UNIVfF!sny of NORTH
BRITISH COlUMBIAERN
. LIBRARY
Prmce George, B.C.

DIFFERENTIATED MATH PROBLEM SOLVING IN THE CONTEXT OF
PERSONALIZED LITERATURE
by

Helena Carolyn Ziefflie
B.Ed., University of Victoria, 2003

PROJECT SUMBITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF EDUCATION
IN
SPECIAL EDUCATION

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
August 2013
© Carolyn Ziefflie, 2013

2

Table of Contents
Acknowledgement
Chapter 1

4
Introduction
Word problems
Provincial Evidence of Difficulty with Word Problems
Personal Observations into Specific Word Problems
Challenges
Performance Differences within my Classroom
Creating a solution to Teaching Deficiency

5
5
6
7

Chapter 2

Justification of Project Based on Research Literature
Differentiation
Progress Monitoring
Value of Personalized Problems
Use of Literature
Reading Comprehension
An Examination of Some Existing Resources

10
10
10
11
15
17
17

Chapter 3

Developing the Word Study
Inspiration for Project
Personal Location
Modifications to Resource
Introducing the Resource
Ethical Considerations

21
21
21
22
22
24

Chapter 4

Pilot Word Study
Introduction
Research
Description
Instructions
Progress Monitoring
Story Segments and Problems

25
25
25
26
27
28
29

Chapter 5

Experiences in the Classroom
Motivation
Differentiation
Progress Monitoring

45
45
45
45

Chapter 6

Discussion
Summary
Limitations
Required Revisions
Implications for Future Research

47
47
47
48
48

7
8

3

Implications for Future Practitioners
Conclusion

51

References
Appendices

49
49

Appendix A (Answer Key and Skill Description)

54

4

Acknowledgement

Instead of saying thank you I first need to apologize. Most importantly I need to
apologize to my children because they have missed so much of my time. Aleena you
were only two weeks old when I started. I will always have great memories of holding
you while in class on the computer. Callum, my heart broke each time I saw your
disappointed face when I told you I could not play because I had work to do. I also need
to apologize to my husband, as I wasn ' t always the happiest person to be around when
the work load was hard.
Tina and Kat, I truly could not have done this without you. I will miss our class
"chats" and look forward to the bonfire.
Mum and Dad, thanks for your support, especially with Callum and Aleena. I
don' t think you will find any spelling mistakes you will be so Proud.
To my class, thank you for your patience and feedback. David Falconer, thanks
for your understanding of my hardships throughout this process.
I would like to thank my supervisor Peter MacMillan, and committee members,
Jennifer Hyndman and Greg Nixon for their time, guidance and support in completing
this project.

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CHAPTER ONE: INTRODUCTION

When reflecting on my teaching, two factors continuously challenge my teaching
practices. The first is the huge range in student abilities in my classroom. There continue
to be cuts to learning support teachers across British Columbia (B.C.). The B.C.
Teachers ' Federation reports that there are fewer special education teachers now than in
2001 , even when declining enrollment is taken into consideration (White, 2008).
Classroom teachers are responsible for addressing the needs of all learners in their
classroom with less support than was provided in the past. I strive to create adaptations
and modifications that are unobtrusive, that is, not obvious to students during classroom
instruction and seat work. For example, I may alter a worksheet by removing some
questions; all the students are working on the same assignment and the fact that some
have fewer questions goes unnoticed by the other students. I am particularly concerned
that self-esteem is often lower in students with learning difficulties and singling out
individuals with special work may add to their negative self-efficacy. The range of
abilities in my classroom is always a significant issue when planning my lessons. There
are children who are performing one or two grades below grade level expectations, as
well as children performing above grade level expectations.
A second factor that challenges my teaching is student motivation. I strive to
create lessons that actively engage students. I have found students ' attention and
understanding is increased when I present lessons in fun and interesting ways .
With differentiated instruction and motivation in mind I created a math word
problem resource for teachers. The resource will aid in developing students' abilities to
identify the correct operation in a given word problem.
Word Problems

6

A word problem can be defined as a problem from the real world that does not
come with an equation ready to be used, rather it requires interpretation to be solved. In
the literature, word problems are also called story problems (Seifi, Haghverdi, &
Azizmohamadi, 2012). Word problems can often be challenging for students. Seifi,
Haghverdi, and Azizmohamadi, (2012) conducted an extensive literature review on real
world math problems in their study of teacher perceptions of student difficulty in solving
word problems. The researchers found that student difficulty with word problems has
been a prominent topic in mathematic research since the 1980's.
Provincial Evidence of Difficulty with Word Problems
In the B.C. math curriculum, multiplication and division are introduced in grade

3. Students now have four options: addition, subtraction, multiplication, and division
when deciding the operation of a word problem. B.C. and Alberta administer province
wide achievement tests annually. Alberta's Provincial Achievement test assesses grade 3
students and the B.C. Foundation Skills Assessment (FSA) tests students in grade 4 and
grade 7. On the 2012 FSA of grade 4 students, one quarter of the students incorrectly
answered a word problem involving multiplication. On a question that related division to
multiplication in order to solve problems, 55% of the students answered incorrectly.
When students were required to multiply, 25% divided instead. The report recognized
that students need improvement in problem solving and recommended more instruction
in problem solving (EducataCanada, 2012). Students in Alberta also struggled in
problem solving. The 2011 test identified that students needed increased instruction on
applying knowledge of basic facts to problem solving situations (Government of Alberta,
2012).

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Personal Observations into Specific Word Problem Challenges

I have taught grade 3 and 4 for five years. My interest to improve students'
ability to solve word problems involving multiplication, division, addition and
subtraction comes from observing my own students' difficulties. When teaching
multiplication and division, I fmd most students are able to understand the concepts of
multiplication, division, and equal groups. Often math worksheets have drill practice
followed by word problems to complete. Students usually figure out that on a
multiplication worksheet they do not need to read the problems; all they need to do is
multiply the two numbers given in the problem. When the operation could be
multiplication or division, students are often confused and will guess. I fmd that my
students do not get enough practice determining the operation when the equation could
involve addition, subtraction, multiplication or division. After I have introduced
multiplication and division, students in my classroom complete three or four word
problems requiring decision making that I prepare on PowerPoint at the start of each
math lesson. After several months I noticed an improvement in the students' ability to
determine the operation.
Although the students improve, they groan whenever they see the word problems. I have
to admit I do not spend a great deal of time writing interesting word problems that the
students would fmd engaging. I have used this opportunity to create a template for a
resource to help students become more engaged and feel personally connected to
problems to increase motivation.
Performance Differences Within My Classroom

8

What I have found to be successful is providing whole class instruction while
altering the work tasks for students depending on ability. For example, some may not be
ready for independent regrouping exercises, so they will continue with addition that they
are capable of, which may be single digit addition or two digit addition with no
regrouping. Students performing above grade levels expectations may be challenged
with three digit regrouping exercises. I believe this is important because the children are
not being blatantly labeled as either below or advanced and are being challenged at their
level. The students get used to each other completing different exercises. I am not naive
to believe that the students do not know who is achieving below or above expectations;
however, it is not continuously being discussed or displayed.
Creating a Solution to my Own Recognized Teaching Deficiency
In spite of my beliefs and desires against obtrusive differentiated instruction and

issues related to student dislike of word problems, I had not yet attempted a solution to
effectively differentiate word problems. I had tended to write the problems for average
students and I know for some students the problems are too hard and for some too easy.
To this end, I created a prototype that I believe addressed this deficiency in my teaching
and will be valuable to other teachers of primary and lower intermediate grades.
Specifically I wrote a ghost story that takes place in an elementary school. Names and
places familiar to the children were used to increase engagement. The story is divided
into 20 small sections with three math word problems accompanying each section. There
are three levels of difficulty, ranging from simple addition and subtraction to multistep
multiplication and division questions. The teacher assigns which questions each student
is to complete.

