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LEARNING FRACTIONS:
A Collaborative Action Research Project
Danielle MacDonald
CAR III
Spring 2007
2
Introduction
My collaborative action research project is to determine whether or not daily repetition
helps students demystify the concept of fractions once and for all. Students are taught fractions in
many different ways since the second grade. I understand that during these years learning
fractions, students are exposed to many different teaching styles and learning styles. Why are
they still having difficulty eight to ten years later?
Before delving into my research of improving high school students fraction usage, I
would like to introduce myself. I began my teaching career at Pinkerton Academy, a semi-private
school in Derry, New Hampshire. After three years of forty-five-minute classes, I made the jump
to teaching four ninety-minute blocks in a small public school on the seacoast named
Winnacunnet High School. During my second and final year there, the administration piloted a
five seventy-minute block trimester schedule. This schedule seems to have made an impression
as they are completing their fourth year with it. Currently, I am finishing up my third year at
Salem High School in Southern New Hampshire where we use a standard four ninety-minute
block semester schedule without study halls.
Throughout my short but vast experience teaching mathematics to teenagers in all kinds
of environments and schedules, I have found that the most enigmatic problem that seems to
consistently arise (besides lack of motivation, of course), is that students do not understand
fractions and how to compute with them. No matter where I have taught or at what level, it has
been my experience that when students see fractions in a problem, they groan and then shut
down. Even in my honors courses, when I expect the students to appreciate the challenge of
working with rational numbers, they complain that they hate dealing with fractions. They feel as
though they do not have any relevance and only make the problems unnecessarily complicated.
3
My usual approach to reaching students is to create a relaxed atmosphere where I speak in an
almost matter of fact way. I like to explain fraction use as if I were giving instructions on how to
wipe down a table after dinner. I try to replace the anxiety with a sense of ease. When I am at the
board it appears to work, but they lack confidence once they are completing individual class
work at their desk.
Upon choosing a topic to research, I explored the abstract and contemplated whether or
not there is a difference between the sexes in math class, among other proposals. After
discussing a few prospects with my mentor, Dr. Charles Ford, we came across the idea of
creating an action plan that might possibly give teachers concrete evidence as to whether daily
repetition of fractions at the high school level would improve students attitude and achievement.
During the early stages of this research, I composed some questions that I would like my
research to answer. Does daily repetition make a difference? How do we help students better
understand the relationship between all rational numbers? Is there really a solution to this
problem? The first question is the focus of this project while the others are secondary.
Description of the Research
I had planned to begin my research by reviewing some articles that may help guide my
research questions. When I first embarked on this journey, I had difficulty finding recent
literature that focused on daily repetition. Most of what I found concentrated on elementary
teachings of fractions. Curricula for Teaching about Fractions (Millsaps and Reed, 1998) and
Fractions and Decimals (Pagni, 2004) describe how to teach rational numbers in the
elementary grades. These articles give excellent ideas on how fractions and decimals should be
presented as equal representations of one another. Both offer their own suggestions on how to
4
teach addition, subtraction, multiplication, and division. They also mention how important it is
for students to understand what fractions are prior to learning how to operate with them.
Not being fully satisfied with my literature review, I continued my pursuit of more
applicable articles. It was after I had formed my research questions that I realized that I required
more focused literature to support my findings. Since I could not find anything suitable in
repetition in mathematics, I decided to investigate some psychology reports. It was there that I
found some interesting material relating repetition as a valid teaching tool. In Skill Learning in
Mirror Reading: How Repetition Determines Acquisition (Ofen-Noy N, 2003), the author wrote
that there was growth after only a single repetition while, contrary to thought, there was no
increased growth in performance after numerous repetitions had reached a certain point. Also the
data collection revealed that more repetitions did not make the subjects learn at a faster pace.
This study intrigued me because there was conflict in the results of the study when it discussed
the line where increased learning turned into futile effort.
Another compelling psychology journal article, Im Different, Not Dumb (Fleming,
1995) discussed the differences among learners. The author writes, as is readily acknowledged,
that there are different modes of learning, i.e. visual, auditory, read/write, and kinesthetic.
