Difficulty of Fraction
Learning: Dilemma in
Mathematical Teaching
Keith Mousley
Christopher Kurz, PhD
SWCED
Houston, TX
July 31, 2012
Newspaper Excerpt:
IDIOT SIGHTING
“We had to have the garage door repaired. The
Sears repairman told us that one of our problems
was that we did not have a "large" enough motor on
the opener. I thought for a minute, and said that we
had the largest one Sears made at that time, a 1/2
horsepower. He shook his head and said, ‘Lady, you
need a 1/4 horsepower.’ I responded that 1/2 was
larger than 1/4. He said, ‘NO, it's not. Four is larger
than two..’ We haven't used Sears repair since.”
Definition of fraction
—  History: Fraction did not originally have a mathematical
sense. Latin word in verb means “to break”.
(thefreedictionary.com/fraction)
—  Formal definition in Algebra is: q(x) = h(x)/g(x) where
h(x) and g(x) are rational numbers and g(x) can not be
equal to zero. ( Any algebra book)
—  A ratio of two expressions or numbers other than zero.
—  Is a part of a whole. ( www.mathsisfun.com/
fractions.html)
Overview
Literature Review
Research Questions
Methodology
Findings
Summary
Q&A
Review of Literature
—  Lag in deaf students’ achievement in mathematics
— 
— 
— 
— 
Basic concepts (e.g., Kritzer, 2009)
Mathematics computations (e.g., Traxler, 2000)
Problem solving (e.g., Qi & Mitchell, 2007)
Curriculum/teaching (e.g., Mousley & Kelly 1998)
Review of Literature
—  Deaf children’s mathematics ability
—  Represent numbers very well…reproduce a set of
objects of the same quantity (Zarfaty, Nunes & Bryant,
2004)
—  There is a lag in knowledge of number sequence but
good conceptual knowledge (i.e., object counting and
cardinality) (J. Leybaert & VanCustem, 2002).
—  Visual-spatial relationship representation was a strong
prediction to conceptual understanding (Blatto-Vallee,
Kelly & Gaustad, 2007).
Review of Literature
—  Deaf children’s fractional concepts
—  ages of 10 and 16 years – on average, were behind
their hearing peers in terms of their understanding
of fractional size (Titus, 1995)
—  Abstract concepts of “comparisons” are lacking…
(Part to whole; miles per hour; ratio)
—  Problems can be language-based (Titus, 1995;
Markey & Booker, 2003)
Purpose of Research
—  To investigate the comprehension of fractional concepts
in deaf children
—  To identify potential factors that promote or delay such
understanding.
—  To determine what age deaf children start to develop
attitudes about fractions
Research Questions:
A. Is there a difference in scores on fraction
test between the identified variables (i.e.,
parents’ demographics variables and
participants’ demographics variables) ?
B. In which representation(s) do deaf students
understand fractional numbers the best?
C. What common misconceptions of fractions
are demonstrated by deaf students?
D. What factors may prevent learning and
understanding?
Mixed-Method Methodology: Data
Collection
Data Collection
Parents: Survey
• Demographics
• Mathematical
exposure at home
Students:
Written Instrument
• Magnitude
• Order
• Equivalence
Students: Interview
•  M.O.E.
•  Attitude
•  Thinking
•  Talking
Research Instruments
—  Interview
—  30-45 minutes , Semi-structured, videotaped
—  Background information, question items, and question probing for
thought process
—  Non-standardized Paper Test
—  20 question items related to fractions
—  Knowledge of fraction sizes, order fractions with like and unlike
denominators, and fraction equivalents
Hold on… wait a minute.
—  Prove this:
12
2
=7
—  Think of the Roman Number:
€
—  XII
—  XII
now divide it by 2. Ummmm
become VII, there you go!
Sample questions
Sample question..
Sample Question…
Sample question..
Participants….
Characteristics
n = 14
Gender:
ª Female
ª Male
7
7
Grade Ranges:
ª -4
ª 5-7
ª 8+
2
8
4
School Settings:
ª Center
ª Mainstreamed
5
9
Parent’s Highest Educational Attainments:
ª Grade 8
ª HS
ª Associate Degree
ª BS
ª MS
ª N/A
Deaf Sibling(s)
ª Yes
ª No
1
5
3
4
1
4
10
Interview Findings: Themes
1.  Attitudes Towards Fractions
2.  Initial Understanding of Fractions
3.  Confusion about Fractional Concepts
4.  “Seeing”, “Understanding,” & “Doing”
Personal Experiences with Fractions
5.  Lack of Real-World Applications
Attitudes Towards Fractions
Sample Responses
Positive:
“Fractions are OK. My dad and I use fractions
for wood construction.”