9

This resource addressed the deficiencies I identified while teaching mathematics.
The problems allow for practice of problems in which students must identify the correct
operation. Using a story that included characters and setting familiar to the children, I
was able to engage the students which increases their motivation to learn. The range of
problem difficulty allows all learners in my class to participate meaningfully without
being segregated or explicitly identified. After my initial pilot some modifications will
be made; however, it is a resource that I and hopefully my colleagues will value and use
year after year.

10

CHAPTER TWO: JUSTIFICATION OF PRODUCT BASED ON RESEARCH
LITERATURE
In preparation for developing this resource I conducted a literature review

focusing on mathematics word problems, differentiation of learning, using literature to
aid mathematics instruction and examined existing resources that are similar to the
resource I developed.
Differentiation

As classrooms become more heterogeneous with different types of learners,
teachers need to alter their instruction to accommodate the needs of all the students in
their class. Differentiated learning can be accomplished with tiering of lessons. Tiering
involves having similar tasks with varying difficulty (Little, Hauser & Corbishley, 2009).
These authors go on to say that in mathematics, differentiation can be accomplished by
reducing the number of steps required to solve a problem. They further caution that
differentiation is not simply changing the amount of questions students are required to
answer. Teachers need to consider the problem and what skills are required to solve the
problem.
Progress Monitoring

A key component to differentiation or any teaching is monitoring the progress of
each student. Levy (2008) describes a situation all teachers will recognize; after teaching
a lesson the teachers asks if anyone has any questions or if anyone is unsure of what to
do. Not one student raises their hand; however, when the teacher walks around the room,
she notices a many students are not sure what to do. Differentiation requires continued
assessment to ensure students are receiving the appropriate instruction. For the purposes

11

of this project, I have limited assessment to the narrow task of determining whether
students are choosing the correct operation and completing the equation.
Value of Personalized Problems

Personalizing word problems as a way to increase student success is well
researched. Hart's (1996) study examined the effectiveness of using personalized word
problems versus standard text-book problems. The study was conducted in a grade six
classroom of21 students over eight weeks. To assess student attitudes, Hart developed a
word problem attitudinal survey that consisted of eight statements to which students
stated their agreement on a scale of 1-5. Hart alternated weeks giving lessons that
included either personalized or standard textbook problems. The personalized problems
were taken from the same text-book, keeping numbers and computations the same,
however, names and context were changed to places and people familiar to the children.
After each week a quiz was administered. The total points on all quizzes for nonpersonalized was 420 and 475 for personalized. Qualitative analysis of statements given
on a questionnaire at the end of Hart' s results indicated that the students had better
attitudes towards math when completing the personalized problems:
As the study progressed, it was clear to me that students preferred solving
personalized problems over "bland" text-book versions. Their enthusiasm and
interest grew as I presented humorous stories or made them applicable to the
students ' lives. I was pleasantly surprised to see the students perk up when
solving problems that involved their immediate environment. The classroom,
other students, teachers, the cafeteria and the entire school were popular topics in
the word problems. (Hart, 1996, p. 504)

12

Only 7% of the students responded that they preferred the textbook problems and 76%
stated that personalized problems were easier to solve.
Hart's study is referenced frequently in the literature; however a limitation of this study is
the small sample of only 21 students. Following this work, Ku and Sullivan (2000,
2002) completed two studies investigating the effectiveness of personalized word
problem instruction. The sample for the first study included seventy-five fifth grade
students in Taiwan. Students were grouped by ability and then randomly assigned to a
personalized or non-personalized word problem instructional program. Results in the
first study did not indicate differences with personalized and non-personalized
treatments. The mean scores for personalized were 10.50 (88%) and Non-personalized
10.20 (85%). Ku and Sullivan declared this study to be limited because of ceiling effects.
Students received high pre-test scores, before treatment which left little room for
improvement. Subsequently these authors redesigned the study to reduce the ceiling
effects by lowering the grade level to grade four and doubling the sample size to provide
more reliable statistical analysis. Researchers gave students an interest survey that they
used to personalize problems according to the interests of the class. Non-personalized
word problems were from the grade 4 text-book. Students completed a pre-test on
solving two step multiplication and division problems with personalized and nonpersonalized problems that determined if they were high or low ability. Students were
then assigned into four classes: (a) personalized high ability; (b) personalized low ability;
(c) non-personalized high ability; and (d) non-personalized low ability. Each group
received the same instructional program that included a four step procedure for solving

13

problems; however, the two classes in the personalized control received problems that
had been altered to reflect the students' interests.
Data analysis was a 2 (Treatment: Personalization and Non-personalization) X 2
(Ability level: Higher Ability and Lower Ability) X 2) (Test Occasion: Pretest and Post
test) X 2 (Problem Type: Personalized and Non-personalized problems) repeated
measures univariate analysis of variance. The mean post-test scores 10.35 for
personalized and 9.32 for non-personalized that proved to be significant on a post hoc
paired sample t test.
Researchers did not completely eliminate ceiling effects for the higher ability
class. They were able to reduce pretest scores by 15% in the second study than in the
first; however, the high ability group still averaged 11.35 of 12 on the post test.
After the treatments, students and teachers completed an attitudinal survey. Both teacher
and student attitude surveys strongly favored personalized instruction regardless of ability
levels. Attitudes were analyzed using a 2 (Treatment) X 2 (Ability) X 8 (Survey items)
MANOVA to test for significant differences. Students who received the personalized
program had higher scores at the P < .001 level on half of the items on the survey.
Students in personalized treatment had a mean of 3.52 and non-personalized 3.31.
Positive results for the effectiveness of personalized problems were similar to results
Sullivan obtained in earlier studies on students in the United States (Lopez & Sullivan,
1991, 1992).
Bates and Weist (2004) completed a study investigating the impact of
personalization of word problems on achievement on fourth grade students in Northern
Nevada. Unlike the previous studies discussed above, personalization only occurred on

14

assessments, not during instructional practice. The sample of 42, 22 boys and 20 girls,
consisted of students who returned permission forms from the grade four classes in one
school. Participants completed an interest survey that was used to personalize
assessment. Two ten-question tests containing five personalized and five nonpersonalized questions were developed. On one test odd numbers were personalized and
even numbers non-personalized, and the second test was reversed. Participants were
randomly assigned one test the first week and received the other test two weeks later.
The two-week period reduced retest effects.
Analysis consisted of four paired t test of 42 pairs of scores. The mean and
standard deviations were calculated. A one-tailed t-test with alpha set at .05 was used to
examine group differences. The frrst test compared the number of correct answers of
personalized versus non-personalized problems. The mean was a difference of 0.3 and
was not statistically significant (t = -10, p = .46). The second was divided by three
predetermined reading levels-low, medium and high. Only the low reading group scored
higher on personalized problems, however, not statistically significantly higher (t = -.84,
p = .20). The third and fourth test used problem types, translation and processes

questions. Neither resulted in statistically significant differences (translation: t = -.84, p

= .20), process: ( t = -.45 , p = .32).
Researchers cautioned that their results should not negate positive studies because
of their small sample size of 42 and the use of only two post-instructional assessments for
data collection. Most previous studies have used instructional strategies when comparing
personalized versus non-personalized problem and factors such as motivation was not
addressed.