Fleming tells of an experiment at Lincoln University where students were able to improve their
grade by focusing on their learning preference rather than attempting to improve their weaker
mode of presentation (which is the opposite of what current university education courses
determine). I found this article to be highly informative because it provided me with another
aspect to incorporate into this research. During this project, I observed a peer who conducts daily
repetition of fraction work. When I conducted my observations, I looked at how the repetition is
presented, how involved the students are, and if repetition is exclusively advantageous to the
5
auditory and read/write learners. How does this type of interaction help the visual and kinesthetic
learners? What can we do to help these learners? Like traditional education in general, do we
have a tendency to ignore the kinesthetic learners?
As another mode of data collection, I created two surveys to distribute to our schools
Geometry and Algebra 2 classes. I chose these two courses because the one major subject of my
research teaches these two courses this semester. I wanted my survey data collection to involve
the same level of students that I will be comparing. I passed these surveys out to each teacher
and asked them to have their students complete them within a few days. I also asked these same
teachers to repeat this process again at the end of the semester. The first survey asked students to
complete eight fraction problems requiring them to add, subtract, multiply, and divide. The
second evaluation is a Likert Scale that asked them to circle Strongly Disagree, Disagree, Neither
Agree Nor Disagree, Agree, and Strongly Agree, for eight questions that relate how they feel
about their competency in dealing with fractions. In my analysis of the data, I discuss the
percentages of correct responses to each problem in the first survey for both the pre test as well
as the post test. I anticipated a greater growth in all four of the model classes taught by the
teacher that implements daily repetition. Through pie charts I represent the responses of students
in these model classes compared to those in all of the others. I also separate the pie charts
according to pre test or post test.
My third method of obtaining data for this research was to observe the model teacher,
Christine Jefferson. Christine has been a mathematics teacher at Salem High School for ten years.
Prior to that, she was a special education teacher in the elementary/middle grades in the same
district. She is well versed and extremely experienced in the area of teaching rational numbers.
She taught Algebra 1 last semester and currently teaches Geometry and Algebra 2. She feels that
6
daily repetition of practice of fractions increases students mathematical success. Christine has a
formatted sheet that she distributes every day that encompasses a weeks worth of student work.
She places five questions on the overhead that students must work through on the paper. The
form is separated by the days of the week and leaves five places for students to label each
response and room for work. After all students have completed and turned in their paper,
Christine asks the students to verbally, from their seat, explain the steps to evaluating each
expression. At the end of the week, each score is tallied up and a biweekly quiz grade is given to
reflect their progress. During my observations, I note what I see happening during this fraction
review. I am also able to obtain ideas for my action plan for implementing this strategy in my
own classroom.
Analysis of the Data
After all of my data collection, I am not so happy to report that the results revealed
conflicting theories. Overall, in the Geometry sections, the surveys show more improvement in
Christine Jeffersons classes than in the other classes. However, in the Algebra 2 results, there is
a huge difference in growth when the pre test is compared to the post test. Christine once told me
at the beginning of the semester that there would be greater growth in the Geometry classes
because they appear to be more interested in learning material rather than just getting a credit.
Do the Geometry students take the material more seriously because they know that there is more
to come? Do most of the Algebra 2 students see the course as their last high school math course
and therefore do not see the relevance of truly comprehending how to deal with fractions?
During my observations, I had the pleasure of watching my peer uncover the pseudo-
mystery of working with fractions. She utilized many methods in order to reach all students.
7
Christine helped the auditory learners by asking some to verbally explain how to do a problem;
through overhead demonstrations, she was also able to appeal to the visual learners. Since
Christine has a very energetic and dynamic personality, she welcomed the kinesthetic learner by
using pictures and real-life visuals. I also noticed that she made the presentations brief so that she
could hold their attention. These observations began first thing in the morning at 7:30 am during
announcements. While the intercom was delivering the daily news, the students worked on the
problems that were displayed on the overhead.
Observations
My first observation took place on Tuesday, February 20, 2007. As soon as the students
entered the classroom, they picked up their pre-made sheet and began working on the following
four problems:
1. 211
43+ 2.
21
75 3.
32
94 4. 4
21
Once they had all been turned in, Christine not only asked a student to explain, but she attempted
to make the student thoroughly analyze how a problem results in an answer by prompting the
student with phrases such as How?, Give me the reason, and Why do we have to have the
same denominator? This method forces the students to think about cause and effect rather than
just following steps. It teaches them to problem solve (this also happens to be one of our schools
standards). After completing the explanation of the addition problem, Christine demonstrated the
problem using circles. She showed a picture of three quarters of a circle shaded and being added
to a full circle shaded next to a half shaded circle. She cut the half shaded in half (resulting in
two quarters), and used one to fill up the three quarter circle and make it a whole. Then, you
could easily see that we now have two wholes and a quarter. The lesson took a total of
8
approximately eighteen minutes (including the five minutes during announcements). I did have
some questions in regards to what happens when a student is absent, and how are these papers
graded.
