathematics educators (Lesh, Post, &
Behr, 1984; Hart, 1984) worried that students only developed the cross-multiplication
algorithm to conceptualize the concept of proportion. These educators thought that most of
these students do not develop a real understanding of the concept. They may never develop

a co-variant or functional relationship between two variables.

144

A Special Case—Bobby

Bobby was a student who came from a low SES family. He studied at a low SES
school and was assigned to a mid-to-low achiever class. He was a mid-achiever at the
school. In the interview, I asked him about his favorite school subject and he told me
science was his favorite. During the interview, Bobby was very quiet. In an informal talk
with his teacher, she told me that Bobby's behavior was not like an average grade 5 student
at times. He often behaved like a grade 2 or 3 student, and she did not find anything special
about him. But, Bobby's former teacher told his current teacher that Bobby was really
good in mathematics.

Bobby was a special case in this study because he showed a consistent strength in
solving non-routine problems. Although Bobby did not solve all the non-routine problems
correctly, he was on the right track on every problem. Bobby correctly conceptualized most
problem situations, but he made computational mistakes in several problems. Bobby only
tried 7 out of 10 quiz problems. As a result, I inferred that he did not have enough time to
finish the last 3 problems in the quiz. Of the 6 finished problems, 2 were routine and 4
were non-routine. He solved 3 of them correctly; two of them were routine and one was
non-routine. He made some computational mistakes on the other 3 problems, but he
approached the problems with the right idea. On the "Lake" and the "Glass House"
problems, he revealed his strength in being able to tackle non-routine and difficult
problems. Although he only solved the "Glass House" problem correctly, on the "Lake"
problem he displayed a correct concept about speed—the co—variant relationship between
distance and time. In the interview, Bobby solved the two assigned non-routine problems.
On the "Train" problem, he used an unexpected model to conceptualize the concept of
speed. On the "Classroom Board" problem, he knew the correct measurement of units for
pieces of paper, and for area and length. He only had difficulty explaining the remainder,

1000 cm2.

145

.‘.'I'

 

In the next section, I will present his work on the non-routine problems, one
problem at a time. Before I go into the detail about Bobby's performance, I would like to

use Table 7.1 to briefly summarize Bobby's performance on the two different sections.

Table 7.1 A Brief Summary of Bobby's Performance

 

Performance Problem Problem Type Solution
EContext Conce tu Com utational
Quiz Multiplication Routine correct correct
Unknown Digit Routine correct correct
Balance Non-routine correct partly correct
Glass House Non-routine correct correct
Coins Non-routine correct mistake
Lake Non-routine correct mistake
Interview Train Non-routine correct correct
Classroom Board Non-routine mostly correct correct

 

The Routine Problems in the Quiz
Th "Mu ' ic ti n" nd "Unkn wn ‘ 't" Problem

"Multiplication" and "Unknown Digit" are the only two routine problems for which
Bobby gave written responses. In the "Multiplication" problem, Bobby multiplied "187"

times "31" correctly, and gave "5797" as his answer.

 

We: Unknown Digit I Problem Type: routine

 

 

mm:
The 3-digit number “57?” is exactly divisible by 16. Find the value of “'2”
Answer: 6 or 576

 

 

Figure 7.7 The "Unknown Digit" Problem

In the "Unknown Digit" problem, Bobby's written responses implied that he
initially tried to divide "57" by "16," and got 9, 57 - 48 = 9. Then, Bobby seemed to try to
see whether "90" was divisible by 16. Then, he tried "96" and he found that "96" was

divisible by "16." So, he gave the answer, "576."

146

The Non-routine Problems in the Quiz
The " " obl m

Only 6 out of 123 students correctly solved the "Lake" problem. Bobby did not
finish the problem because there was no given answers on the sheet. However, Bobby was
the only student who showed in detail what he thought about the "Lake" problem logically
and mathematically. He tried to figure out how far the couple walked together in a minute.
He found the couple walked 83.33 m a minute. So, he thought he could subtract 83.33 m
from 10 km continuously to see how many times 83.33 m were used. He would then know
how many minutes it took the couple to walk around the lake-track. The walking time of
the couple also gave the dog's running time. He then multiplied the time used by the couple
times the running speed of the dog (466.6 m/min). Mr. and Mrs. Chen did not walk 10 km
individually; each of them only walked 5 km on the lake-track. 80, using 10 km of total
distance was a mistake. I do not have proof as to whether it was a conceptual or a careless
mistake. Additionally, Bobby made two computational mistakes. The first one resulted
when he subtracted 83.33 from 10000 and got 916.67. Bobby missed a digit, the correct
answer should have been 9016.67. The second one occurred when he tried to compute the
dog's running speed per minute. The correct speed should have been 466.6 m/min, not
46.6 m/min, but Bobby lost track of his decimal point. Figure 7.9 shows Bobby's work.
Bobby's strength in the problem was that he grasped the co-varying relationship between
distance and time and the co—varying relationship between the distance walked by the

couples and that of the dog. This co-varying relationship is one of the core concepts of rate.

147

 

 

WEN:

Mr. and Mrs. Chen walk with their dog, Bobby, every morning. They walk at the same speed, 5 km per
hour. The distance of the walking path around the lake is 10 km. They start and walk in opposite
directions at the starting point and meet each other midway, then go to have their breakfast. Bobby starts
his morning run with Mrs. Chen and runs toward Mr. Chen. When he meets Mr. Chen, he runs back to
Mrs. Chen. He continues like this until Mr. Chen and Mrs. Chen meet. His running speed is 28 km per
hour. How far does Bobby run, by the time the couple meet?

 

 

Mrs. Chen
and Bobby

 

 

 

 

 

Starting 7
point
Answer: 28 km

Figure '73 II'he Lake ProbTem and Its Solution on Form A

 

 

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Figure 73.9 Bobby's Work on the "Lake" Problem

The "Ba1a_nce" Problem

The "Balance" problem was one of the most difficult problems in the quiz. Only 3
students correctly solved the problem. Although there was no explicit evidence to explain
Bobby's solution, Bobby seemed to only make a computational mistake, rather than

experiencing conceptual difficulties in solving the problem.

149

 

B l Pr 1 A :
All marbles are the same weight, what are the heaviest and the lightest weight to a marble?

 

 

 

 

Answer: the li htest is 112 or 111.12 ; the heaviest is 199g
Figure 7.101 'The "Balance" Emblem and Its Solution on Form A

Quiz Form A
r armafiafi4i:amaafiafifinm£§rgI~gm
fiiqmififi?“q_. J
-wy

 

 

I: 0/ 11'3”:
3

Figure 7.11 Bobby's Work on the "Balance" Problem

150

_ . .1. '\ Jun—1-

 

Several aspects of Bobby's written responses showed that he might have only made a
computational mistake. First, Bobby used the signs, "<" and ">" correctly. This implied
that he had a clear idea of the problem, knowing that the weight of a marble can be
expressed in a range of weights. Second, Bobby added 1 gram to 110 grams and thought
that it was the lightest weight of a marble, and that adding an additional gram would make
the 9 marbles heavier than the 1000 gram weight. Using the same logic, he subtracted 1
gram from 200 gram and thought it was the heaviest weight of a marble. This subtraction
made the weight of the 5 marbles lighter than the 1000 gram weight. Adding and
subtracting gave more information about his thinking and supported the inference that he
only made a computational mistake in the problem. Conceptually, Bobby was on the right
track in thinking that he should find the range of the weight of a marble. The only mistake
he seemed to make might have been related to dividing 1000 grams by 9 marbles. The
quotient of the division should have given him 111 grams, not 110. And, since each marble
weighed 111 grams, combined, the nine of them made the total weight equal to 999 grams.
So, their total weight was less than one kilogram and the balance would not be like the
demonstrated picture on Figure 7.8. Unfortunately, he did not write down the
computational procedures on the quiz sheet, so I do not have a valid verification of this
computational mistake. However, his mathematical sign usage and his logic of using
addition and subtraction to the quotients again revealed that he might have only made a

computational mistake.

Th "G a H use" le

The "Glass House" problem was the most difficult problem on Form A of the quiz.
None of the students gave a correct solution to "Part a," listing all the possible rectangles
with different perimeters and with the same area of 24 m2. Only 2 out of 123 students gave

the correct answer on "Part b," the floor plan that costs the least to build the glass house.

151

* “W. In

 

WW:
Your school is planning to build a glass-house for the science classes. The area of the floor must be 24
square meters. The building will be 3 meters in height. The builder estimates the house costs $1200 for
every 1 meter of the perimeter.

a. List out all the possible lengths, widths, perimeters, areas, and the cost of walls for different rectangular
shapes. (You only need to consider rectangle with length and width equal to whole numbers of meter)

 

 

 

 

 

 

 

 

 

 

 

Answer: -
Width Length Perimeter Area Cost of the wall I
1 24 50 24 60000
2 12 28 24 33600
3 8 22 24 26400
4 6 20 24 24000

 

 

 

 

b. Based on the cost of walls, which rectangular shape would cost least to build?

 

 

Answer: 4 mx 6m
Figure 7.12 The "Glass-House" Problem and Its Solution on Form A

In Part a, most students did not understand what was being asked in the problem. These

students filled in some numbers in the table, but the numbers did not make sense within the

context of the problem. For instance, one student wrote 5 meters for width, 6 meters for

length, 22 meters for perimeter, 30 square meter for area, and $36,000 for the cost. More

than 20% of the students gave up on this problem. In part b, more than 65% of the students

did not give any answer, and about 25% of the students gave non-contextual answers.

Most of the cases with non-contextual answers showed that students might have had

difficulty understanding the context of the problem, but might experience no difficulty

memorizing the formula for area and perimeter. Here are the two typical examples of

students' work:

152

 

Table 7.2 The First Case's Response on the "Glass House" Problem

 

Width Length Perimeter Area Cost of the wall
4 meter 2 meter 12 meter 8 meter2 $400
2 meter 1 meter 6 meter 4 meter2 $200

 

Table 7.3 The Second Case's Response on the "Glass House" Problem
Width Length Perimeter Area Cost of the wall
24 3 54 72 $86400

 

The first case revealed that the student did not understand the context of the problem.
Her/his response to the "Area" column revealed her/his lack of understanding of the
problem. The area should be a constant quantity, 24 meterz, but s/he did not complete the
cells with the constant quantity. Also, the student seemed to have no idea about how s/he
could figure out the cost of the wall. The numbers s/he added to the cells did not make any
contextual sense. However, s/he do not have any difficulty figuring the area and perimeter
when s/he knew the width and length. The values given in the Perimeter and Area columns
corresponded to the values of width and length. The second case revealed similar difficulty
by that student. The student had no difficulty in using the formula for area and perimeter,
as evidenced by the numbers that were filled in the "Width," "Length," "Perimeter," and
"Area" columns. However, the numerals s/he completed in the cells did not make any
contextual sense. For example, the quantities filled in the Width and the Length columns
could not result in an area of 24 metersz. This evidence revealed that s/he did not
understand the problem context. Also, the quantity in the "Cost of the Wall" did not make
any contextual sense. The answer, $86400 [$1200 x 72], was the product of the area and
the building cost, $1200 per meter.