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Use of Literature
During my research I noticed that literature in math consisted of two types of
stories. There were stories that contained mathematical concepts in which the characters
demonstrated the concepts and there were books that were never intended to be used in
mathematical instruction; however, the teacher used them to engage the students.
Regardless of which type, including literature in mathematics lesson was seen as
beneficial.
When language skills are embedded in meaningful contexts, they are easier and
more enjoyable to learn. In the same way, numbers and their operations, when
embedded in meaningful real world contexts, give children the opportunity to
make sense of mathematics and to gain mathematical power. (Moyer, 2000, p.
248)
The use of books to introduce or teach particular math concepts has been in the literature
for a while; however, few teachers regularly use literature in mathematics lesson (Moyer,
2000).
Clark's (2004) study compared the use of stories to teach fractions and integers
verses lessons that did not include stories. The researcher completed the study over two
years in grade seven classrooms. Both classes came from similar socioeconomic
backgrounds and academic achievement. One year the integers unit instruction included
a story frame and the fractions unit did not. The following year the fraction unit included
a story frame and the integers unit did not include a story. The researcher wrote two
story frames. The integers unit revolved around two worlds that were at war with each
other and the fraction unit was based on pirates. Attention was given to ensure that the

16

units were taught the same with the inclusion of a story being the only difference. Exam
results were compared to determine if there was a difference between the two instruction
methods. Quantitative analysis, including a two tailed t-test for the comparison on
fractions, resulted in no statistical difference. ( M= 84, t= 1.85; t-Critical for two tailed
test= 2.01 , alpha= 0.5). The fraction unit within a story resulted in a mean of 84 and

standard deviation (SD) of 10 versus the unit with a story with mean of 72 and SD 28.
Conversely, there was a difference in the integers unit. (M= 89, t= 3.24, t-Criticalfor a,
two tailed test= 2.01 , alpha= .05 . The mean for story was 89 and SD 11 , and without

story, mean 76 and SD 16. Qualitatively, the researcher reported that students enjoyed
the story framed instruction and most felt that it improved their learning. On a
questionnaire completed at the end the units less than 10% responded that the story didn ' t
help at all to understand. For the integers unit 58% felt the story helped a lot. Students
were critical of the length of the stories and the researcher reflected that smaller chunks
would be preferable.
I learned from my literature review that I could use the story as springboard and
interest grabber that would engage the students. Jenner and Anderson ' s (2000) article
about a child in grade 2, who they named Neil, demonstrates how using literature can
enhance learning. Neil was able to participate in literature discussions with confidence;
however, during math he was reserved and unwilling to share his thinking. When a math
lesson was focused around a popular book, Neil felt at ease in discussing math problems
based on the characters in the book. The author reflected that "through the gift of a story,
Neil was given a rare opportunity to shine and to think and act like a mathematician"
(Jenner & Anderson, 2000, p 547).

17

Reading Comprehension

While conducting the literature review, reading comprehension strategies was a
prominent topic. Arthur Hyde' s article Mathematics and Cognition (2007) discusses the
disconnect between math and reading cognition strategies. To understand word problems
students must first be able to comprehend what is being asked. Reading research has
shown that cognitive strategies such as predicting, asking questions, inferring, visualizing
and making connections are just as important when reading word problems as when
reading other materials. Too often these are left out of the mathematics classroom even
though they increase mathematics achievement. "[I]t is essential that we infuse language
and thought into mathematics" (Hyde, 2007, p. 46). Teachers need to ask the right
questions that help students make connections and see patterns.
An Examination of Some Existing Resources

One might question the need for yet another problem solving resource as there are
so many on the market. In the past ten years I have seen and used many resources for
problem solving: however, none have addressed my need to aid in determining the
correct operation. I reviewed several resources that have similar aspects to the resource
that I am proposing to create.
Dan Greenburg has authored several resources that use humour and stories for
teaching math. His book Mega-Funny Division Stories (2002) contains 24 stories with
math exercises to accompany the story. It starts with the concept of division and
progresses in difficulty with questions involving remainders and decimals at the end.
This resource does not fulfill my needs because it is all division and is mostly related to

18

the process of division rather than problem-solving. I feel this resource will be effective
for introducing new concepts of division.
My initial thoughts when I looked at another Dan Greenburg resource titled,
Fractured Fairy Tales: Multiplication and Division (2005), was that I needed to come up

with another idea because it appeared to be what I was looking for. After trying it in my
classroom I have found it does not meet my needs for the following reasons. First is the
length of the stories. Some of my weaker students were not able to read the stories or it
took a significant amount of time to complete the reading. The most notable flaw was the
challenge to differentiate or tier the task. Often tasks or problems were embedded in the
story that prevented me from reading the story to the students. It was too difficult to
work through as a class because the completion rate varied so greatly. I also had trouble
tiering because of the organization of the book. The task and problem difficulty increases
from the start of the book to the end and not within a story. For some students, the stories
and tasks at the beginning were too easy and the stories and tasks at the end of the
resources were too difficult. I also found that problems did not match the curriculum
outcomes in B.C. For example, multiplication and division of fractions included in this
resource are not introduced until grade seven.
Dan Greenburg' s Comic Strip Math Problem Solving (2010) addresses the issue
of too much reading, with short comics and five word problems to go with them. The
book is organized into chapters based on skills, such as fractions or measurement. There
are eight comic strips for multiplication and division, which I have used in my classroom.
The students enjoyed these lessons; however, my students need more practice than what

19

this resource offered. As with previous resources it is difficult to include all learners in
my class, often my struggling learners were unable to participate in these activities.
Betsy Franco's resource, 20 Marvelous Math Tales (2000), is very similar to Dan
Greenburg's 30 Wild and Wonderful Math Stories (1992). Both of these resources have a
one page story and then word problems to accompany the story. The book is organized
according to which problem solving strategy is appropriate for solving the problem, such
as guess and check, draw a picture or find a pattern. I have used this resource when
teaching those specific strategies and have found them successful in modeling how to use
a particular strategy to solve a problem. Once again the reading required is a barrier to
some students.
Evan Moor Education books are familiar to most teachers. I have used Daily
Word Problems (Wurst, 2001), which has a word problem for each day of the week. The

problems increase in difficulty as the week progresses, with Friday's problem involving
multiple steps. Each week contains a mixture of skills, such as multiplication, perimeter,
time, and measurement. In Daily Math Practice (Tuttle, 1999) there are five questions
for each day including one word problem. As with daily math practice there is range of
skills addressed each day.
Although I have used some of these resources on occasion, the wide range of
skills required prevents me from using the resource as intended. Students may not know
how to complete a particular skill because I have not covered a certain topic yet that year.
For instance, I may not yet have covered perimeter in the school year. Even when it
should be a review from the previous year, I fmd my students need extensive review
before they are ready for word problems. Another reason for student difficulty is many

20

published resources are American and the curriculum outcomes do not match the
curriculum outcomes in B.C.
The literature confirmed my experiences in the classroom. Students have
difficulty solving math word problems, but personalizing word problems and embedding
them in literature activities can help improve student achievement.