The second observation occurred on Tuesday, March 6, 2007. After the sheets had been
picked up, I did notice that the boy next to me had been absent on Monday. Christine had written
the four problems on his sheet and he was to complete them along with the current days set.
This process answered my question on absence noted in the previous observation.
Christine began her demonstration with a little humor to catch their attention, Fractions
are our friends. Every student was quiet as the teacher called on different individuals to describe
their method. Some students raised their hands while others did not. This response did not appear
to dissuade Christine from calling on whomever she desired. During one of her explanations, I
noticed how she thoroughly explained that when borrowing, one full unit is being taken away
from the whole number and actually adds some form of 11 to the fraction. For example, in the
problem 971
954 , one needs to change
954 into
9143 in order to subtract
971 . Christine
demonstrated that you needed to borrow one from the four to make it three and then represent
that one unit by adding 99 to the fraction part.
Many students mistake the method of borrowing with mixed numbers with borrowing
when dealing with integers and just make the numerator a fifteen but putting a one in front of the
five. Sure, fourteen and fifteen are really close together, but do we really want our medical
technicians to have that philosophy? All too often, we take for granted that students know why
we are adding the denominator to the numerator in order to obtain a new numerator, when in
reality, we need to discuss it more. Does it take maturity to fully understand this process? If so,
9
then why do we drill these skills in the elementary grades? It is helpful that people write articles
such as Curricula for Teaching about Fractions, but maybe we should be collaborate with them
on developing unique strategies for high school students. I did also notice that once a student
leads Christine in a certain direction, she allows him to complete the process without adding her
opinion. She then asks if anyone else has another way of attacking the problem. For example, a
student wanted to multiply 25 by
54 by multiplying the numerators together to obtain twenty,
multiplying the denominators together to obtain ten, and then simplifying the fraction 1020 to two.
I thought quietly to myself that there was an easier way by cross reducing. Christine never
offered up this option; instead, she patiently asked if someone wanted to share their method. One
student did describe how he used the cross reduce process. I liked the fact that she didnt try to
teach them all different methods at the same time. She took the time to develop an idea and solve
the problem. She then moved on to look at the problem through the eyes of a different learner.
This lesson took approximately twenty minutes.
My third and final observation was on May 24, 2007. This review lesson was quite quick
because they were about to commence a test review for the following day. The four problems
were a standard addition, subtraction requiring borrowing, mixed number multiplication, and
division of fraction and integer. In the addition problem, 51
32+ , Christine actually showed how
we create like denominators. She asked a student to tell how the two turns into a ten and how
the one turns into a three. She prompted the student by saying If I multiply anything by one, do
I change the amount? The student stated that since we need to multiply both fractions by some
form of 11 and both denominators need to be fifteen (which is the least common denominator),
10
equivalent fractions need to be created. So that the original fractions could result in like
denominators, the first fraction needs to be multiplied by 55 , and the second fraction by
33 . I
could tell that, by the words that the student used, Christine had been working on this reasoning
quite a few times.
During the division problem, 281 , a student commented that taking half of something
is the same thing as dividing by two. This is another aha moment that I observed in this class.
I also observed that many more students raised their hands to show Christine that they
knew the material and were proud to get a chance to respond. This type of confidence is what we
need to build up within our students! It is also great to see that even though they have been doing
the same task and type of review for four months, they still remained attentive and involved. This
response says a great deal about Christines effort and impact in making a mundane topic
interesting and tolerable. This observation took approximately ten minutes.
Pre Test Surveys-Geometry
I collected data from Geometry and Algebra 2 students this spring semester in the form of
two surveys, a Likert Scale and sample problems to complete. For the pre test, all teachers of
these two courses administered the surveys in their classes. I will compare Christine Jeffersons
classes (model population) to the other classes (general population) when relating results of each
question. Forty-six students completed the pre test surveys in the model population and thirty-
seven students completed the pre test surveys in the general population.