For Part a, Bobby wrote down the information for three different rectangular
shapes—width, length, perimeter, area, and the cost of the wall. The only missing

rectangular shape was the one with a 2 meter width and a 12 meter length. The information

153

did not help me make an inference as to whether he understood that a 2 meter by 12 meter
rectangle was also a candidate. I do not know whether be mentally missed this possibility,
or he intentionally skipped this rectangular shape because he knew this shape could not
offer the lowest building cost. Bobby also made a computational mistake on Part a. In the 3
meter by 8 meter rectangle row, he wrote 24 meters in the perimeter column. In part b,
Bobby gave a numerically correct answer, he wrote width was 4 (without the unit label,
"meter") and length was 6 (also without the unit label).

Bobby's response revealed his understanding of the problem situation. Most
students did not reveal their understanding of the problem. Bobby also showed that he had

no difficulty in distinguishing between the two geometrical concepts, area and perimeter.

154

Quiz Form A

4 - fitéflfifiiilifiR-E-‘Efifi/i ’ “35$: E1 realm ’ Bifiliéfififl‘
rfifiiZ4-‘I‘7‘5‘IAR ’ arena/amen ° Itfifitfiiffii-fifile‘flr‘ffi
IAK‘I‘Hfi ’ E-fiifi $120072. 0

A) W‘Tfl'fi‘i‘lzzF‘lfiz‘r‘lfiWfifi'fififififi/fififitfifi ’ fig‘imfiififllrh
saaxeaa~r~mr~aa~zaaso ;.
(¢R%%#&&fifiéfifififififli) Féfif

 

 

 

 

 

 

 

 

 

 

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m #aeaas,flfiwaaxeaaaaaaaaafiaao

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Figure 7.13 Bobby's Work on the "Glass House" Problem

N I
1‘ ‘

155

The "going" Emblem

Only 7 out of 123 students were able to solve the "Coins" problem. Bobby did not
produce a correct answer on this problem. There were two constraints to this problem. The
first one was that the total should be more than 10 dollars. The second one was that no
subset of the coins should be equal to 10 dollars. Most students violated the two constraints
simultaneously, their answers were coin-combinations equal to 10 dollars. Bobby's
answer, one 5-dollar coin, three 2-dollar coins, one 50-cent coin, three 20-cent coins, and
one 10-cent coin, violated the second constraint. The coin-combination given by Bobby
was more than 10 dollars, but one combination could be built to equal 10 dollars. From the
written protocol, it was quite difficult to infer what the cause was of Bobby's mistake. On
the one hand, he might have made a computational error. He did not discover that one of
his coin combinations was equal to 10 dollars. On the other hand, he might not have
understood the second constraint for the problem. Bobby only violated one constraint for

the problem, rather than two.

156

 

 

MW:

You know you have more than ten dollars in you pocket (all are coins, $5, $2, 50 ¢, 20¢, or 10¢) but you

find you are unable to give someone a change for a ten dollar note. List out the combination of coins that

you have.

Answer: any combination with all different coins, and the combination is not equal to $10 and is more
than $10

Possible Combinations are:

 

Type\Quantity option 1 option2 option3
s 5 l 1 1
$2 3 4 4
50 cent 1 l 1
20 cent 1 l l
10 cent 1 l 2

 

 

 

Figure 7.14 The "Coin" Problem and Its Solution on Form A

Quiz Form A

& fi~k,mwm&¢miahmasfilmaloawfiaflasa
rmefilfirlra$5~m~m¢~mcxmeesfilfirl’mm
are{Mk}maaflflmavM£xsaema’0flawam&&%
mififilfifil°$£9%Wifli}%mio

fifltfifififlfléfi$flfi$fi[fifi}sfieymo

Figure 7.15 Bobby's Work on the "Coins" Problem

Non-routine Problems in the Interview
1 he " 1 rain" Problem

Most students who solved this problem successfully found the speed per minute by
dividing 66 km by 60 minutes. Then, they multiplied 1.1 km per minute by 2.5 minutes.
Finally, they multiplied 2.75 km by 2 because 2 trains were traveling at 2.75 km. This

157

resulted in the correct answer, 5.5 km. Bobby solved this problem with a novel strategy.
Although he did not give a good explanation (the verbal protocol he reported) as to why he
did it, he seemed to have a special implicit model (Figure 7.16) for interpreting the concept
of speed. Bobby might have a co-varying relationship model between time (minutes) and
distance (km) in his mind. According to the verbal protocol, Bobby thought first about the
ratio relationship between 2 minutes and 30 seconds (2.5 minutes) and 60 minutes (1
hour). He partitioned 60 minutes into 2.5 minute segments. Originally, he thought that the
quotient (60 minutes -:- 2.5 minutes) could be divided by 66 km, and this would give him
the distance run by a train. He then reasoned that by doubling it, he would be able to
determine the distance between two trains. His formula, 60 + 2.5 + 66 + 60 + 2.5 + 66,
confirmed his original thinking. However, he changed his mind when he was asked about
the number, 66. He immediately reported that he made a mistake on the formula. He wrote
a new formula,

66 + (60 -:- 2.5) + 66 + (60 + 2.5).
From the new formula, I inferred that Bobby thought he had some number of partitions (in
this case, 24) in the time ratio by dividing 60 minutes by 2.5 minutes, he then reasoned that
he could partition the 66 km into the same number of partitions. Each partition in the
distance ratio represents 2.75 km. Moreover, the central idea of his thinking was that each
partition of the two ratio lines (time ratio and distance ratio) corresponded to each other.
So, he knew that a train could run 2.75 km in 2.5 minutes. Doubling the distance run by a
train gave him the distance between two trains. Figure 7.16 shows diagrammatically how I
interpreted Bobby's use of ratio models (time ratio and distance ratio) to think about the

concept of rate (speed).

158

 

 

‘ 60 minutes >

2.5 minutes

i i 4' minutes

60 minutes can be divided by 2.5 minutes with 24 sectors

 

 

‘ 66km f

l l l
hf 1 km

2.75 km corresponds to the 2.5 minutes that the train
runs.

Figure 7.16 Covarying Relationship Between Time and Distance

The following section of the interview protocol gives us more information about

how Bobby thought about the problem and the concept of speed. He spent quite a long time

thinking about the problem situation before he wrote his formula. His extended thinking on

this problem and his novel approach might explain why he did not have enough time to

finish all the quiz problems.

(Bobby thought about the problem for about a minute, after reading)
Interviewer: What are you thinking?

Bobby: Think about how to write a formula.

(Bobby did not say anything for more than a minute)
Interviewer: Then, what do you think first?

Bobby: First, think about the numbers.

(Bobby did not say anything for about 1.5 minutes)
Bobby: I think two numbers at the same time.
Bobby: What is the relation?

(Bobby did not say anything for minutes)

Bobby: I am thinking about 60, now.

Interviewer: What is it, 60?

159

Bobby: I think about 60 minutes and 2 minutes and 30 seconds, their relationship.

Bobby's talk about the 2 numbers relationship (60 divided by 2.5) showed that he tried to
think about the time ratio—the ratio relationship between one hour and 2.5 minutes. He
transformed one hour to 60 minutes and thought about the ratio relationship between 60

minutes and 2.5 minutes.

(For about 2 minutes, Bobby did not say anything)
Interviewer: What are you thinking now?

Bobby: I think about the formula, but not sure.

Interviewer: Can you tell me more about your unsure formula?
Bobby: Can I write it down?

(Bobby wrote, 60 + 2.5 + 66 + 60 -:- 2.5 + 66)

The form of this expression (quantity + quantity + quantity) and the time it took Bobby to
produce it strongly suggests that he did not retrieve it from memory. He chose to elaborate

his thinking gradually.

Interviewer: What is 60, here?

Bobby: Minutes in an hour.

Interviewer: What is 2.5?

Bobby: 2 minutes and 30 seconds.

Interviewer: Why do you divide 60 by 2.5?

Bobby: I don't know.

Interviewer: Why do you divide 66 here?

Bobby: Oh! I made a mistake.

(Bobby rewrote his formula, 66 -:- (60 + 2.5) + 66 -:— (60 + 25 )

160

 

Interviewer: What is the addition?

Bobby: Distance between two trains.

Bobby could not articulate the ratio concept when he was asked why he did a division
between 60 and 2.5. He realized that the formula was not correct when the interviewer
asked him about the number "66" in the formula. Although he did not report why he
changed his mind to create a new formula, his instant response about the mistake he made
in the original formula revealed that he had a clear idea of what he was doing. In general,
there was a watershed point where Bobby behaved quite differently in two stages. When he
generated his first formula, he spent a lot of time thinking about the problem. He needed
one or two minutes to think before he reported his ideas to the interviewer. However, after
Bobby was better able to conceptualize the situation (the quantities and how they relate), he
replied to the interviewer's questions instantly. That pattern revealed that Bobby could
answer the questions easily when he built a mental model to represent or to understand a

problem situation.

 

Two trains (A and B) are starting out in two different directions at the same time with the
same speed, 66 km per hour, from Mongkok Station. 2 minutes and 30 second later, how
far apart distance are the trains?

 

lTrain B l

‘ l ’

IMongkok Station I

 

 

 

 

fTrain]
(NR)

Figure 7.17 The "Train" Problem in the Interview

 

161

 

 

I J

i .. . . I .
J1 .9 " T fit? ' . ' .
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.., ,. . r r .0 ..,...»»:~—+z- I.
uw‘

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x l .
r
i

.l l L,” I l ' -,_ r“ I" ' (”7 .., ”I gr- l
L" IMP/l Db» 16b- 425/
Figure 7.18 Bobby's Work on the "Train" Problem

I. 1'
a"; 1 a "’-
X- L‘ I

If we compare the two problems involving speed (the Train and the Lake) solved by
Bobby, we might find a common mental model about speed that Bobby has created in his
mind, although he did not report the model verbally in the quiz or the interview. In the
"Lake" problem, Bobby might also have been thinking about distance and time (Figure
7.19) using a model he created in the "Train" problem. According to what he wrote down
on the quiz, he tried to build two corresponding lines. He used the known quantity, 5 km
per hour, to figure out how many minutes were needed by the couple to finish their walk
on the trail. He subtracted 83.3 meters continuously from 10 km. He then multiplied this
quantity (minutes used by the couple in walking) times the dog's running speed of 466.6
meters per minute to find out how far the dog ran in the same amount of time walked by the
couple. This co-varying relationship between distance and time appears to be the central
idea in Bobby's work in the two problems. Although Bobby did not explain directly how

he conceptualized these problems, his performance supports this interpretation.