Negative:
“I hate fractions. I do not understand them.”
“My mind goes blank every time I see a
fraction.”
Participant Responses
More Sample Responses
Explain why everyone should learn about fractions.
—  “Yes, it is important for future jobs.”
—  “Yes, it is something you use for work.”
—  “Yes…because my teacher told me so…I don’t know.”
Initial Understanding of Fractions
—  “I remember we did 1/2 first, then 1/3, 1/4 and
1/5. After that, we did not do much.”
—  “In class, I made a fraction bar paper
construction where we use 1/2 and 1/2 for one
bar. And, 1/3, 1/3, and 1/3 for the next bar…I
then compare fractions like 1/2 and two 1/4s to
see if they are the same….yes, the width is all
the same for the bars.”
—  “I have to relearn fractions every year.”
Confusion about Fractions
—  “4/4 is like 0, because it means all pieces
were eaten, so it became zero.”
—  “2/7 is like, 2 pieces are shaded and seven
are not.”
—  “1/3 is smaller than 1/7 because the size of 7
is bigger than the size of 3. You see, if I draw
a bar of seven pieces, it is longer than a bar
of three pieces.”
—  “I don’t know what it is. Isn’t it a fraction?”
Confusion: continue
There are twelve children going on a field trip.
If half of the children are girls, how many girls
are going on the field trip?
More on confusion:
Response Samples
A girl and her brother have some money to spend. The
girl spends 1/3 of hers and her brother spends 1/3
of his. Is it possible that the girl and her brother
spent the same amount of money? Tell me what you
are thinking.
Responses:
“Yes, they spent the same because they spent 1/3 of their
money.”
“It depends on how much they earn for allowance. If the
girl earns more allowance than her brother, she would
spend more.”
Response Samples
Four children want to share three candy bars so that
each child gets the same amount. Show how much
one child can have?
“Each child gets ½ of the candy bar…” “Well, they all can
have ½ of the candy bar so that means it is equal for all of
them.”
“They can go to store to get one more candy bar.”
“Seeing,” “Understanding,” and
“Doing” Fractions
“I see fractions everywhere…like shopping. You
know ½ off the price of something…I do not
use it to figure how much it would cost after
they take off the half.”
“My father has a shop, so we use fraction
measurements for cutting woods. I feel
comfortable using fractions.”
“I see fractions in school only. Usually we spend
3 weeks working with fractions.”
Real-Life Applications
“My teachers does not give word problems with
fractions.”
“My mother loves to cook, so I would help her
sometimes. We have to follow the ingredients in
recipes and we use measuring cups that have
fractions.”
“I use a ruler to measure things in class, but I do
not use it at home.”
“We do not usually talk math or fractions at home
or with friends.”
How about this?
—  A chef at the famous cooking school was teaching his
son how to make chicken noodle soup…
—  The Chef first said…
—  Start with ½ of a pot of water.
—  Then… ¼ of a pot of chicken broth
—  Then…. 1/3 of a pot of cream of chicken
—  Dad, Dad, Dad, wait wait…that is more than one pot..
—  Chef said, huh, what.. Just get a bigger pot.
Fractional Number
—  They all could identify
numerator and denominator
except for the fraction bar.
—  “I don’t know what it is. Isn’t
it a fraction?”
—  “It is a line that separates
numerator and denominator. I
don’t know.”
—  “I only know what the numbers
mean, but I don’t know why we
use that one.”
3
4
Parent Survey Findings
—  Half of the parents who filled out the survey did not
discuss fractions with their child(ren).
—  Another half gave examples:
—  Mathematics on the road:
1.  How many miles or hours are left to arrive at destinations? We
compute how many minutes to arrive at a specific distance.
—  Mathematics while shopping or dining:
1.  Cost of food ( Cost per unit)
2.  Figure out how much change will be received
—  Fractions at home:
1.  We measure cutting woods with fraction measurements.
2.  We use measure cups for cooking.
Additional Findings
—  “Mathematics rarely occurs in our daily
conversation, but only when we talk about
money, such as spending or how much does
an item cost. This is to get our kids to
understand we cannot always spend money.
In fraction, we do not need to discuss at
home. Is there?”
– A deaf parent
Research Question A- Answered
— 
Is there a difference in scores on fraction test between
the identified variables (i.e., parents’ demographics
variables and participants’ demographics variables) ?
Findings
Parents' hearing status
Concept/
Magnitude
Fraction in
Order
Equivalence
Total (correct
answer)
.80
.53
.52
.61*, **
Hearing
.40
Parents (n=5)
.48
.44
.44
One Deaf/
One Hearing .39
Parents (n=3)
.38
.47
.40
Deaf Parents
(n=5)
Findings
Parents with highest education level
Highest ed.
attainment!