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CHAPTER THREE: DEVELOPING THE WORD STORY
Source of Inspiration for this Pilot Program

The idea for this project came not from mathematics teaching literature but from a
language program. Specifically, it was from my success in using a program called
Caught'Ya (Kiester, 1993) to teach grammar. Caught 'Ya is a story that is written on the
board one sentence per day. In each sentence there are grammatical and spelling errors
the students correct. The story is about a classroom with a mean teacher. I give the
students five minutes to write the correct sentence and then as a class we correct the
sentence and I teach any new skills that arise. The students correct the errors they missed
in pen allowing me to quickly see what skills they understand and which need more
practice. The students look forward to this activity each day and even complain if we do
not complete it. The students' ability to find errors and correct them improved quickly
and the skills transferred into their writing. In the past when I taught grammar, it usually
included a lesson and a worksheet to follow and it was not something I or the students
particularly enjoyed.
Personal Location

I used my current class of grade 3 and 4 students to pilot the story and word
problems. My current classroom configuration is 18 boys and five girls. The boys are
very interested in sports such as hockey and basketball. The school is in a rural area and
many children have horses. Keeping these interests in mind I wrote a ghost story that
took place in our school with interests such as hockey and basketball in the story.
As the case with many classrooms, I have students with a great range of abilities.
It is often very difficult and time consuming to come up with activities and lessons that
address the needs of all the learners in my class. I created this story to allow all learners

22

an opportunity to participate in the classroom activity. Appendix A is a table that
indicates which operations are required to solve the problem, the answer, and if there is
specific mathematical knowledge required, such as regrouping.
Modifications To Resource

When I first thought of creating this math resource, I initially thought my story
would have the problems embedded in the story. After starting the literature review and
reflecting on my current class's needs, I realized that doing so would make differentiation
difficult. If the problems were embedded in the story, how could I make them
appropriate for all the learners in my class?
Initially I wanted to write problems that were more directly related to the story. I
found when writing the problems that this was difficult, especially given the short length
of the story segments. The original plan was to make thirty segments. After completing
ten segments in the classroom I combined some segments to make twenty. Completing
ten segments took three weeks as we didn't always have time to complete this activity
every day. I felt that two or more months would be too long and I worried the
engagement with the story would not be sustained over that period of time.
Some of the low achieving students completing the A questions were having
difficulty with the reading level of the problems. I re-wrote some questions to simplify
them and changed the number words to numerals to make it easier for the students. Most
students completing the A questions struggle with learning and I wanted to ensure they
could succeed with these questions.
Introducing The Resource

23

Before starting I discussed the different learning needs in our classroom and that
not all students would be completing the same questions. I discreetly let each child know
which question I wanted them to complete. I did not have to change assigned questions
too often. It occurred to me while completing the trial that the grade three students could
complete more difficult questions if given the opportunity to use a calculator. For
example, grade 3 students are not taught two-digit by one digit multiplication; however, I
could assign the problems and allow the grade three students to use a calculator to
complete the equations.
Time constraints would not allow me to mark each child' s work each day. The
self-monitoring check list was designed for students to keep track of their own progress
and allow me with a quick glance to monitor their progress. Students would indicate
whether they completed the problem and correctly identified and carried out the
operation. Figure 1 is an example of the student self-assessment.

Figure 1. Student Self Evaluation

The Ghost Upstairs
Date:
Problem
Problem
A

Problem
B
Problem

c

I wasn't assigned this
problem.

I got it right!

I chose the correct
operation, but I made
a small error.

I did not choose the
correct operation.

24

Ethical Considerations

This project was approved by the University ofNorthem British Columbia's
Research Ethics Board. The Principal of the pilot school and parents of students were
informed of the project. The lead characters were not children in the class; however, I
used the names of our Principal and custodian. It is important to personalize the story
and problems; however, it is important to avoid inadvertently causing problems.
Teachers should let all people who will be part of a story read the story and consent to the
use of their name as a character.
Teachers should also be aware of gender and ethnicity when deciding on names.
The pilot classroom was predominately boys and a community with very little
multiculturalism. Gender and names were picked to represent the classroom in which it
was being used. Teachers using this resource need to take this into consideration and
ensure there is a balance of representation of their classroom.
The following chapter is the resource which was developed to be used by
teachers. As well as the story and problems there is a short introduction that briefly
describes the research, and provides instructions for use.

25

CHAPTER FOUR: THE PILOT WORD STORY
Introduction

Teachers in today' s classroom are challenged with addressing the needs of diverse
learners in their classroom. It can be difficult for teachers to differentiate lessons that
allow all children to be included in the classroom instruction. This resource was
designed to meet the needs of different learners in math problem solving.
When teaching my own class, I noticed my students ' ability to determine the
correct operation of a given word problem was difficult after multiplication and division
were introduced. My students needed more practice answering word problems where
they have a choice of addition, subtraction, multiplication or division. I wanted to
present the problems in an interesting manner. My students love when I incorporate them
or staff at the school into word problems but it was challenging to continually come up
with my own problems. I looked for a resource, but couldn' t find one that fit my needs.
My students, especially the boys in my classes are often interested in ghost and mystery
stories. This gave me the idea to write a ghost story to accompany the word problems
which required the students to choose the correct operation.
Research
Personalizing word problems.

Education researchers have looked into the role of personalizing word problems
to increase student success. Several studies have indicated that when word problems are
personalized student success increases. Hart' s study (1996) compared personalized word
problems to generic text-book questions. Hart's results indicated that students had better
attitudes towards math when completing the personalized problems:

26

As the study progressed, it was clear to me that students preferred solving
personalized problems over "bland" text book versions. Their enthusiasm and
interest grew as I presented humorous stories or made them applicable to the
students' lives. I was pleasantly surprised to see the students perk up when
solving problems that involved their immediate environment. The classroom,
other students, teachers, the cafeteria and the entire school were popular topics in
the word problems. (Hart, 1996, p. 504)

Using literature in mathematics.
I have always enjoyed using literature to aid my lessons. Mathematics class,
however, was one where I did not incorporate much literature. Research has shown that
including literature in mathematics lesson is beneficial.
When language skills are embedded in meaningful contexts, they are easier and
more enjoyable to learn. In the same way, numbers and their operations, when
embedded in meaningful real world contests, give children the opportunity to
make sense of mathematics and to gain mathematical power. (Moyer, 2000, p.
248)

Using This Resource
Description
This resource is designed for students in grade 3-5. The story titled "The Ghost
Upstairs" is presented to students in twenty segments. After each section, three math
word problems based on the characters and events in the story are provided. The
difficulty of the problems increases with the third problem being the most complex.
Educators decide which problems each student is to complete and can discreetly assign

27

one, two or three problems to a student. A description of the how the problems are
organized is described in Table 1. Problem A questions are simple addition and
subtraction questions. These questions would be appropriate for grade 2 students or for
older students performing below grade level expectations. Problem B questions are for
students in grade 3 after instruction in multiplication and division. Problem C questions
are more difficult and are designed to challenge grade 3 students and be appropriate for
grade 4 students.
Table 1

Problem Descriptions
Operations

Single or

Regrouping

Multistep
Problem

Extraneous
Numbers

-,+

single

no

no

+, x,-,+

Mostly single

possible

possible

+, x,-,+

single or multistep

possible

possible

A
Problem

B
Problem

c
Instructions

It is important that names of places and characters are changed to personalize the
problems. The school, teachers, principal, custodian and students names should all be
changed prior to using this resource. All of those involved should read the resource and
approve the use of their name. Another option is to use names that are similar to the
person but not exactly the same. In addition to names and places, the students' interests
should be considered. There are problems about hockey and basketball, but, if the

28

students are in soccer, educators can easily change the questions to be about soccer
instead.
After students have had time to complete the problems, educators should model or
think aloud how to solve the problem or have students explain to each other and or the
whole class how they thought to solve the problem. The modeling of correct completion
is important especially for the learners who were not assigned a particular question. Even
though not all students will be given independent practice for each problem, they are still
exposed to those problems.
Although this resource is not designed to teach problem solving strategies such as
drawing a picture, guess and check, or fmd a pattern, these strategies can be discussed
during the modeling completing the problem. When appropriate, show or ask students
how they could have solved the problem another way using a problem solving strategy.
If students use these strategies instead of an equation, for example, a student may have
solved the problem 2 X 3 by drawing a picture of two groups of three. This should be
acknowledged as an acceptable strategy and then discussed how the student can use their
picture to determine the correct equation.
Progress Monitoring

While going over the problems the students should mark their own work. Having
the students complete their own assessments develops ownership of their own learning.
Students should complete the assessment record that is included in the appendix. This
record also allows the educator to quickly assess how each student is doing by simply
glancing at each child's record sheet. When students are having difficulty, their work can
be checked to determine what further instruction the student needs. Students should

29

show all work, which allows the educator to see what errors they are making. For
instance, a student may have understood the problem and chosen the correct operation but
have forgotten how to regroup over zero.
Students need daily practice solving word problems. When multiplication and
division are introduced it becomes increasingly complicated. This resource will provide
practice in a way that engages the students with literature. The three problems of varying
difficulty allow all students to be included in the activity at a level that is appropriate for
them. All students will benefit from teacher and student modeling of how to solve the
problems. This resource aims to engage students to improve their ability to solve
mathematic word problems.
Story and Problems
Segment 1

"Oh no!" groaned Robert, "Mrs. Smith is going to be so mad at me, this is the
second day in a row I forgot my homework at school." Robert was in grade 3 at Sparrow
Elementary. Mrs. Smith, his teacher was nice most of the time, but she got really upset
when you didn't do your homework. Robert was going to be in a lot of trouble unless he
walked back to school to get his homework. "I really don't want to walk back to school,
it's too cold and it's starting to snow, but who knows what Mrs. Smith will do if I don't,"
grumbled Robert.