162

 

 

‘ 60 minutes 5

2.5 minutes

} i : minutes

60 minutes den be divided by 2.5 minutes with 24 sectors

 

 

 

 

 

 

‘ 66 km 5
2.75 km
L r J
I l 1 "m

2.75 km corresponds to the 2.5 minutes that the train runs.

Speed Model for the "Train" Problem

 

 

 

 

 

 

‘ 10km >

ll {km

 

83.3 meter is the distance walked by the couples in a minute

4 ? +

466.6 m
l l
l l

 

 

{km

466.6 meter is the distance run by the dog in a minute

Speed Model for the "Lake" Problem

 

 

 

Figure 7.19 Two Similar Speed Models for Two Speed Problems

Th "Classr mB ar " Pr blem

In solving this problem, students needed to go through three main procedures, (1)
find the area of each paper sheet, (2) divide the area of the board by the area of each paper

sheet, (3) round up and subtract the remainders from the last whole paper sheet to find the

163

. Tl

' ‘

 

waste. Bobby solved this problems like most of the high achievers. First, he thought that
he needed to know how many pieces of poster paper were needed for covering the board
when there were two different kinds of poster paper. He figured out he could find out how
many pieces were needed by dividing the board area by the area of two different kinds of
paper [2.5 m2 -:- (30 x 50) cm2 and 2.5 m2 -:- (40 x 50) cmz]. Second, he tried to find the
area of the paper and the board. Then, he thought that by subtracting the number of pieces
of paper (17 pieces - 16.666 pieces and 13 pieces - 12.5 pieces) needed from the quotient
he got by dividing the board area by the area of the papers he would know which kind of

poster paper wasted less.

 

You and 3 other classmates are assigned to design your classroom board this semester. The
board is a rectangular shape, 2.5 meters by 1 meter. First you think about the background.

1) You decide the background color (light blue), and find there are two sizes of that color
paper. One is 30 cm by 50 cm, and another is 40 cm by 50 cm. You want to buy the size
that waste the least paper

a) Which will you buy?

b) How many pieces do you need to buy to cover the whole board?

[Classroom Board]
(NR)

 

 

Figure 7.20 The "Classroom Board" Problem in the Interview

164

‘5!

t 3.9.4.,

 

Problem

"Classroom Board"

.21 Bobby's Work on the

Figure 7

165

The strengths that Bobby revealed in this problem were his understandings of the
problem context, his understanding of units (m2, cm2, and, meter, cm) and their
relationship, and his understanding of the quotient provided by dividing the area of the
board by the areas of different kinds of poster paper. Bobby's solution was similar in some
respects to his solution to the "Train" problem. He spent quite a long time figuring out the
formulas for representing the situations at the beginning. The formulas he wrote down (2.5
m2 + (30 x 50) cm2 and 2.5 m2 -:- (40 x 50) cm2) revealed how he conceptualized the 2
problem. And, the algorithm he wrote, 2.5 x 100 x 100 = 25000 cmz, revealed his
understanding about the relationship between square meters and square centimeters, and his

understanding of 2 dimensional measurement. Subtracting the quantity of paper (17 and 13

 

 

pieces) he needed to buy from the quotient (16.6 and 12.5 pieces) provided by dividing the
area of board and the areas of different kinds of paper revealed his understanding of the
representation of the quotient and his understanding of the meaning of waste (after the
prompt given by the interviewer). The only weakness that Bobby revealed was his
interpretation on the remainder, 1000. By dividing the area of the board by the area of a
piece of paper, the net result was 2 quotients—16.6 pieces and 12.5 pieces, and two
remainders, both of which are 1000 cm2. Bobby reported that he did not know what the
remainder represented.

The following verbal protocol shows how Bobby solved the problem. First, we
will see that Bobby spent quite a while conceptualizing the problem and trying to represent

the problem situation in two formulas.

Bobby: First, I think how I can build a formula.
Bobby: The numbers.
(For about a minute, he did not say anything.)

Interviewer: What are you thinking?

Bobby: Can I write?

166

Bobby wrote, 2.5 + (30 X 50) and 2.5 + (40 X 50)

Interviewer: What do you get from them?

Bobby changed his formulas to, 2.5m -:- (30 X 50) cm and 2.5m + (40 X 50) cm

Interviewer: Do you know that all of them are areas? What unit for area?

( I introduced the idea, area here, and he did not say anything about area previously.

So, I inferred that he might not have the correct unit for thinking about the problem at

that time.)

Bobby: I forgot.

Bobby changed his formulas to, 2.5 m2 + (30 X 50) cm2 and 2.5 m2 + (40 X 50) cm2

Interviewer: Can you tell me what 2.5 m2 is?

Bobby: The area of the board.

Interviewer: So, the others are the areas of the paper?

Interviewer: Can you tell what you get when you finish all the computation? I mean
when you finish the computation of your formula, 2.5 m2 -:- (30 X 50) cm2?

Bobby: cm.

About 10 seconds.

Bobby: Pieces.

Bobby began to write on the working sheet. He wrote,

40 30
x 50 x 50

2000 1500

 

 

100
x 2.5
250
Interviewer: Can you tell me what it is?
Interviewer pointed at 100 X 2.5 = 250
Bobby: Area.
(At that time, Bobby was still working on 100 X 2.5, but, the interviewer did not

notice.)

167

Interviewer: Do you think multiplying by 100 will give you the area?

Bobby continued his work, he wrote,
100
X 2.5

250 cm
100

25000 cm sq

Interviewer: Oh! I made a mistake, you wanted to do the transformation here.

i
3
Transforming the unit from meters to centimeters revealed that Bobby understood the units '
very well. This suggested that he simply forgot to add the units to the two formulas he
wrote at the beginning. Many middle and low achieving students confused linear and l

 

square measurement and their units. Some of them were confused with the transformation
relationship between meter and cm, and some of them were confused by the units for linear
measurement and the 2-dimensional measurement. Bobby's performance revealed his firm

understanding about the unit transformation and unit measurement.

Bobby continued to work and wrote,

 

 

 

16.666
1500 25000
1500
—— 1500
10000 15009 6
9000
1000 13500 9000

Interviewer: What are you thinking? Why do you stop when you get the 1000?
Bobby: Checking.

Bobby continued his work, and wrote,

12.5
2000 I 25000
2000
5000
4000
10000
10000

 

 

Interviewer: What are you thinking now?

168

Bobby: Which one saves more?

Interviewer: How do you think which one saves more?

Bobby: From different views.

Interviewer: Can you tell me about your views?

Bobby: Think what it means, "do not waste any paper?"

Interviewer: Can you tell me what you think about, waste, what is waste?
Bobby: Si (a Cantonese dialect mean waste).

Interviewer: What does s1 mean?

Bobby: It is not used.

 

Interviewer: Not used.

Bobby: And, buy more.

Interviewer: Buy more. Let's see, how many pieces you buy in this size?

Bobby: 17 pieces.

Interviewer: This one.

Bobby: 13 pieces.

Interviewer: You know, you need to buy more of this one (17 pieces). But, if you use
most of them and very less is left. 0r say, the first 16 or the first 12 pieces

put on the board. Do you waste any of those I 6 and 12 pieces?

Most students were confused about the meaning of "waste;" they thought that if more was
purchased, then more was wasted. Bobby also interpreted "waste" in that way, so he
thought the class would waste more when they bought more pieces. I tried to clarify what I

meant by "waste" to him in order to help him re-think about how to solve the problem.

Bobby: No.
Interviewer: So, how many pieces are used that do not tell you about the waste?

Bobby: Yes.

169

Interviewer: Now, how do you think?

Bobby: Use this (pointed at I 7 pieces). Subtract and get the approximation.

When he used the word, approximation, Bobby might have been referring to the result of
subtracting the quotient by dividing of the area of the board and the area of the poster paper
(16.6 and 12.5 pieces) from the whole numbers of paper (17 and 13 pieces) that were
purchased. This was a correct way to infer which kind of poster paper wasted less. The

smaller the difference, the smaller the amount that was wasted. Bobby's talk of subtraction

 

and approximation revealed his understanding of what the quotient represented in that
situation. However, since I provided direct support for this way of thinking about "waste," 7

I cannot claim that he reached this understanding completely on his own.

Interviewer: Get the approximation, hmm. Then, which one saves more?

Bobby pointed at 16.666.

Interviewer: What is left-over from the subtraction? The approximation?

Bobby: The paper that was wasted.

Interviewer: 0k! Bobby, do you think about this (the remainder 1000). What is it?
Bobby: No.

Interviewer: Do you draw picture when you solve a problem like this one?

Bobby: Seldom.

Although Bobby had no problem making sense of the quotients, he could not conceptualize
the remainder. He could not figure out what 1000 as the remainder of 25000/1500
represented. This difficulty was very common among the students who participated in the
interview. This result might reveal that Bobby and many students did not have a pictorial
representation for the problem situation. Most of them seemed to rely on logical

propositions to help them to infer the answer. But Bobby and some students had a sensible

170

approach: The quotients (2.5 m2 + 30 cm x 50 cm and 2.5 m2 + 40 cm X 50 cm) were
equal to how many pieces were used. Using this argument, they inferred that the decimals
represented how large the partition was used on the last piece of paper. So, they figured out

which poster wasted less, the one with a larger decimal.

Summary and Implications
First, I would like to clarify why I considered the solutions from Bobby and other
students to be novel representatives of students' strengths. I do not deny that students who

solved the problems with traditional and school-taught methods understood mathematical

 

concepts, such as speed, ratio, and proportion. However, I believe that students who can
think hard about the situation and construct their own models display greater understanding
of the concepts than other students. If we agree that understanding is important to the
quantity and quality of different perspectives in a situation, students who develop novel
solutions may develop a deeper understanding of the specific concept.

There are several characteristics of novel solutions worth mentioning. First, on the
most general level, all students who provided those solutions were middle achievers, based
on their school's mathematics performances. Second, the students used their knowledge to
develop such solutions in constructive ways. I have talked about the reasons why I think
these solutions were not learned from traditional schooling. Only a few students solved the
problems using such unexpected solutions and the novel solutions were distributed across
different classes and different schools. This distribution pattern suggests that it was highly
unlikely that students learned such models or rules in school. Third, the solutions were
related to students' prior knowledge. The additive model and the intuitive functional model
of proportional reasoning were typical examples. In the case of the additive model, students
used their knowledge of addition to think about the problem situation. They found that
there was a replicate additive relationship between two entities (10 = 4 + 4 +2), the image

height of the sculpture and the image height of him/her on the photograph. Fourth, the

171

functional model used by several students to solve the problems was intuitive. They had
not learned the function concept in their classrooms, but they developed an intuition about
the concept through their daily experiences. I think students have a lot of experiences that
foster the development of such intuitive ideas. The most obvious case is the vending
machines they use every day and every where to buy snacks and tickets. Finally, students
who constructed the novel solutions were trying to conceptualize the problem situation
rather than just focusing on the numbers or key words in the problems when they solved
them. I did not interview all the students who generated novel solutions, but Bobby's case
revealed that he spent quite a good deal of time conceptualizing the problem situation before
providing a solution.