The Likert Scale revealed how students felt in regards to their confidence in computing
fraction operations. In the model population, when asked to comment on adding fractions, the
11
majority of the model population felt that they either agreed (41.3%) or strongly agreed (47.8%)
that they can add fractions (see appendix B, figure 1). After correcting the operations survey,
69.6% correctly answered 98
75+ , and 76.1% correctly answered
214
853 + (see appendix A,
figure 1). I do not fully comprehend why more students scored higher on the second question as
it required more steps and involved mixed numbers. In general population, the Likert Scale
revealed more spread out feelings in regards to adding fractions. 10.8% disagreed, 18.9% neither
agreed nor disagreed, 43.2% agreed, and only 27% strongly agreed that they could add fractions
(see appendix B, figure 3). Also lower were the results from the general population with 54.1%
answering 98
75+ correctly and 56.8% answering
214
853 + correctly (see appendix A, figure 2).
As for subtraction, some parallels with addition in the model population as well as in the
general population. The model population boasts that 47.8% agreed and 41.3% strongly agreed
that they could subtract fractions (see appendix B, figure 1). On the operations survey, 76.1%
correctly answered 31
76 , while only 37% answered
654
219 correctly (see appendix A, figure
1). The second subtraction question required students to borrow correctly, thus making it a
challenging pre test question. In general population, 10.8% again disagreed, 21.6% neither
agreed nor disagreed, 43.2% again agreed, and 24.3% strongly disagreed (see appendix B, figure
3). There was a startling change from the model population when I noticed that only 48.6%
correctly answered 31
76 (see appendix A, figure2), although the results from the borrowing
question reveals about the same percent of correctness (37.8%) (see appendix A, figure 2).
For the model population, multiplication seemed to follow in suit. While 41.3% agree and
47.8% strongly agree (see appendix B, figure 1) that they can multiply fractions, the operations
12
survey exposed conflicting results as 80.4% correctly answered 31
94 , while only 37% answered
11101
518 properly (see appendix A, figure 1). This outcome makes sense because most forget
that mixed numbers need to be converted into improper fractions before multiplying. Most
students that multiplied incorrectly multiplied the two whole numbers together, the two
numerators together, and the two denominators together. The general population was once again
more spread out with 10.8% disagreed, 24.3% neither agreed nor disagreed, 51.4% agreed, and
13.5% strongly agreed that they can multiply fractions (see appendix B, figure 3). Sadly, only
48.6% responded correctly to 31
94 , and 37.8% answered
11101
518 correctly (see appendix A,
figure 2).
For division in the model population, 32.6% agreed and 45.7% strongly agreed that they
could divide (see appendix B, figure 1). Appropriately, 78.3% answered 31
65 correctly and
50% correctly answered 32
726 (see appendix A, figure 1). As for the general population,
13.5% disagreed, 29.7% neither agreed nor disagreed, 40.5% agreed, and 13.5% strongly agreed
that they could divide fractions (see appendix B, figure 3). Still lower, but not drastically lower,
56.8% responded correctly to 31
65 , and 40.5% suitably responded to
32
726 (see appendix A,
figure 2).
Some questions on the Likert Scale required the students to comment on converting
fractions to decimals, converting fractions to percents, and interchanging between mixed number
and improper fraction. There were no questions on the operations survey that contained decimals
or percents. Some problems did ask students to change mixed numbers into improper fractions
13
and some problems gave the option to change from improper to mixed numbers. I did not correct
these conversions, but in questions 11101
518 and
32
726 , students needed to change the mixed
numbers into improper fractions in order to answer correctly. However, after putting them into
improper fractions, a student could have scored incorrectly due to not multiplying properly.
In the model population, when asked if they can change an improper fraction into a mixed
number, amazingly, 2.3% strongly disagreed, 4.3% disagreed, 6.5% neither agreed nor disagreed,
30.4% agreed, and 56.5% strongly agreed (see appendix B, figure 1). In the general population,
13.5% neither agreed nor disagreed, 43.2% agreed, and 37.8% strongly agreed (see appendix B,
figure 3). The reverse of that process seemed to be in better favor with students, yielding 34.8%
agreeing and 54.3% strongly agreeing with the fact that they can change a mixed number into an
improper fraction (see appendix B, figure 3). Only 37% answered 11101
518 correctly in the
model population (see appendix A, figure 1) and only 37.8% in the general population (see
appendix A, figure 2); however, many didnt realize that they needed to put the mixed numbers
into improper fractions to even begin the problem. I would say that the low percentage of
correctness was due to the fact that students didnt know the rules for multiplying with mixed
numbers.