A)

Robert forgot his homework twice this week. He also forgot his homework 3
times last week. How many times has he forgotten his homework in the last 2
weeks?

30

B)

Robert passed five fields on his way back to the school. In each field there were
three horses, which he stopped to feed some grass. How many horses did Robert
feed?

C)

Robert decided to stop at the Canyon store on his way back to school to buy some
candy. He bought three chocolate bars which cost $ 2.00 each. If he gave the
clerk twenty dollars, how much did he get back in change?

Segment 2
When Robert got back to the school, all the teachers' cars were gone, only Mr. J,
the custodian' s car was there and snow was starting to cover it. Robert knocked on the
door wishing Mr. J would hurry up so he could get inside and warm up.

A)

Earlier that day there were 3 blue cars in the parking lot and 4 red cars. How
many cars in total were in the parking lot?

B)

Mr. J has to clean eight classrooms each night. How many classrooms does he
clean in five days?

C)

Each day Mr. J needs to fill the toilet stall washrooms with toilet paper. There are
five stalls in the girl ' s washrooms and two in the boy' s washrooms. The toilet
paper comes in packs of nine. How many packs of toilet paper will Mr. J use in
five days?

Segment 3
Suddenly someone grabbed his shoulder from behind! "Ahhhh! " screamed
Robert. "Hey don ' t worry kid." It was just Mr. J, who had been out back to put the
garbage in the dumpster.

31

"Yikes Mr. J, you scared me! I just need to get my homework from my desk" said
Robert.
"No problem," Mr. J replied, "you just go on down to your class, I will be in the
library if you need me." Robert's class was at the end of the hall, next to the stairs that
led to the school attic. Students were not allowed up there and all the students said it was
haunted up there.

A)

Robert had 5 books checked out the library, he returned 2. How many books does
he still have checked out?

B)

In the library there are twelve computers, three on each table. How many tables

have computers?
C)

Mrs. Baker, the librarian had rearranged the Captain Underpants books. She has
forty eight books. Six books will fit on each level of the rack. How many levels
will be Mr. J see when he spot them in the library?

Segment 4
Robert grabbed his math book and headed back down the hall. As he was passing
the stairs, he thought he could hear a groan coming from the attic. Robert stopped and
listened again, yes he definitely could hear a groan. Roberts's heart started to race, what
could that be? "I am getting out of here!" screamed Robert.

A)

For homework Robert has to do 30 problems. He did 10 at school. How many
questions does he have left to do?

32

B)

Robert's homework includes thirty questions including subtraction and addition.
There are eighteen subtraction questions. How many are addition questions?

C)

Robert has completed six addition and nine subtraction questions of his thirty
questions for homework. How many more does he need to complete?

Segment 5
That night Robert could not sleep. He didn't tell his parents about the groaning he
heard, he thought they would make fun of him for being a baby and being scared of
ghosts that didn't exist. He knew he heard a noise and he needed to come up with a plan
to figure out what it was.

A)

When Robert finally got to sleep, he slept for 2 hours and then woke up. After he
got a drink he slept for 3 more hours. How many hours of sleep did Robert get?

B)

When Robert finally fell asleep he dreamed about dilly bars at Dairy Queen. He
had found six boxes in his freezer. Each box has eight dilly bars in it. How many
dilly bars did Robert find?

C)

Before bed Robert took his vitamins and ate four cookies. He takes two vitamins
each night before bed. How many vitamins does he take in two weeks?

Segment 6
Robert found his best friend Chris as soon as he got to school the next day. "Hey,
I was at the school last night and I heard something or someone groaning from the
upstairs attic." Chris started to laugh.
"Oh come on Robert, are you being serious?"
"Yes!" insisted Robert, "Come with me tonight and you can hear for yourself."

33

A)

At recess, Robert and Chris played basketball. Robert scored 12 points and Chris
scored 9. How many more points does Chris need to beat Robert?

B)

Robert and Chris organized a soccer game at lunch. There were sixteen players,
how many players were on each team?

C)

Chris, Robert and Tyler were the only ones who scored during a basketball game
at lunch. Tyler scored three, Chris two, and Robert scored twice as many goals as
Chris. How many goals did Robert score?

Segment 7
Later that night Robert and Chris returned to the school. Even though Chris
didn't believe there was anything in the attic, he felt a little nervous. The boys walked to
the front door and peeked through the windows and knocked loudly. Down the hall a
man appeared, but it was not Mr. J, this person was much shorter and had gray hair that
was all frizzy and covered up most of his face. The person did not look at them, but
walked slowly upstairs to the attic.

A)

Last week 3 school windows were smashed. 15 windows were not broken. How
many windows are in the school?

B)

The school did a bake sale to help pay for the repair of windows that were
smashed. They wanted to raise $100.00. They made $25.00 on cupcakes and
$37.00 on cookies. How much money do they still need to make?

34

C)

The class made eight dozen cookies and ten dozen cupcakes to sell at the school
bake sale. They used 32 cups of flour to make the cookies. How many cups of
flour does one dozen call for?

Segment 8

The boys froze in fear. Chris yelled, "Get out of here!" The boys turned and raced
back towards Chris' s house, stopping only when their lungs felt like they were going to
burst open. Neither boy said a word as they caught their breath. "I told you," Robert
finally said.
"We need to tell someone," said Chris. "Do you remember what you thought of
me when I told you I thought there was a ghost upstairs? You thought I was crazy!"
exclaimed Robert.

A)

Chris ' s house is 4 blocks from the school. They ran for 3 blocks before taking a
break. How much farther to Chris's house?

B)

Chris ' s house is four blocks from the school. Robert' s house is twice as far from
the school than Chris's house. How far is Robert' s house from the school?

C)

Chris lives four blocks from the school. It takes Chris 3 minutes to walk a block,
how long does it take for Chris to walk home?

Segment 9

Back at home Robert and Chris tried to think of plan. They couldn ' t use their
excuse of forgetting homework again or their parents were going to get angry. "How
about I forget my skates at school tomorrow?" asked Chris.

35

"We are going skating with my class and I have hockey practice after dinner so I
will have to go back to get them."
"Great idea," replied Robert.

A)

At hockey practice Chris scored 3 goals in the first period and 2 in the second
period. How many goals did he score?

B)

There are twenty-eight games in Chris ' s hockey season and he has already played
nine games. How many games does Chris have left?

C)

For a hockey drill at practice the coach needed four teams and there were twenty
seven players. How many players will be on each team? Will there be any players
left?