Based on these characteristics, I propose some implications for teaching. If we
agree that mathematical knowledge in elementary school can be constructive, by which I
mean students can use their daily knowledge and experiences to generate model(s) or rules
for the mathematical concepts they learn at schools, we need to rethink how we teach
mathematics in elementary school. This study shows that some students can construct their
own models to solve proportional problems; they do not need to rely on algorithms taught
by the teachers in school. If we agree that this statement about the constructive nature of
students' arguments are advantageous, we can see three implications for teaching. First,
listening is important in classroom teaching. If students construct their own models for
understanding and interpreting the problem situations, their models may be quite different
from the models their teacher had in mind. So, teachers need listening skills to hear what
students are saying. This is quite different from standard approaches. They should not
deny the non-standard solutions, because they do not understand them. These listening
skills should be based on the findings of developmental studies about mathematics
learning. Second, teachers should provide enough time for students to think about
problems. Based on Bobby's performance, we see that students need time to think and

conceptualize problems before they find constructive solutions. Teachers should not rush

172

A. . . \"’.V'Ll"
I'

through solutions or strategies quickly in order to move on. If classroom work and
discussion are rushed, teachers will eliminate chances for their students to conceptualize
and construct their own understanding of the concepts embedded in the problems. Third,
evaluation should not be based solely on standard solution and final answers. If teachers
want to help their students develop understanding or overcome conceptual obstacles, they
need to hear and understand the procedures given by students. Also, they need to explore
the non-standard solutions given by some students. How do the students construct those
non-standard solutions? Do those solutions show a deep understanding of basic concepts?
Or, Are they flawed and/or do they represent obstacles for future development? For
example, the additive model for proportion may not be efficient and generalizable for
students in the long-run. The model may only be applicable in specific problem situations
where numbers can be easily related additively. Likewise, the intuitive function model may
be inefficient for students to use to solve all proportion problems because the form of the

function may be difficult to determine by examining the input and output values.

173

Chapter 8

CONCLUSION

In this final chapter, I return to my research questions, consider how my results
address them, and discuss some issues related to the findings of this study. I will compare
the performances on two different kinds of problems, routine and non-routine. In
particular, I will focus on exploring the possible difficulties that the students experienced
on three non-routine problems, the "Photo-Robber," "Photo-You," and "Extended
Square". I selected these 3 problems because they were where the students provided the
most complete data on their reasoning, allowing me to consider and weigh the effects of
different sources of difficulty. I am concerned with these possible source(s) of difficulty.
Were their poor performances on the non-routine quiz problems related more to conceptual,
computational, linguistic, or developmental difficulties? I will try to use the collect data to
interpret their difficulties. I will also consider whether or not the students' social

background influenced their performances on the quiz.

In this section, I would like to answer my research questions by briefly
summarizing the results which were presented in Chapter 5 to 7. Then, I will present a
more detailed interpretation of the students' performance in this study.

1. Do Hong Kong students have difficulties solving non-routine mathematical
problems?

In general, the students from both schools performed well on the routine problems,
but poorly on the non-routine problems. Tables 8.1 and 8.2 reveal their difficulties solving
the non-routine problems. The difficulty rankings (column 2) were based on the frequency
of correct and numerically correct responses. In determining these rankings, the first
criterion was the correct frequency. If the correct frequencies on two problems were the

same, I considered the numerically correct frequency. If two problems had the same correct

174

frequency, the one with more numerical correct responses had a lower difficulty rank. In
column 1, [N] indicates non-routine problems, and [R] indicates routine problems.

Table 8.1 Performance on Routine and Non-Routine Quiz Problems
Form A (N=58)

 

 

Frequency,

Problem Difficulty Percent Correct Correct and

Rank (Frequency) Numerically

(Total=12) Correct

Glass House (Part A)[N] 12 0% (0) 0
Balance [N] 11 3.4% (2) 3
Glass House (Part B)[N] 10 3.4% (2) 4
Lake [N] 9 10.3% (6) 7
Extended Square (Part B)[N] 8 10.3% (6) 9
Coins [N] 7 12.1% (7) 7
Extended Square (Part C)[N] 6 19% (11) 11
Photo-Robber [N] 5 19% (1 l) 14
Saving [R] 4 43.1% (25) 38
Unknown Digit [R] 3 74.1% (43) 43
Rectangle Width [R] 2 79.3% (46) 52
Multiplication [R] 1 84.5% (49) 49

 

On Form A, the non-routine problems were the most difficult. In addition, there was a
large performance difference between the non-routine problems (considered as a group)
and routine problems. Among the non—routine problems, the highest percentage correct was
19%, where the lowest percentage correct among the routine problems was 43%. Clearly,
on Form A the non-routine problems were much more difficult for the 5th graders than the

routine problems.

175

Table 8.2 Performance on Routine and Non-Routine Quiz Problems
Form B (N=65)

 

 

Frequency,

Problem Difficulty Percent Correct Correct and

Rank (Frequency) Numerically

(Total=14) Correct

Balance [N] 14 1.5% (1) 2
Lake 13 13.8% (9) 9
Number Pattern (Part D)[N] 12 23.1% (15) 15
Extended Square (Part B)[N] 11 23.1% (15) 16
Extended Square (Part C)[N] 10 23.1% (15) 17
Number Pattern (Part C)[N] 9 41.5% (27) 27
Photo-You [N] 8 44.6% (29) 31
Number Pattern (Part B)[N] 7 55.4% (36) 36
Magazine [R] 6 55.4% (36) 47
Cake (Part B) [N] 5 63.1% (41) 45
Unknown Digit [R] 4 69.2% (45) 45
Rectangle Width [R] 3 74.4% (49) 49
Cake (Part A) [N] 2 76.9% (50) 51
Multiplication [R] 1 81.5% (53) 53

 

On Form B, the students also found the non-routine problems more difficult. The only two
exceptions were the "Cake" and "Number Pattern (Part B)" problems. More than an half of
the students solved these problems correctly. However, the "Cake" problem was actually
more “routine” than the other non-routine problems. Although it had a very complicated
story structure and a lot of words, the problem situation was familiar to the students. It
asked the students to partition the whole cake into pieces. This multiplicative concept of
division, partitioning, was a familiar story problem to the students. On the other non-
routine problems, the students' performances revealed a difficulty pattern similar to that on
Form A. The non-routine problems had the highest difficulty rankings, but—on
average—were easier for students than the Form B non-routine problems. One possible
reason for this overall difference is that 3 non-routine problems—"Cake," "Number
Pattern," and "Photo-You"——-were comparatively easier for students than their

counterparts—"Glass House," "Coins," and "Photo-Robber"—on Form A.. “Cake”

176

presented a familiar situation, and “Number Pattern” did not require thinking about as many
possibilities as “Coins.” It is unclear to me why “Photo-Robber” was more difficult than
“Photo-You.”

All these data revealed that the students had difficulties solving the non-routine
problems. Form the Table 8.1 and 8.2, we can see that the non-routine problems earned the
lower correct percentages and frequencies. Why did the students find the non-routine

problem more difficult? I will try to investigate this question in the next section.

2. What are the general characteristics of their difficulties—conceptual,
computational, linguistic, or developmental?

In general, the students' performances revealed their difficulties solving the non-
routine problems. In this section, I will consider some possible sources of their difficulties.
I consider four general factors—conceptual, computational, linguistic, and developmental
challenges. Although the students' incorrect answers can be interpreted in different ways,
and different sources of difficulty could have their impact on student's mistakes, I will try
to use the data to make my argument—in general, the conceptual difficulties with several
mathematical concepts had a strong impact on performance. However, I do not deny that
other factors had their effects.

Before I start to discuss the effects of four factors, I would like to introduce them
briefly. I hope that this introduction will provide a common ground to explore and
investigate the sources of the students' difficulties.

1. Conceptual Factor: Different psychological traditions—Empiricist, Rationalist,
and Sociohistoric—have their own interpretation to the psychological construct,
conception (Case, 1998). In my discussion, I refer the conceptual sources as
conceptual entities—abstract rules and mental models—are used by children to
interpret their surroundings and solve problems. For example, Bobby (Chapter

7), seemed to acquire an abstract rule or mental model about the mathematical

177

concept, speed. He seemed to see speed as a co-varying relationship between
two variables, distance and time. This co—varying relationship is a conceptual
entity that a student can use to solve the relative unfamiliar non-routine problems.
Students can solve non-routine problems if they can correctly conceptualize the
mathematical concepts which they study in school. Though they might not
conceptualize them in the same way that those concepts are taught. The models
or rules for area, perimeter, and proportion are considered under this dimension.

2. Computational Factor: This is the most straightforward factor. It refers to
students' ability to correctly complete the four arithmetic operations—addition,
subtraction, multiplication, and division—on the numbers given in the problems.
In other words, could the children correctly manipulate the numbers with
different operations? I want to consider whether the students did not develop this
competence well, so they performed poorly in the quiz.

3. Linguistic Factor: This is a more complicated dimension, combining 3 different,
but related elements. The first one is the number of words in the problem text.
The students had more experiences with simple, one or two sentence story
problems. They seldom solved problems with long and complicated stories. The
second is the familiarity of the story structure. The routine problems were taken
directly from the students’ textbooks, and some non-routine problems (e.g.,
“Cake”) were adapted from textbook problems. These problems were more
likely to be familiar to students from their school work. The third is the number
of quantities (variables) in the problem. This element lies in the gray area
between the conceptual and linguistic dimensions. More quantities in a problem
meant higher cognitive load so it is related to the cognitive domain. I include
number of quantities under the linguistic factor because I want to consider the

effect of specific multiplicative concepts under the conceptual factor.

178

4. Developmental Factor: Developmental psychologists, educators, and cuniculum
designers (Gelman & Brown, 1986; Mumbauer & Odom, 1967; Olser &
Kofsky, 1966; Renninger, 1998) have focused on the match between children's
development and the materials taught in school. A common consideration has
been the match between the cognitive/psychological maturity of children and the
learning tasks that they work on in school. Have children matured enough
(psychologically) to learn the abstract knowledge and apply the knowledge? The
developmental factor refers to the students' psychological age when they
participated in this study. I consider whether the students were psychologically
ready to carry the abstract knowledge (mathematical concepts) that they learned

in class to apply to unfamiliar situations (non-routine problems).