Magnitiude
Order
Equivalence
Total (correct
answer)
*High School
.77
or Drop-out
.63
.33
.60
AAS or less
.00
.19
.3
.16
BS
.17
.38
.30
.29
MS
.67
.55
.63
.61
Research Question B- Answered
—  In which representation(s) do deaf students understand
fractional numbers?
FINDINGS
Concept/
Magnitude
Fraction in
Order
Equivalence
Questions with
words:
.4762
.2143
.5714
.4642
Questions with
pictures:
.5952
.0714
.3810
.4186
Compare with yes/
no questions
----------
.8095
-----------------
Determine the size
----------
.4048
-----------------
Total
FINDINGS
—  No significant difference in scores between texts and
pictures…
Research Question C-D Answered
What factors may prevent learning and understanding?
—  Common misconceptions:
— 
— 
— 
— 
Counting in denominators
Whole as one as opposed whole as zero
Size misperception
Numerator and denominator are perceived as two separate
entities.
SUMMARY…
Discussion Points…
—  n is small.
—  Answers varied, depending on previous knowledge and/or
understanding of fraction concepts.
—  During the interview,
—  Two participants changed their perceptions of fractional
concepts
—  Fuzzy knowledge à clear knowledge or misconceptions
—  One participant correctly answered the problems thereafter
Discussion Points…
—  Fractions are more complex than we initially thought
—  Including Chris and Moose
—  English language may add complications to understanding
—  Lack of incidental learning with fractions (at home and
school)
—  Negative disposition towards fractions
—  Started at very young..
—  Fractions are taught annually, but for a short time
Standards for
Learning Fractions
—  3.N.10 – Develop an understanding of fractions as part of a whole
unit and as parts of a collection
—  3.N.11 – Use manipulative, visual models, and illustration to name
and represent unit fractions as part of a whole or a set of objects.
—  3.N.12 – Understand and recognize the meaning of numerator and
denominator in the symbolic form of a fraction
—  3.N.13 – Recognize fractional numbers as equal parts of a whole
—  3.N. 14 – Explore equivalent fractions
—  3.N.15 – Compare and order unit fractions
—  5.S.6 – Record experiment results using fractions/ratios
Application-Hands On Activity
—  EXAMPLES— 
— 
— 
— 
— 
Miles Per Hour (MPH)
Resolution
Field Trip (Problem Solving)
Slope 7% grade
Service Tip (15%)
What are the learning challenges? How can we address
these challenges? Teaching strategies?
Group Discussion
Resources..
—  www.conceptuamath.com
—  www.khanacademy.org/
Recommendations:
—  Mathematics Curriculum and Instruction for the Deaf
—  Math education at elementary level is sorely
lacking due to lack of training at the teacher
preparation programs (Titus, 1995)
—  Teacher’s sign usage in class could determine the
content knowledge (Lang & Pagliaro, 2007)
—  Keep sign usage/vocabulary consistent (parents)
Recommendations…….
—  Use different representations (concrete
concepts → abstract concepts)
—  Comparison (ratios)
—  Diagrams/Illustrations
—  Get more understanding what fraction is
—  Encourage students to develop relationships
between fractions and personal experiences
(e.g., mpg, size, ratios, discount/sale, sets)
Recommendations…
—  Personalize math problems.
—  Use your students’ names
—  Involve them
—  Role-play
—  Increase length of study on fractions
—  Consistency of using fractions for different topics
—  Incorporate fractions in school and class daily
—  time, size, ratio, grade, specific characteristic
—  In every subject.
Recommendations…….
—  Deaf children are left out of interactions
happening within the family (Evans, 1998;
Kritzer, 2009)
—  Involve family fraction activities
—  Daily use of fractions (cooking, measurements,
shopping, ratios)
Recommendations (Strongly)
—  Thinking out loud during problem solving. (Mousley and
Kelly, 1998)
—  Dialogue, Dialogue, Dialogue
Future Research…
—  Compare learning experiences
—  Deaf parents with deaf children
—  Hearing parents with hearing children
—  Examine what parents/teachers think deaf
children know regarding initial fraction concepts,
and compare this with what the children actually
know
—  Investigate factors that are associated with
fractional number understanding in deaf children
What would our world be like without fractions?
Our language would certainly change!
—  You could never tell a friend to break a cookie in
"half" to share with you. You could only tell them
to break it into two pieces. —  A glass containing water could never be described
as "half full." How could you describe this glass? —  There would be no such thing as "half past the
hour" with timekeeping. You could never say you
are "halfway" there when traveling.
Thanks!
—  Keith Mousley “Moose”
—  [email protected]
—  Dr. Christopher Kurz
—  [email protected]