Segment 10
The next day at school, Robert and Chris glanced anxiously up the stairs to the
attic every time they passed. Straining their ears they couldn ' t hear any sounds coming
from the attic. They wanted to tell the other kids, but they were worried about being
made fun of.
Just as planned, Chris left his skates at school. When he got home, he told his
Mom he needed to go back, and Robert would be going with him. They met at the store
and walked to rest of the way to school together.
A)

To go skating each child had to pay 3 dollars for skating and 2 dollars for the bus.
How much did it cost to go skating?

B)

Three busses came to take the whole school skating. Each bus took forty-eight
kids. How many students went skating?

36

C)

The rec centre parking has ten rows. Each row has fourteen spaces. Eighty-seven
spots were empty. How many cars were parked?

Segment 11
Neither wanted to admit it, but their stomachs were churning. "Maybe this isn' t
such a good idea," Robert suggested.
"Don 't be such a baby," replied Chris, not wanting to let on that he was nervous
as well. At the school, they were relieved to see Mr. J outside taking the garbage out.
Mr. J let them inside and told them to let themselves out when they were done. They
walked down the hallway and stopped at the bottom of the stairs to the attic.

A)

There are 6 classes in the hallway. 4 classes have their doors open. How many
doors are closed?

B)

There are forty-five students in grade three at Sparrow Elementary. Mrs. Smith
has twenty four students in her class, the rest are in Mr. Simpson' s grade 3/4
class. How many grade three students does Mr. Simpson have?

C)

Sparrow Elementary has 175 students. There are four classes of intermediate
students. If there are twenty-eight students in each class, how many intermediate
students go to Sparrow Elementary?

Segment 12
Slowly they walked up the stairs, with every creak of the stairs they grew more
anxious. When they reached the top of stairs, they could see that the door was slightly
open. "You go first," whispered Robert.

37

"No way gizzard breath, you go first," hissed back Chris. Robert pushed the door,
open just a little.
"Hey, what are you doing! " The boys jumped and screamed. It was Mr. J "You
two are not supposed to be up there, what you are up to?"
"We thought we heard something," stammered Robert.
"Well there is nothing up there but a bunch ofboxes, you two better get on home
now." Mr. J grumbled back. The boys turned and walked back down the stairs and out
of the school.

A)

In the attic there are elf costumes. 7 are red and 8 are green. How many elf

costumes are there?
B)

There were nine strings of lights in the attic. Each string has 8 lights on them.
Thirteen lights were missing. How many lights were there?

C)

The Christmas tree ornaments are stored in the attic. There are thirteen boxes of
ornaments and each box has eight ornaments. There are six boxes that are
missing four ornaments each, the rest have all of their ornaments. How many
ornaments are in the attic?

Segment 13

"Maybe it is not a ghost, maybe Mr. J and whoever that other guy was kidnaps
kids and keeps them prisoners up there," whispered Robert.
"Well no one has gone missing and he could have just easily captured us, so I
don ' t think so," replied Robert.

38

"Now what should we do?" They couldn't think of anything, so they walked
home.
That night neither boy could sleep. They were out of ideas to figure out what was
going on in the attic.
When the boys got to school the next morning, they decided it was time to tell
their other friends, even ifthey got made of. "You guys are so stupid," laughed Karen,
"ghosts in the attic, why don't you guys go home to your mommies for a nap?"
"I told you, we shouldn't have said anything," whispered Chris.
"Yeah you were right, now everyone thinks we are crazy and we're the laughing
stocks ofthe school," groaned Derrick.

A)

In gym class Mrs. Smith made the class run 6laps of the gym, skip 3 laps and

walk backwards I lap. How many laps did they do of the gym?
B)

At school that day Robert had a pack of gummy bears. He shared the pack with
four friends and there were no gummy bears left over. How many gummy bears
could have been in the bag? Give three possible answers.

C)

At recess Robert ate thirty candies. The candies came in packages of nine. How
many whole boxes did he eat and how many candies does he have left?

Segment 14

Later that afternoon as the class was silent reading, they could a faint thumping
coming from upstairs. All the students glanced nervously at each other. Tanner asked
Mrs. Smith what the noise was but she just said it was nothing and to get back to reading.

39

The next morning, Derrick was anxiously waiting for Chris and Robert at the
school gate. "You will never believe what I saw last night. I was here playing hoops with
my older brother. A big truck showed up and Mr. J came outside. When he saw me he
looked all nervous and whispered to the other men. Then they went into the truck and
came out carrying a huge box, it took all three of them to carry."
"What was it?'' asked Chris.

A)

During silent reading Robert read 5 pages of his book. His book has 15 pages.
How many pages does Robert have left to read?

B)

Chris and Roberts are trying to reach a 500 minutes reading goal. Chris has read
56 minutes and Robert seventy-eight. How many minutes do they have left to go?

C)

Chris's book for silent reading has six chapters. There are between eight and
twelve pages per chapter. About how many pages are in Chris ' s book?

Segment 15
"Well that was the strangest part," replied Derrick. The box was covered by a
blanket. "Why on earth would they have to cover it up? I think you guys were right there
is defmitely something going on here and I am getting a little nervous. I think we should
come tonight, hide and see if we can figure out what is going on."
After dinner the boys met at the store. "Ok I think we should hide under the slide,
that way we can see the front of the school and if we go around back we will be able to
see in the window by the attic stairs," suggested Chris.

40

A)

The playground at the school has 14 swings, but 3 are broken. How many swings
are left?

B)

The school wants to build a new playground. Each child is supposed to raise
$5.00. So far nine children have brought their money. How much have they
raised so far?

C)

At a bake sale fundraiser for the playground, cookies cost $1.00, cupcakes cost
2.00, and a piece of cake was 3.00. Chris had $9.oo to spend. What combinations
of goodies could he have bought?

Segment 16

The boys watched and waited but saw nothing. Mr. J came out of the school a
couple times to put garbage in the dumpster but that was it. Just as they decided it was
time to go home another truck turned into the school yard. Surprisingly, Mr. Hooper,
their principal pulled into the yard just after. The boys watched as another huge box was
carried into the school. This time they didn' t use a blanket but the box didn 't have any
writing on it.

A)

Mr. J brought 2 bags of garbage and then later 9 more bags. How many bags of

garbage did he bring outside?
B)

Garbage bags are sold in packs of seventy-two. Each day Mr. J uses nine bags.
How many days does one package last?

C)

The garbage truck empties the garbage bin twice a month. In December, May and
June it is emptied three. How many times in a year is the bin emptied?

Segment 17

41

The boys hurried to the window and watched as the box was carried up the stairs
into the attic. Mr. J and Mr. Hooper didn't come back down the stairs for a long time,
when they did the boys heard Mr. Hooper laugh, and clap his hands, "I can't wait until
tomorrow and see the looks on the kids' faces when they see what is in store for them."
He had such an evil grin his face. The boys couldn't believe Mr. Hooper was involved in
whatever was going on.

A)

There are 14 girls in Grade 3 and 13 boys. How many students are in the Grade 3
class?

B)

Mr. Hooper is forty five years old. He has been a principal for fifteen years. He

was at a different school for six years before coming to Sparrow Elementary.
How many years has he been at Sparrow Elementary?
C)

Mr. Hooper holds a floor Hockey tournament every Friday at lunch. One the last

day of the tournament he is having a pizza party for the 26 players. Each player
can have two pieces of pizza, A large pizza costs $7.00 and has 8 pieces. How
many pizzas does Mr. Hooper need to buy?
Segment 18
On the way home, the boys decided it was time to tell their parents what was
going on, there was no way they would be going to school in the morning. Unfortunately
for the boys, their parents wouldn't listen to them no matter how much they pleaded and
the boys were forced to go the school in the morning.
Just before lunch, Mr. Hooper came into the class and told the class he had a
surprise for them. He asked the class to line up and then he walked down the hall

42

towards the attic stairs. Derrick, Chris and Robert were at the very back of the line.
"Make sure you don't let the door shut behind us, if something happens we can escape
back down the stairs, "whispered Derrick.