First, I would like to discuss the developmental dimension. The weak performance
of these students could be taken as a developmental effect. Maybe teachers cannot do
anything for students, because they are not ready to abstract the knowledge they learned
and transfer it to a totally unfamiliar situation or new problem with too many new
conceptual dimensions. I think we do not need to worry too much about the developmental
factor among the students, except two cases who were identified as the mildly mentally
retarded. According to the studies done by Mumbauer and Odom (1967), Olser and Kofsky
(1966), students older the 7 years old should be capable to transfer their knowledge to
different situations if they understand the lmowledge (or rules) they learned. Only children
less than 7 years old have problem to abstract the rule(s) from what they learned and apply
these rules to different but applicable situations. All subjects in this study are older than 10
years old, I think they are developmentally ready for solving the non-routine problems.

Second, I would like to explore the conceptual and computational factors together. I
did conceptual analysis on 2 non-routine problems—"Ratio" (Photo-Robber and Photo-

You), "Extended Square." I chose these problems because they were the only problems

179

that offer me enough data to compare the relative effects of conceptual and computational
difficulty. I will examine the students' performances on the "Ratio" and "Extended Square"
problems in next paragraph.

On the two "Ratio" problems, more than 60% of the students solved the problems
incorrectly on Form A, and 44% on Form B (Table 8.3). Among the 67 students who gave
the incorrect answer to the problems, at least 53 of them made one or more conceptual
mistake. Only 14 incorrect responses were due to computational errors.

Table 8.3 Incorrect Responses Related to Conceptual and Computational
Errors on the Two Ratio Problems

MinimumJIncorrect Incorrect

 

Frequency, Response Response
Form Problem Incorrect Frequency related Frequency related
(N=) Response to Conceptual to Computational
Mistakes Errors
A Photo-Robber 38 28 10
(58)
B Photo-You 29 25 4
(65)

 

On the "Extended Square" problems, 72.4% of the students solved the problem
incorrectly in Part B, and 69% in Part C on Form A (Table 8.4). In Part B, among the 42
students who gave an incorrect answer, at least 30 of them made one or more conceptual
mistake. In Part C, among the 40 students who gave an incorrect answer, at least 37 of
them made one or more conceptual mistake. On Form B, 67.7% of the students solved the
problem incorrectly in Part B, and 58.5% in Part C. In Part B, among the 44 students who
gave an incorrect answer, at least 33 of them made one or more conceptual mistake. In Part
C, among the 38 students who gave an incorrect answer, at least 32 of them made one or

more conceptual mistake.

180

Table 8.4 Incorrect Responses Related to Conceptual and Computational
Errors on the Extended Square Problems

Minimum Incorrect Incorrect

 

 

Frequency, Response Response
Form Problem Incorrect Frequency related Frequency related
(N=) Response to Conceptual to Computational
Mistakes Errors
A Extended Square 42 30 12
(58) (Part 13)
Extended Square 40 37 3
(Part C)
B Extended Square 44 33 11
(65) (Part B)
Extended Square 38 32 6
(Part C)

 

On these two problems, students' incorrect answers were clearly related to
conceptual mistakes they made. The main factor for their poor performance was more likely
related to their understanding of the main concepts embedded in the problems—area and
proportionality. Many students thought about ratio with the additive model. They seemed to
carry an inefficient or inappropriate abstract rule to the problem situations. On the problem
related to area and perimeter, many students were confused by measurements for two
different dimensions, area and length. They thought that they could divide the area by 4 to
get the length of a side. They applied the one-dimensional measurement rule on the problem
that needed the rule for two-dimensional measurement. Students were able to any and
transfer abstract knowledge from one situation to another. However, they seemed to
conceptualize an inefficient or inappropriate rule.

Finally, I consider the linguistic factor. We can easily link the conceptual difficulty
to another psychological construct—linguistic ability. Different psychologists (Cummins,
Kintsch, Reusser, & Weimer, 1988; Cummins, 1991; Okamoto, 1996; Riley & Greeno,
1988) have their own theoretical stances on the importance of linguistic or conceptual

influence on solving mathematics word problem. Cummins, et al. (1988) and Cummins

181

(1991) were inclined to consider the linguistic factor as the main one associated with
students' difficulties in solving the arithmetic word problems. Many psychologists with the
same stance cite Hudson's (1983) study to support their position: When the problem such
as, "How many more birds are there than worms?" is recoded as "How may birds won't
get worm?" is presented to the kindergartners, their performance improved from 25% to
96% correct. Cummins and others psychologists argued that children found the first
version more difficult because they could not interpret key words and phrases in the
problem text. Okamoto (1996) and other psychologists argued that this kind of change in
problem statements might possibly lead students to construct different problem
representations from which students needed to activate their mathematical knowledge to
solve the problem. So, there are two possible difficulties, the first one may be related to
students' ability to construct the representation for the situations. The second possibility is
related to a lack of relative mathematics knowledge. The second one is more related to
conceptual than linguistic competence.

There is evidence that linguistic factors had an influence on problem solving. If we
examine the data carefully, we find that there was a higher percent correct on problems with
fewer words. In other words, students performed better on problems (Multiplication,
Unknown Digit, Rectangular Width, Saving, and Magazine) with fewer words. More than
62% of the students gave correct or numerically correct answers to these problems.
However, language and problem type interacted with each other. The simpler story
structures and fewer words were associated with routine problems. In part, the word
problems that were familiar to the students were the problems in their textbook—that is, the
routine problems. Most of these problems were with simple story structure and fewer
words. Under this condition, the non-routine word problems I designed naturally contained
more words and had more complicated structures. Among the non-routine problem, there
was a different story. The story structure and the number of words were not always the

significant factor.

182

 

Was the linguistic factor the main obstacle to students' successful performances?
The Cake problem and the Glass House problem were the two most complicated story
problems on the quiz. They had a lot of words and a complicated story structure. However,
more than 65% of the students solved the "Cake" problem while less than 3.5% of the
students solved the "Glass House" problem. The main difference between the two
problems seems to be the mathematical concepts needed to solve the problems. The "Cake"
problem asked students to apply arithmetic operations (addition, subtraction, multiplication,
and division), to the partitioning and aggregation of familiar objects—cakes and pieces of

cakes. The story structure and the number of words did not have an obvious impact on

 

performance when the students had well-developed conceptions on the mathematical
concepts in the problem. The "Glass House" problem asked for an understanding of area
and perimeter. And, we lmow that some of the weakest conceptions were these concepts.
This weakness did contribute to the poor performance on this problem. The difficulty also
showed up on another problem, "Extended Square," that required thinking about area and
length together. If linguistic difficulties were primary, students should have difficulties
with both of these complicated story problems—"Cake" and "Glass House"—and they did
not.

This does not mean that the linguistic factor was unimportant. As we can see the
students' performances on the two non-routine problems—"Glass House" and "Extended
Square"— which students needed to understand the same mathematical concept of area, the
students performed slightly better on the problem that had a simpler story structure and
fewer words—"Extended Square." On the "Extended Square" problem, the students
achieved a higher percentage correct—10%, compared to the percentage correct, 3.4%, of
the "Glass House" problem. So, I prefer to argue that conceptual and linguistic elements
had an interactive effect on difficulty.

To conclude, the difficulties that students had were more likely related to their

conceptual difficulties with the multiplicative concepts, area and perimeter, and ratio. Most

183

of the students did not have computational difficulties, and were developmentally ready for
learning the concepts. Although linguistic factors, like the story structure and number of
words, had influence on students' performance, it did not appear to be the most significant

cause for the students' performances.

3. Are these difficulties related to social classes?

Students from both schools, serving the low SES and middle class communities,
performed well on the routine problems. The students solved the routine problems correctly
about 65% the time (combining both correct and numerically correct responses). The
students from the low SES school solved the routine problems correctly more than 57% the
time. This result is consistent with the study done by Oakes (1990). Teachers who taught
in schools in low SES communities were more likely focus on "computational basics" in
their lessons. This may be the reason why the students from the low SES school did well
on the non-routine problems. However, students from middle class outperformed their
counterparts on the non-routine problems. This performance difference may be related to
students' social background. Although the relation was revealed in this study, there is no
any conclusive explanation, based on the data. I did not carry out any observations of
classroom instruction, nor collect the data about parents' educational background, so it is
difficult for me to make any conclusive statement about how differences in social
background exerted an influence.

Students in the middle class school outperformed their counterparts on most of the
non-routine problems. The only exception was the "Cake" problem. The differences
between the two groups on performance in the two sets of parallel problems, "Ratio" and
"Extended Square" problems, were the most obvious. Ameng these problems, the
difference on the percentage correct between two groups were more than 20%. The only

exception was the "Extended Square" Part B on Form A. You can read the differences in

184

Table 8.5. When the differences in students' performance between two schools were
greater than 20%, the percentage correct is given in italics.

Table 8.5 Two Schools Performance on the Non-Routine Problems

 

Problems Form Low SES School Middle Class School
Ratio A 9.1% 32%

B 28.6% 63.3%
Balance A 0% 8%

B 0% 3.3%
Lake A 9.1% 12%

B 8.6% 20%
Extended Square A 6.1% (B), 9.1% (C) 16% (B), 32% (C)

B 8.6% (B), 8.6% (C) 40% (B), 40% (C)
Glass House A 0% (A), 0% (B) 0% (A), 8% (B)
Coins A 9.1% 16%
Numberpa‘mm B 48.6% (B), 31.4% (C), 63.3% (B), 53.3% (C),

14.3% (D) 33.3% (D)

Cake B 77.1% (A), 60% (B) 76.7% (A), 66.7%(B)

 

In addition to the level of general correctness, the students from the low SES school
made more conceptual mistakes than the students from the middle class school on the
"Ratio" problems. Tables 8.6 shows that the students from the low SES school
experienced more conceptual difficulty on both "Ratio" problems. The incorrect responses
given by the students from the low SES school and related to conceptual mistakes were
greater than that of their counterparts. On the "Extended Square" problem on the two
forms, the students from the low SES school might have experienced more conceptual
difficulty than their counterparts. However, I cannot make a valid inference on their
performance, because at least 6 of the students out of a total 68 gave unclassified written
responses on one part of the problem, and at least 14 of the students gave up one part of the
problems. With these kinds of responses, it is difficult for me to make any valid inference

about their thinking.

185

Table 8.6 Students' Incorrect Frequency on Two Ratio Problems

 

 

 

inimum Incorrect
Incorrect Response
Frequency, Response Frequency
School Form Problem Incorrect Frequency related to
(N=) Response related to Computational
Conceptual Errors
J Mistakes
Low SES A Photo-
School Robber 24 I9 5
(33)
Middle Class A Photo-
School Robber 14 9 5
(25)
Low SES B Photo-You
School 20 I7 3
(35)
Middle Class B Photo-You
School 9 8 1
(30)

 

In general, the performance between of these two groups were substantively different on
the non-routine problems. Although the students from the middle class school did not do an
excellent job on these non-routine problems, they out-performed their counterparts in the
low SES school. And, the mistakes made by the students from the low SES school were
highly related to conceptual difficulty. These results seemed to reveal that these two groups
of students might have learned differently in two different schools, or they had different
kind of support after schooling. In particular, their experiences in leaming and
understanding the multiplicative concepts—area and perimeter, and ratio—might be
enlarging their performance differences on the quiz. This study cannot explain why there
was this obvious difference between the two groups, but it raise the educators' and social
services providers' attention to the social class issue that influences how students learn in

and out schools.

186

APPENDICES

187

APPENDIX A

QUIZ PROBLEMS IN FORM A & FORM B

Form A Problems and Their Conceptual Analysis

 

 

Preblem Name: Multiplication I fighlem Type: routine
fieblem (frem Pom; A):
187 x 31 =

Answer: 5797

 

Methemetjes gzeneepte Embedded:
multiplication

Quantities.
W:

 

 

 

 

188

 

 

floblem Name: Unknown Digit I Wee: routine

 

W:
The 3-digit number “57?” is exactly divisible by 16. Find the value of “?”

Answer: 6 or 576

 

Mathem ic nc ts Em d
multiplication, division

Quantities:

W:

1. Try to put the even numbers, 2, 4, 6, 8, 0 in to the unknown digit position, and try to divide it with
16. Or, try all digits, 1, 2, 3, ...9, 0 in to the last digit position.

2. Divide 57 by 16, and get the remainder 9, then think about what number is the product of 16 in a 2-
digit number beginning with 9.

Comm

 

 

W
Competatienel Error

 

189

 

 

Emblem—Name: Balance I mum: non-routine

 

Erublcmitromfiormfl:

All marbles are the same weight, what are the range of the possible weight that each marble can be?

 

Answer: the lightest is 112g or 111.12g; the heaviest is 199g

 

WWW:

multiplication: unit weight x number of units = total weight

order: many total weights less than 1 kg... < i kg. < many total weights greater than 1 kg.

Qumitissr

unit weight, number of marbles, total weight of marbles in Balance 1, total weight of marbles in Balance
2.

WW:

1.Turn1kgt01000g

2. Divide lOOOg by 9 and by 5

 

3. Think about the concepts, less and larger.

Walrus

- Understand how the balance works, the heavier side will be down.
- Relationship between quantity of marbles and 1 kg; 1 kg / # of marbles can estimate the weight of a
marble.

 

 

 

- Understand the unit, kg.
- The concept of the heaviest and the lightest weight.
Comm

- unit confusion: kg = g

- using 7 to divide 1 kg, because they thought 7 is between 5 and 9.
- multiply 5 or/and 9 with 1 kg

- divide 1 kg with 9 - l. and5 +1

- divide 1 kg with 9 + 5

Computatinnalfimr

- Miscount the quantity of marbles

 

190

 

 

We: Glass House I W: non-routine

 

Emblem (frem Fem; A):

Your school is planning to build a glass-house for science classes. The area of the floor must be 24 square
meters. The building will be 3 meters in height. The builder estimates the house costs $1200 for every 1
meter of the perimeter.

a. List all the possible lengths, widths, perimeters, areas, and the cost of walls for different rectangular
shapes. (You only need to consider rectangle with length and width equal to whole numbers of the meter)

 

 

 

 

 

 

 

Answer:
Width Length Perimeter Area Cost of the wall I
1 24 50 24 60000
2 12 28 24 33600
3 8 22 24 26400
4 6 20 24 24000

 

 

 

 

 

 

 

b. Based on the cost of walls, which rectangular shape would cost least to build?
Answer: 4 m x 6m

 

WM:

3-dimensional, 2-dimensional and linear thinking (to have a model about the building in 3-d, understand
that 2-d information, area, is fixed in the problem, and linear unit, perimeter, is needed to solve the
problem), the relationship between area and perimeter.

Quamties:

'cost for the length of the wall, square meters, meter

W:

1. Factors of 24

2. Perimeters of different rectangles

3. Costs for building the wall with different perimeter

 

Comm
- Relationship between a fixed area and a perimeter with several dynamics.
- Understand that the 3rd dimension (height) is not useful for solving the problem.

- Understand different units (m3, m2, m) for measuring different dimensions.

 

 

Wm

- the area is not fixed at 24 m2.

- using area to calculate the cost

- incorrect way to get the area

- incorrect way to get the perimeter

- multiply the cost $1200 with the length of two sides.

Cnmnutamnalfirmr

 

 

191

 

Emblernflame: Coins I

Embleml‘xpe: non-routine

 

Emblemfiromiqnnmz

you have.

than $10
Possible Combinations:

 

 

You know you have more than ten dollars in you pocket (all are coins, $5, $2, 50 ¢, 20¢, or 10¢) but
you find you are unable to give someone a change for a ten dollar note. List the combination of coins that

Answer: any combination with all different coins, and the combination is not equal to $10 and is bigger

 

Type\Q.rantity Option 1 Option2 option3

$ 5 l l 1
$2 3 4 4
50 cent 1 l 1
20 cent 1 1 1
10 cent 1 l 2

Mathematicsfionmflmhedded:

multiplication, addition

0 . . :

number of each coin; value of each coin

WW:

Concentuamnalxsis

- understand there are different kinds of coins in the pocket
- understand you have more than 10 dollars
- understand the sum of the coins cannot be equal to 10 dollars

Starting from coin for $5, establishing all the possible combinations.

 

Concentnalfimr

- the sum of the coins is less than 10 dollars

- one of the coin combinations is equal to 10 dollars
- do not list out the different kinds of coins

- no unit given

W

 

 

192

 

 

Ernblemfiame: Lake I Wn-mufinc

 

Emblemlfrnmfionum:

Mr. and Mrs. Chen walk with their dog, Bobby, every morning. They walk at the same speed, 5 km per
hour. The distance of the walking path around the lake is 10 km. They start and walk in opposite
directions at the starting point and meet each other midway, then go to have their breakfast. Bobby starts
his morning run with Mrs. Chen and runs toward Mr. Chen. When he meets Mr. Chen, he runs back to
Mrs. Chen. He continues like this until Mr. Chen and Mrs. Chen meet. His running speed is 28 km per
hour. How far does Bobby run, by the time the couple meet?

 

 

 

 

 

 

p h
Lake
A. Chen
‘—
Mrs. Chen .
and Bobby St‘j‘m"' I
pomt
Answer: 28 km
MW:
ratio
0 . . :
distance, time-spending
W:

1. Read the relationship about the time spent by the couples
2. Using the time—spending by the couple to get the distance run by the dog
t A i
- understand the couple need one hour to finish the morning walk
- the dog will run an hour on the path

 

W
- multiply 28 km with 2. In this case, I suspect that the students think 10/5 = 2, and 30 km multiply by

2, then they used all the numbers in the problem.
- no unit is given

WEED;

 

 

193

 

 

39mm: Photo-ratio (robber) I W: non-routine

 

Problem Ifrem Fogn A):

A hidden camera took the picture below during a robbery. In the picture, you can see the robber and a
service desk. The height of the desk is 3 cm, and the height of the robber is 4.4 cm. The real height of the
desk is 1.2 meter. Can you figure out the height of the robber? How?

 

[service desk I 4.41:
T [ I
3cm \

Answer: 1.76 m or 176 cm

 

 

 

 

 

 

 

 

 

 

#3 .. vermin

 

 

Mathemaucsfiancmfimbrddrd

ratio, equal ratio, shrinking and stretching

Quantities:

Actual height of the desk, height of the desk on the photo; height of the robber on the photo, actual
height of the robber

1. Get the relative actual height to 1 cm on the picture, and then multiply 4.4 cm to the relative actual
height of 1 cm.
2. Cross-multiplication

1.2 x

3 4.4
3. Functional thinking, find the function by turning 3 to 1.2 or 3 to 120, then applying that function to
4.4.

 

ConceptualAnalxsis
- Proportional relationship between actual height and the height on the picture
- Unit sense; how long is a cm and a m?

 

 

W

Type A: unit confusion,
Example 1:4.4 cm - 1.2 cm (it should be 1.2 m) = 3.2 cm; 3.0 - 1.2 = 1.8
Example 2: 3 m - 1.2 m = 1.8m (3 should be cm, not m)

Example 3: 1.2m x 4.4 cm = 52.8 cm (it should be 0.528m2)

Type B: seeing ratio as an additional relationship between two entities
Example 1: 4.4 - 3 + 1.2 = 2.6 cm

Type C: others
Example 1: 1.2 / 3 = 0.4, => (4.4 - 3) x 0.4 = 0.56 m;
Example 2: 4.4 cm - 3 cm =1.4, 3 x 0.4 :12 m => 1.2 +1.4 = 2.6 m
Example 3: 4.4 l 3 = 1.46

QommrtatronaLEmr
Example 1: 4.4 / 5 x 2 = 17.6 (should be 1.76)

 

 

194

 

 

Preblem Name: Extended Square ] Emblemlxnez non-routine

 

BrnblemifrnmfannAlt

The area of a square is 196 cm2, now you lengthen one dimension by S cm to form a new shape.

a. Draw the new shape.
Answer: a rectangle
b. What is its area?

Answer: 14 cm x 19 em = 266 cm2
c. What is its perimeter?

 

Answer: (14+191cm x 2 = 66 cm
W:

area: the unit for measuring surface, perimeter: the unit for measuring the length

Quantum:

initial length; new length, perimeter of initial and new shape, area of initial and new shape
x l ' m :

1. Get the length of one side of the square

2. Add 5 cm to two opposite sides

3. Add the length of two adjacent sides and mulmly the sum by 2

 

Ceneeptusl Analysis

- Relationship between the area of a square and its length on a side
- Area concept

- length concept

— Formula of area (width x length)

- Formula of perimeter [(width x length)xfl

 

 

W

Type A: unit confusion
wrong unit is used, for instance, cm for area
Example 1: 19cm x 19cm = 361cm

Example 2: 196 + 5 + 5 = 206cm2 (196cm2 cannot be added to 5cm)
Type B: confusion on area and length,
Example 1: 196/4 = 49 => 49 x 54 = 2646 => (49 + 54) x 2 = 206;
Example 2: (196/4 + 5 x 2) x (196/4) = 59 x 49 = 2891 => (59 + 49) x 2 = 216
Example 3: (196/4 x 5) x (196/4 x 5)
Type C: confusion between area and perimeter
Example 1: 196/4 = 49 => 49 + 49 + 54 + 54 = 206 (area), 206 (perimeter).
Type D: add a new area to the given area

Example 1: 196 + 5 x 5 = 221cm2
Type E : others
Example 1:5x5=25=>5+5= 10
Example 2: 5/4 = 1.25cm
Example 3: 196/2 = 980m
Example 4: 16 x 12 = 196, 16 + 12 + 16 + 12 = 56 (re-assemble a product for 196)
Example 5: 196/4 x 5 = 245cm square => 196/2 + 5 = 103cm
Example 6: (196 x 5)cm = 980cm (area), (196 x 5)cm = 980cm (perimeter)
Example 7: (196 + 5) x 2 + 196 x 2 = 694 (area), 196 + 5 = 201 (perimeter)
CommfimaLErLQr
- unable to find the square root of 196
Mimeading
- extending 4 sides
Example 1: 19 x 19 = 361cm square, 19 x 4 = 76cm

- extending one side
Example 1: a trapezoid was drawn; (196 + 201) x 196/2 = 4962.5 => 4962.5 x 21196 = 50.6

 

195

 

 

Emblem; Saving I humanism: routine

 

WA):

Every day, you save $2.20, and your brother saves $1.70. Four weeks later, how much
money do you and your brother save?