A)

Chris has 2 brothers and 1 sister. How many siblings does he have?

B)

Two houses on Robert' s street have three children each, one house has four
children, two houses have two children each. How many children live on
Robert' s street?

C)

Robert and Chris play ball for one hour on Tuesday and Thursday, and two hours
on Saturday. How many hours do they play in six weeks?

Segment 19
As the kids in the front of the line reached the top of the stairs and into the attic,
they heard them gasp and scream. "Run!" yelled Robert, the boys turned and raced down
the hall and out of the school as fast as they could. Mr. Hooper was fast on their tail,
yelling at them to come back at once but they knew better and continued to run. Mr.
Hooper was pretty fast for an old guy and he quickly caught up to them.
"What on earth is going on with you boys?" he asked.
"We know there is something going on in that attic we have seen and heard
strange things going on" replied Chris.
"You are right boys, there is something going on, but there is nothing to be scared
of, come let me show you."

43

A)

There are 12 girls on the school's soccer team and 15 boys on the boys' team.
How many children play soccer on the school teams?

B)

There are forty children that play chess, if twenty children can play in one day,
how many children can play in six days?

C)

There are 9 children on each baseball team. There are two girls' teams and three
boys' teams. How many children play baseball?

Segment 20
When they came back into the school the boys thought it was end of them. They
walked slowly up the stairs. When Mr. Hooper opened the door, they could not believe
their eyes, the attic has been turned into a huge playroom with a ping pong table, foosball
table and places to play board games. "You boys almost caught us several times, we
really wanted it to be a surprise. Now when it is raining you can have a great place to
hang out at lunch. I want you to thank Mr. Kangle. He is the one who donated his labour
to fix this place up" said Mr. Hooper. The boys turned and saw the man with the crazy
hair who they thought had been a ghost. Boy did they feel silly, they shook the man's
hand and thanked the man.
"No problem, now why don't you get busy trying this ping pong table out," he
said. The boys grinned at each other and laughed.

A)

The first player to 10 wins the foosball game. Chris has 6 points. How many more
points until he wins?

B)

The new floor cost two dollars a square foot. The attic space was five hundred
square feet. How much did the new floor cost?

44

C)

The School raised four thousand dollars. The foosball table cost five hundred
dollars and the ping pong table cost two hundred and fifty dollars. How much did
they have left for board games and furniture and a new floor?

45

CHAPTER FIVE: EXPERIENCES IN THE CLASSROOM
Motivation

The class was very receptive to the story and problems. They were engaged with
the story right from the start. I randomly drew two students ' names for the main
characters and then changed all names appropriately for our school. They thought it was
really funny to have myself, the principal and the custodian as characters in the story.
Every student reported on the questionnaire that they enjoyed completing the word
problems that went with the story versus text-book or problems I had previously written.
Several students commented that they really enjoyed the story, that it was exciting and
they liked how they didn 't know what was going to happen next. This increase in
motivation is important. Having students hooked at the beginning of a lesson keeps the
students engaged in the activity and increases their ability to focus and learn.
Differentiation

Another goal of the resource was to differentiate practice under a common
assignment. As expected some C questions were challenging for the students, especially
the multistep problems. After the initial ten questions, I reduced the difficulty of some,
as I could see some were getting frustrated and simply not even wanting to try the C
questions. When reporting the difficultly level for themselves, the results were varied.
One student reported, "I feel the word problems are just right sometimes and sometimes
really hard." Another student wrote, "Some were too hard, but some were good, like the
A and B questions."
Progress Monitoring

The assessment record, described in Chapter Three, worked well. I could quickly
walk around the room and see how each student did. I did not have students hand in

46

work; however, I sometimes students were not honest and were checking that they got the
question right when in fact they had got it wrong. To prevent this from occurring I told
the students that I would periodically have them hand in their work and assessments
records to ensure they were being honest. I checked once and I did not observe further
instances of dishonesty on assessment records.
The modification to shorten to twenty assignments was beneficial. Classroom
routines inevitably get disrupted by activities going on in the school- assemblies, field
trips, etc. It took just over five weeks to complete the twenty segments. This amount of
time allowed the students to get into a routine but was not too long that they started to
lose interest. Another two or three weeks to complete would have been too long.

47

CHAPTER SIX: DISCUSSION
Summary of Study

The Ghost Upstairs was created to increase motivation, offer more practice with

word problems, and differentiate learning. The personalized ghost story increased and
maintained student attention. The three problems of different difficulty level allowed all
students to participate and practice problem solving. There are some changes that will be
made and opportunities to expand on this project.
Limitations

Teaching students to effectively problem solve is a complex process. This
resource is not meant to be a complete guide to problem solving. The goal of this
resource is for children to identify the correct operation. There are several problem
solving strategies such as drawing a picture, creating a table, or finding patterns that are
not explicitly addressed in this resource.
This resource is narrow in focus intentionally. As previously discussed I found
resources that were too broad in content areas difficult to use. Word problems that
address other content areas such as geometry, ratios, fractions, are equally important, so
teachers will need to develop or look to other resources for these topics.
During my literature review on problem solving I found several studies related to
reading comprehension and problem solving. In an effort to keep the focus narrow, I
neglected to address this. I regret this omission and think that more of an effort to
include language cognition would have improved the resource.
Because the story segments so short it was difficult to write problems that were
more related to events going on in the story. The students did not make any comments

48

about this, but it would be an improvement to have them more imbedded in the story
events.
In the resource Caught 'Ya, described in chapter 2, all students are eventually part

of the story. Students look forward to hearing what their character is going to say and do.
If I were doing it over I would be to have enough characters so all students would be
named at least once. Using names of students in the class increases the personalization
and motivation.
Required Revisions

Before using or sharing this resource some revisions need to occur. There are
several grammatical errors that need to be addressed. Several questions have ambiguous
phrasing and need to be re-worded. For example in segment 12 question A, "In the attic
there are elf costumes. 7 are red and 8 are green. How many elf costumes are there? It is
unclear if there is any other color of costumes; the reader has to assume there is not.
Ambiguity can be eliminated by re-wording the question. "In the attic there are two
colors of elf costumes. 7 costumes are red and 8 costumes are green. How may elf
costumes are there?" I regret that I did not have someone with a strong mathematical
background review the story and problems before the pilot.
Implications for Future Research

Ideally I would have wanted to complete a study that determined if this resource
increased student ability to determine the correct operation in a word problem,
unfortunately that would be a project on its own. As is stands I do not have any data
except for the qualitative data that my students enjoyed this method versus ways I taught
problem solving previously. A study similar to that of Ku and Sullivan (2002) described

49

in chapter 2, randomly assigning students to receive practice using this resource or
standard text questions could be completed. My school district has opportunities for
action research projects; therefore, this is something that could be completed in the
future.
Implications for Future Practitioners

Teachers can use this resource as is or use is as a start to create their own which
reflects the interests of their own students. My current school is small which results in
spilt classes. Often students remain with the same teacher two years in a row which
restricts its use. Using this resource as a template, teachers could collaborate on
professional development days to create another story and problems for use in subsequent
years. Having the class collectively work to create their own story would be an
interesting assignment. The class could work as a whole or in small groups to write the
story and the accompanying problems.
Conclusion

Word problems can cause frustration for both students and teachers. The resource
developed aimed to increase motivation, give students more practice, and allow all
children to participate. The pilot completed in my own classroom was successful.
Qualitatively the results were similar to research reviewed and my expectations. Students
reported that they enjoyed the story and problems and preferred it over previous
instructional methods. Students looked forward to finding out what was going to happen
next. A quantitative study is needed to determine if using this resource results in greater
ability to solve word problems.

so
Using this resource allowed me to give students daily practice solving word
problems with little preparation. It is very difficult to address all the learners in one
classroom, but the three levels of problem difficulty allowed all learners to participate.
To maintain self-esteem in learners who have more difficulty than others, it is important
that they are not always singled out with activities different from the rest of the class.
After making some modifications, my colleagues and I will fmd this a valuable
resource to use in the classroom. By using it as a template, there is an opportunity to
create more stories further enhancing its benefit.