Answer: 109.2 dollars

 

M mat' m
multiplication, addition

Quantities.

W:
(2.2+1.7)x7x4or2.2x7x4+1.7x7x4

 

CummLAnalxm

 

 

QQnQQRmaLEnQr

'na rrr

 

196

 

 

 

 

 

 

mm: Rectangle Width I - Wm: routine
BrotflerrLthQmEQmA):

The area of a rectangle is 72 cm2, its length 9 cm, what is width?

Answer: 8 cm

 

Mathematics Coneepts Embglfigi:
division, area, length

0 . . :

cm, cm2

Ex t

72 / 9

 

 

 

Cenceptuel Errer

 

197

 

 

Form B Problems and Their Conceptual Analysis '

 

W: Multiplication
le fr '
167 x 27 =

l

REQIEE £3. routine ;

 

Answer: 4509

multiplication

Quantities-
W:

W:

 

Wei:

 

W
ComantatinnaLEnsu

 

 

 

198

 

W! Unknown Digit I Breblem 13m: routine

 

Brnblemtfrumiemfll:
The 3-di git number “55?” is exactly divisible by 12. Find the value of “?”.

Answer: 2, or 552

 

h ' t E d
multiplication, division

Quantities:

WW:

1. Try to put the even numbers, 2, 4, 6, 8, 0 in to the unknown digit position, and try to divide it by 12.
Or, try all digits, 1,2, 3, ...9, 0 in to the last digit position.

2. Divide 55 by 12, and get the remainder 7, then think about what number is the product of 12 in a 2—
digit number beginning with 7.

 

ConcentmLAnalxm

 

 

Wider
W

 

199

 

 

Wm: Number Pattern L Ereblem 13m: non-routine

 

W:
a) Fill the following empty cells with the pattern you find,

 

 

 

 

 

 

 

 

 

 

lst row 1 2
2nd row 4 3
3rd row 5 6
4th row 8 7
5th row 9 10
6th row

7th row

8th row

9th row
10th row

 

 

 

 

b) What are the two numbers on 21st row?

Answer: 41, 42

c) What are the two numbers on the 73th row?

Answer: 145, 146

(1) With the above table, we say 8 is in Row 4 and Column 1. What do we say about 87 relative to Row
and Column?

Answer: Row 44 , Column 2

 

M th ma i n Em .
multiplication, division, function, odd/even numbers

Quantities.
N/A
W:

1. Complete all the cells, until they get the answer
2. Find out the pattern of number distribution
* In odd number rows, numbers are ascending (col-2-numbers are one larger than Col-l-numbers)
* In even number rows, number are descending (col-2-numbers are one less than Col- l-numbers)
* Multiply 2 with the n of nth row, the product is one number on that row.
3. List the numbers in 20th Row and its following two rows, list the 70th Row and its following 3- 4
rows, list 40th Row and its following 5 rows.

 

Qeneepteal Analysis

- follow the given pattern, list all the number in the cells and extend the pattern until the answer is
achieved
- understand the pattern, use multiplication to achieve the answer

 

 

ConcentuaLErrm

- have a wrong number arrangement pattern

WM

 

 

200

 

Brnhlernl‘lame: Lake I Emblemlene: non-routine

 

WM:

Mr. and Mrs. Chen walk with their dog, Bobby, every morning. They walk at the same speed, 4 km per
hour. The distance of the walking path around the lake is 8 km. They start and walk in opposite directions
at the starting point and meet each other midway, then go to have their breakfast. Bobby starts his
morning run with Mrs. Chen and runs toward Mr. Chen. When he meets Mr. Chen, he runs back to Mrs.
Chen. He continues like this until Mr. Chen and Mrs. Chen meet. His running speed is 30 km per hour.
How far does Bobby run, by the time the couple meet?

 

 

 

and Bobby

Mrs. Chen l

Starting
point

 

Answer: 30 km

 

W:

ratio

0 . . :

distance, time-spending

WW:

1. Read the relationship about the time spent by the couples

 

2. Use the time-spending unit to get the distance run by the dog.

- understand that the couple needed one hour to finish their morning walk
- then, the dog will run an hour on the path

 

 

W

- multiply 30 km by 2. In this case I suspect the students think 8/4 = 2, and 30 km multiply by 2, then
they used all the numbers in the problem.

- try to adding from 30 to 1

- no unit is given

Computaunnalfinor

 

 

201

 

 

MM: Cake [ Breblem 13m: non-routine

 

Emblem Ifrem Page 3):

A cake-shop owner found that some customers like to buy a slice of cakes and others like to buy whole
cakes. So he cuts some of his cakes into 8 equal slices and sells them for $6 per slice. He sells the whole
cakes for $45.

a) One day, he sold 12 whole cakes and 62 slices. How much money did he earn that day?

Answer: $ 912

b) On other occasion, he had $963 in his cash-register at the end of the day. He knew he sold 11 whole
cakes. How many slices did he sell on that day?

Answer: 78 pieces

 

WW:

multiplication, addition, division, subtraction

Quantities:

price for a whole cake, price for a slice, total sales, sales for sold cakes, sales for sold slices
Ex ed 1 ti m :

A. Multiply 12 with $45 and multiply 62 with $6, adding two products

B. Multiply 11 with $45, subtract the product by $963, then divide it by $6

 

CementuaLAnabrsu

- unit price x quantities = sales

 

 

9.9mm

Type A: confused with the quantities or unit
Example 1: $963 - 11x $45 = 468 pieces (this algorithm can only offer the $ unit, not pieces)
Example 2: 963/11/6
Example 3: 963/ll = 87... 6

Type B: others
Example 1: 12 x 6 = 72 => 62 x 6 = 372 => 372 -72 = $300.
Cnmnutatienalfium
Example 1: 963- 495 = 568 (should be 468)
I Ii 1'

 

202

 

 

Emblemflame: Photo-ratio (you) ] mm: non-routine

 

WW:

Last Sunday, you stood beside a sculpture and had your picture taken. Your height on the photo is 4 cm,
and the height of the sculpture is 10 cm. You know your real height is 1.4 meters. Can you figure out
the real height of the sculpture? How?

 

 

 

 

 

 

 

Answer: 3.5 m, or 350 cm

Mathematicsfionmfimbeddedt

ratio

Quantities:

Actual height of the sculpture, height of the sculpture in the photo; your height in the photo, your actual
height.

1. Get the relative actual height to 1 cm on the picture, and then multiply 10 cm to the relative actual
height of 1 cm.
2. Cross-multiplication

1.4 x

4 10
3. Functional thinking: find the function turning 4 to 1.4 or 4 to 140, then applyig that function to 4.4.

 

W

- Proportional relationship between actual height and the length on the picture

- Unit sense; how long is a cm and a m?

- 10cm is relative to 1.4m+ l.4m+ 1.4m/2(10cm=4cm+4cm +2 c119

 

 

Wm
Type A: unit confusion

Example 1: 1.4 m x 4 cm = 5.6 cm

Example 2: 10 cm x 4 cm = 40 cm (it should be 40 cm2)
Example 3: 10 cm x 4 cm = 40 cm
Type B: seeing ratio as an addition relationship between two entities
Example 1: 1.4 + 6 = 2 m
Example 2: 1.4 m +1.4 m + 0.2 = 3 m
Example 3: 1.4 x 6 = 8.4 (6 is produced by subtracting 4 from 10)
Type C: others
Example 1: 10+ 4/4 =11
Example 2: 1.4 / 2 + 1.4 / 2 =
Example 3: 10 cm x 4 cm = 40 cm
Example 4: 1.4 + 1.4 + 0.2 = 3 m
Computationatfinur
Example 1: 1.4 / 4 = 0.7 (should be 0.35)
Example 2: 1.4 m +1.4 m + 0.2 = 3 m

 

203

 

 

Memflame: Balance 1 W: non-routine

 

W:

All marbles are the same weight, what are the range of the possible weight that each marble can be?

 

 

 

 

Answer: the lightest is 126 g and the heaviest is 199 g
MW:

multiplication: unit weight x number of units = total weight

order: many total weights less than 1 kg < i kg. < many total weights greater than 1 kg.
Q . . :

unit weight, number of marbles, total weight of marbles in Balance 1, total weight of marbles in Balance
2.

Wham:

1.Turn1kgt01000g

2. Divide lOOOg by 8 and by 5

3. Think about the concepts, less and larger.

 

 

Emma:

- - Understand how the balance works, the heavier side will be down.

- Relationship between quantity of marbles and 1 kg; 1 kg / # of marbles can estimate the weight of a
marble.

- understand the unit, kg.

- The concept of the heaviest and the lightest weight.

Wm

- unit confusion, kg = g; or no unit is given

- multiply 5 or/and 8 with 1 kg

- confused with what is the heaviest and the lightest, Heaviest = 125 and Lightest = 142.

- take a number between 125 and 200.

Qcmnumnalflmr

- Miscount the quantity of marbles

 

204

 

 

Memflame: IILagazine I Problemjxmr routine

 

393W:

The listed price for the magazine, Electronic Games, is $14 a copy. You can also pay
$114.20 for a 12-issue subscription. How much do you save by purchasing a
subscription?

Answer: 53.8 dollars

 

a m i E
multiplication, subtraction
Quantities:
dollars, issues
ExpectrAAlgnnthms:

12 x 14 - 114.20

 

Shamanism

 

 

mandamus:

 

205

 

 

We: Extended Square I W: non-routine

 

Erablemifmmfionnfliz

The area of a square is 121 cm2, now you lengthen one dimension by 3 cm to form a new shape.

a. Draw the new shape.

Answer: a rectangle

b. What is its area?

Answer: 11 cm x 14 cm = 154 cm2
c. What is its perimeter?

Answer: (1 l+14)cm x 2 = 50 cm

 

WWW:

area: the unit for measuring surface , perimeter: the unit for measuring the length
Qualities:

initial length; new length, perimeter of initial and new shape, area of initial and new shape
Wattles:

1. Get the length of one side of the square

2. Add 5 cm to two opposite sides

3. Add the length of two adjacent sides and multiplythe sum by 2

 

Ceneeptual Analysis

- Relationship between the area of a square and its length of a side
- Area concept

- length concept

- Formula of area (width x length)

- Formula of perimeter [(width x length)x2]

 

 

Cenoeptea! Errer
Type A: unit confusion & mistake
Example 1: (121/2 + 3) x (121/2 + 3) (121/2 = 60.5cm2, cannot be added to a length)
Example 2: (121 + 3)(121)
Type B: confusion on area and/or length
Example 1: (121/4 + 2) x (121/4): 9983.3125cm2 (the correct computation should be 1005 .8125
cm2)
Example 2: dividing the area by 4 and think they can get the length of a side.
Type C: confusion about area and perimeter
Example 1: (121/4 + 3) x 4 = 133 (for area)
Example 2: 121 + 3 x 4 = 133 (for area), 133/4 x 4 = 133 (for perimeter)
Type D: Add a new area to the given area

Example 1: 121+ 3 x 3 =130cm2

Example 2: 121 + 60.5 x 3: 302.5cm2, (63.5 + 60.5) x 2 = 124 cm
Type E: others
Example 1: 3 x 3 = 9cm square => 3 x 4 =12 cm
Example 2: 123/3 = 41 => 41x 41 =1681cm square => 41x 4 =164 cm

Example 3: 121 x 3 =363cm2, 121 + 3:124 cm

Example 4: 60.5 x 3 + 121 = 302.5cm square => (63.5 + 60.5) x 2 = 124 cm
ComputafianaLError

Example 1: 121/4 = 3.25

Example 2: 6.25 x 4 = 2500
mutating
- extending one side

Example 1: a trapezoid was drawn: (11 + 14) x 11/2 = 137.5cm square => 11 x 3 + 14 = 47cm
- extending four sides

Example 1: 14 x 14 = 196 cm square => 14 x 4 = 56 cm

 

206

 

 

 

Mariam: Rectang
W-
The area of a rectangle is 72 cm2, its width is 6 cm, what is the length?
Answer: 12 cm

le Length 1 1196mm: routine

 

WWW:
division, area, length
0 . . :
cm, cm2
x Al
72/6

 

 

 

 

207

 

APPENDIX B

INTERVIEW PROBLEMS

 

Johnny wants to decorate his new apartment.

 

3m

 

 

.I' l

 

2m

 

 

 

 

Hm—l

 

 

——4m

 

a) He wants to buy wallpaper and put it onto the walls (he is not going to put wallpaper on
the ceiling). What does he need to find out, the perimeter or the area?

b) He goes to a construction-material shop, and finds that the wallpaper packed in rolls, has
50 cm x 600 cm of material. How many rolls does he need to cover all the walls, except
for the one with door?

[Apartment]
(NR)

 

 

208

 

 

 

You and 3 other classmates are assigned to design your classroom bulletin board this
semester. The board is a rectangular shape, 2.5 meters by 1 meter. First you think about
the background.

1) You decide the background color (light blue), and find there are two sizes of that color
paper. One is 30 cm by 50 cm, and another is 40 cm by 50 cm. You want to buy the size
that waste the least paper

a) Which will you buy?

b) How many pieces do you need to buy to cover the whole board?

2) The small paper costs $3 per piece, and the larger paper costs $5 per piece. One of your
classmates suggests comparing the cost because your budget is very tight.

a) Compare the cost of using each size paper and decide which is cheaper.

[Classroom Board]
(NR)

 

 

209

 

 

Recycling newspaper is one way to save trees. For every 2 meter high stack of newspaper
that is recycled, about one tree 14 meter tall is saved.

a) Your family can accumulate a 2 meter high stack in a month. How many trees 14 meter
tall do your family save in a year?

 

b) Your school subscribes to several different kinds of newspaper, it only takes 5 days to
accumulate 2 meter high of newspaper in a stack. How many tree 14 meter tall does your
school save in a year?

[Newspaper Recycling]
(NR)

 

 

 

 

 

210

 

 

 

A rope 30 centimeter long is formed into a rectangle. If the widthojthe rectangle is 6
centimeter, what is its length?

[Repel
(R)

 

211

 

 

 

Following figure shows a rectangle consists of 4 equal squares. If the area of the rectangle
18 ,

 

 

 

 

 

 

 

a) What is the area of each square?
b) What is the perimeter of each square?

c) What is the perimeter of the rectangle showed in the figure?

[4 Squares]
(NR)

 

212

 

 

Your school plans to hold a table-tennis competition

a) There are 64 students applying for the singles competition, how many pieces record-
paper do you need to prepare, from the starting to the final of that competition?

b) If you want to set up a table-tennis competition, can you have any number of
participants? Or, will only some number of participants work? (Which number(s)?

[Table Tennis]
(NR)

 

 

213

 

 

 

All marbles are the same weight, what are the range of the possible weight that each marble
can be? .

 

[Balance]
(NR)

 

 

214

 

The listed price for the magazine, Electronic Games, is $14 a copy. You can pay $288.88
for a 24-issue subscription. How much do you save by purchasing the magazine by
subscription?

[Magazine]
(R)

 

 

 

215

 

 

Using the pattern given in Figure A and B, try to figure out the value of “ .” is in the
following Figure C.

o o a
0‘9 one 6‘6

8 C

[Numbers Relation]
(R)

 

216

 

 

 

You have 150 rectangular tiles. The length of each tile is 30 cm, and the width is 8 cm. You
want to build a square platform from that set of rectangular tiles. How many tiles will you
use?

 

8cm

 

 

 

300m

(This picture is not the actual size of the tile)

[Tiles]
(NR)

 

 

217

 

 

Table of Moon Facts
The moon is smaller than the earth.
People weigh 6 times as much as on Earth as on the moon.

The moon goes around the earth once in 28 days.
The moon is about 240,000 miles from the earth

a. Peter figures that he would weigh 14 pounds on the moon. What does Peter weigh on
Earth?

b. Peter’s mother weighs 108 pounds on Earth. How much would she weigh on the moon?

c. About how long does it take the moon to go around the earth four times?

[Moon]
(NR)

 

218

 

 

Mr. and Mrs. C-hen walk with their dog, Bobby, every morning. They are walk at the same
speed, 5 km per hour. The distance of the walking path around the lake is 10 km. They
start and walk in opposite directions at the starting point and meet each other midway, then
go to have their breakfast. Bobby starts his morning run with Mrs. Chen and runs toward
Mr. When he meets Mr. Chen, he runs back to Mrs. Chen. He continues like this until Mr.
Chen and Mrs. Chen meet. His running speed is 28 km per hour. How far does Bobby
run, by the time the couple meet?

 
   

/|\;. Chen

 

<—

Mrs. Chen ,

and Bobby 3‘5“” l
pornt

 

 

[Lake]
(NR)

 

 

 

219

 

 

A poster designer just finished a poster on her computer. The size of the poster on her
computer is 27.5 cm by 21 cm. She wants to make the length and the width 3 times larger
and print the poster on a plotter to see how it works.

1) What is the area of the poster she prints?

2) The printed poster is larger than the poster on the computer screen, how many time
larger?

[Poster]
(NR)

 

220

 

 

Two trains (A and B) are starting out from Mongkok Station in two different directions at
the same time with the same speed, 66 km per hour. 2 minutes and 30 second later, how
far apart distance-wise are the trains?

¢ I ’

Mongkok Station I

 

 

 

 

 

[Train]
(NR)

 

 

221

 

 

 

[Fraction Division]

(R)

 

222

 

 

APPENDIX C

SCORING SHEETS

Scoring Sheet for Non-routine Problems

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Student ID: Rater:

Item Att Procedures Read Answer Answer Conceptual Type of Computing

provided Provided correctness Error Error Error

A3 NA Yes No MR Yes No C I 0 1 2 0 1 2
balance NC
A4a NA Yes No MR Yes No C I 0 1 2 0 l 2
_glass house NC
A4b NA Yes No MR Yes No C I 0 1 2 0 1 2
_glass house NC
A5 NA Yes No MR Yes No C 1 0 l 2 0 1 2
coins NC
A6 NA Yes No MR Yes No C I O 1 2 0 1 2
lake NC
A7 - NA Yes No MR Yes No C I 0 1 2 A B C 0 1 2
ratio NC
A8b NA Yes No MR Yes No C I 0 l 2 A B C 0 l 2
extended sq NC D E
A8c NA Yes No MR Yes No C I 0 l 2 A B C 0 l 2
extended sq NC D E
B3b NA Yes No MR Yes No C I 0 1 2 0 1 2
# pattern NC
B3c NA Yes No MR Yes No C I 0 1 2 0 1 2
# pattern NC
B3d NA Yes No MR Yes No C I 0 1 2 0 l 2
# pattern NC
B4 NA Yes No MR Yes No C I 0 1 2 O 1 2
lake NC
BSa NA Yes No MR Yes No C I 0 1 2 A B 0 l 2
cakes NC
B5b NA Yes No MR Yes No C I 0 1 2 A B O 1 2
cake NC
B6 NA Yes No MR Yes No C I 0 l 2 A B C 0 l 2
ratio NC
B7 NA Yes No MR Yes No C I 0 1 2 O 1 2
balance NC
8% NA Yes No MR Yes No C I 0 1 2 A B C 0 l 2
extended sq NC D E
B9c NA Yes No MR Yes No C I O 1 2 A B C 0 l 2
extended sq NC D E

 

 

 

 

 

 

 

 

 

 

223

 

Scoring Sheet for Routine Problems

 

 

 

 

 

 

 

 

 

 

 

Student ID: Rater:
Item Att Procedures Read Answer Answer Conceptual Type of Computing
provided Provided correctness Error Error Furor
Al NA Yes No MR Yes No C I 0 1 2 0 1 2
multiplication NC
A2 NA Yes No MR Yes No C I 0 1 2 0 1 2
unknown digit NC
A9 NA Yes No MR Yes No C I 0 1 2 0 l 2
saving NC
A10 NA Yes No MR Yes No C I 0 l 2 0 1 2
rectangle NC
width
Bl NA Yes No MR Yes No C I 0 1 2 0 l 2
multiplication NC
B2 NA Yes No MR Yes No C I 0 1 2 0 l 2
unknown digit NC
B8 NA Yes No MR Yes No C I 0 1 2 0 1 2
magazine NC
810 NA Yes No MR Yes No C I 0 1 2 0 l 2
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BIBLIOGRAPHY

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