51

References
Alberta Government of Education (2012) Alberta provincial achievement testing:

Assessment highlights grade 3. Retrieved from Education Alberta
website:http: //www.education.alberta.ca/media/6807592/04%20math3%20as
sess%2020 12_ book%20signoff.pdf
Bates, E. T., & Wiest, L. R. (2004). Impact ofpersonalization of mathematical word
problems on student performance. Mathematics Educator, 14(2), 17-26.
Clark, P. G. (2004). The effectiveness of story-framed instruction in mathematics
(Unpublished masters's project). University ofNorthem British Columbia,
Prince George B.C.
Edudatacanada (2012) FSA 2012 item level response report. Vancouver B.C:
University of British Columbia. Retrieved from Edudatacanada website:
http: //www.edudata.ca/apps/fsa_itern/en/dO/sO/t0/2012/NUE/4/MC/
Franco, B. (2000). 20 marvelous math tales. New York: Scholastic.
Greenburg, D. (1992). 30 wild and wonderful math stories to develop problem

solving skills. New York: Scholastic.
Greenburg, D. (2005). Fractured fairy tales: Multiplication and division. New York:
Scholastic.
Greenburg, D. (2002). Mega funny division stories. New York: Scholastic.
Greenburg, D. (2005). Fractured fairy tales: Multiplication and division. New York:
Scholastic.
Greenburg, D. (2010). Comic-strip math problem solving. New York: Scholastic.

52

Hart, J. M. (1996). The effect of personalized word problems. Teaching Children

Mathematics, 2(8), 504.
Hyde, A. (2007). Mathematics and cognition. Educational Leadership, 65(3), 43-47.
Jenner, D. M., & Anderson, A. G. (2000). Experiencing mathematics through
literature: The story ofNeil. Teaching Children Mathematics, 6(9), 544-547.
Kiester, J. B. (1993). Caught 'Ya again!: More grammar with a giggle. Gainesville,
FL: Maupin House
Ku, H., & Sullivan, H. J. (2000). Personalization of mathematics word problems in
Taiwan. Educational Technology Research and Development, 48(3), 49-59.
Ku, H., & Sullivan, H. J. (2002). Student performance and attitudes using
personalized mathematics instruction. Educational Technology Research and

Development, 50(1), 21-34.
Levy, H . M. (2008). Meeting the needs of all students through differentiated
instruction: Helping every child reach and exceed standards. Clearing House:

A Journal of Educational Strategies, Issues and Ideas, 81(4), 161-164.
Little, C. A., Hauser, S., & Corbishley, J. (2009). Constructing complexity for
differentiated learning. Mathematics Teaching in the Middle School, 15(1),
34-42.
Lopez, C. L. , & Sulivan, H. J. (1991) Effects of personalized math instruction for
Hispanic students. Contemporary Educational Psychology, 16(1 ), 95-100.
Lopez, C. L., & Sulivan, H. J. (1992) Effects of personalization of instructional
contexts on achievement and attitudes of Hispanic students. Educational

Technology Research and Development, 40(4) , 5-13.

53

Moyer, P. S. (2000). Communicating mathematically: Children's literature as a
connection. The Reading Teacher, 54(3), 246-255.
Seifi, M., Haghverdi, M., & Azizmohamadi, F. (2012). Recognition of students'
difficulties in solving mathematical word problems from the viewpoint of
teachers. Journal of Basic and Applied scientific Research, 2(3), 2923-2928.
Tuttle, W. (1999). Daily math practice-grade 4. Monterey, CA: Evan-Moor.
White, M. (2008) Changes in specialist teaching positions and student
enrolment;2001-02 to 2007-08. Retrieved from British Columbia Teachers'
Federation website:
https://www. bctf.ca/uploadedFiles/Publications/Research_reports/2008WLC
Ol.pdf
Wurst, S., & Wurst, D. (2001). Daily word problems-grade 3. Monterey, CA: EvanMoor.

54

APPENDIX A

Skill description and Answer Key
Answer

Operations

Single or
Multistep

Addition/
Subtraction
Regrouping

Extraneous
Numbers

+

+-

Single
Single
multi

no
no
yes

no
no
no

Single
Single
Multi

no
no
no

no
no
no

Segment
Question
Question
Question

1
A
B

c

5
15
14

Segment
Question
Question
Question

2
A

7

+

40
3

+, X,-:-

Segment
Question
Question
Question
Segment
Question
Question
Question

3
A
B

Segment
Question
Question
Question
Segment
Question
Question
Question
Segment
Question
Question
Question
Segment
Question
Question
Question
Segment
Question

B

c

X

X

-

c

3
4
8

Single
Single
Single

no
no
no

no
no
no

c

20
12
15

+ '

Single
Single
Multi

no
yes
yes

no
no
no

5
48
28

·+

Single
Single
Single

no
no
no

no
no
yes

-

c

4
8
4

X

Single
Single
single

no
no
no

no
no
yes

A
B

18
38

+
+, -

Single
Multi

no
yes

no
no

c

4

Single

no

yes

no
no
no

no
no
no

no

no

4
A
B

Add itional
knowledge
required

Borrow over
0

remainders

Borrowover
0

5
A
B

c

X
X

Days in a
week

6
A
B

7

8
A

-

c

1
8
12

X
X

Single
Single
Single

A

5

+

Single

B

9

Borrow over
0
Dozen =12

55

Question
Question

B

c

Segment 10
Question A
Question B
Question c
Segment 11
Question A
Question B
Question c
Segment 12
Question A
Question B
Question c
Segment 13
Question A
Question B
Question

c

Segment 14
Question A
Question B
Question

c

Segment 15
Question A
Question B
Question c
Segment 16
Question A
Question B
Question c
Segment 17
Question A
Question B
Question c
Segment 18
Question A

Answer

Operations

Single or
Multistep

Extraneous
Numbers

19
6,3
left

-

Addition/
Subtraction
Regrouping

Single
Single

yes
yes

no
no

5
144
53

+

Single
Single
Multi

no
no
no

no
no
no

Single
Single
Single

no
no
no

no
no
no

Single
Multi
Multi

no
yes
no

no
yes
no

Single
Single

no
no

no
no

Single

no

no

+ I

Single
Multi

no
yes

no
no

X

Single

no

no

11

-

45
vario
us

X

Single
Single
Single

no
no
no

no
no
no

Single
Single
Multi

no
no
no

no
no
no

no
yes
no

no
yes
yes

2
21
112
15
59
80

10
10,15
,20
3,3
left
10
366
About
60

X

X I

X

+
+,X,-

-

-

+

+

11
8
27

x,+

27

+

9
7

+,x

Single
Single
Multi

3

+

Single

-

Additional
knowledge
required

Remainders

2 digit by 1 X

2 digit by 1
digit X

remainders

Regroup over
zero
Estimate

Months in a
year
Siblings
remainders

56

Question
Question

B

c

Segment 19
Question A
Question B
Question c
Segment 20
Question A
Question B
Question c

Answer

Operations

Single or
Multistep

Extraneous
Numbers

14
24

Addition/
Subtraction
Regrouping

+
+,x

Single
Multi

no
no

no
no

27

+

Single
Single
Multi

no
no
no

no
yes
no

Single
Single
multi

no
no
no

no
no
no

120
45

X
X,+

4
1000
2250

X
+ -

'

Additional
knowledge
required

Days in a
week