In the model population, 32.6% neither agreed nor disagreed, 41.3% agreed, and only
13% strongly agreed that they can change a fraction into a decimal (see appendix B, figure 1).
Even more startling, 21.7% disagreed, 17.4% neither agreed nor disagreed, 41.3% agreed, and
13% strongly agreed that they could change a fraction into a percent (see appendix B, figure 1).
In the general population, 10.8% disagreed, 24.3% neither agreed nor disagreed, 35.1% agreed,
and 27% strongly agreed that they could change a fraction into a decimal (see appendix B, figure
14
3). 13.5% strongly disagreed, 18.9% disagreed, 27% neither agreed nor disagreed, 24.3% agreed,
and only 16.2% strongly agreed that they could change a fraction into a percent (see appendix B,
figure 3).
As a whole, based on the results, the geometry model population started the semester
with an advantage over the geometry general population. I do not know why this is the case but I
will keep it in mind when I discuss the growth in the results from the post test.
Post Test Surveys-Geometry
Fifty students took the post test in the model population and thirty students took the post
test in the general population. Four students were absent for the post test in the model population.
Id also like to note that a teacher relayed to me that seven students from the general population
refused to complete the survey.
In the area of addition, from the model population, 30% agreed and 66% strongly agreed
that they could add fractions (see appendix B, figure 2). According to the operations survey, 72%
answered correctly to 98
75+ , representing a 2.4% growth, and 68% correctly answered
214
853 + ,
representing an 8.1% deficit (see appendix A, figure 1). As for the general population,
confidence remained at the same level as 16.7% neither agreed nor disagreed, 46.7% agreed, and
33.3% strongly agreed that they could add fractions (see appendix B, figure 4). 70% correctly
answered 98
75+ , showing an increase of 15.9%, and 46.7% responded correctly to
214
853 + ,
displaying a decrease of 10.1% (see appendix A, figure 2).
It seems as though students in the model population felt more confidence in the post test
with subtraction as 36% agreed and 60% strongly agreed that they could subtract fractions (see
15
appendix B, figure 2). Unfortunately results from 31
76 decreased 4.1% as 72% correctly
responded (see appendix A, figure 1). However, surprisingly, 44% answered 654
219 , the
borrowing question, correctly (see appendix A, figure 1), showing an increase of 7%. In the
general population, 23.4% neither agreed nor disagreed, 40% agreed, and 33.3% strongly agreed
that they could subtract fractions (see appendix B, figure 4). Results from the operations survey
showed an increase of 11.4% with a score of 60% for 31
76 , while a 4.5% decrease occurred
from 33.3% correct responses from the problem that required borrowing (see appendix A, figure
2).
An increase in confidence was apparent with multiplication within the model population
when 36% agreed and 60% strongly agreed that they could multiply fractions (see appendix B,
figure 2). This category showed the most improvement with students scoring 82% on 31
94 ,
representing a 1.6% growth, and 52% on 11101
518 , representing an outstanding growth rate of
15% (see appendix A, figure 1). Even though confidence grew with the general population when
multiplying fractions with 13.3% neither agreed nor disagreed, 43.3% agreed, and 36.7%
strongly agreed, both problems on the operations survey showed a decrease in correct responses
(see appendix B, figure 4). 36.7%, 11.9% lower than the pre test, answered 31
94 correctly; and
26.7%, 11.1% lower than the pre test, answered 11101
518 correctly (see appendix A, figure 2).
16
Not as spectacular were the results from division in the model population. While the
confidence was high, 32% agreed and 54% strongly agreed, 72% correctly answered 31
65 , and
50% responded correctly to 32
726 (see appendix A, figure 1). The results showed a 6.3%
decline in the first problem and no growth or decline in the mixed number problem. Following a
similar pattern, in the general population, division showed the most decline with a combined
decrease in correct responses by 40.6%. Still remaining ambivalent, 10% disagreed, 23.3%
neither agreed nor disagreed, 26.7% agreed, and 40% strongly agreed that they could divide
fractions (see appendix B, figure 4). Frankly, I am surprised that so many felt strongly that they
knew the rules. 36.7% correctly answered 31
65 , and only a mere 20% responded correctly to
32
726 (see appendix A, figure 2).
In the model population, the confidence continued to build as 36% agreed and 54%
strongly agreed that they could change an improper fraction into a mixed number (see appendix
B, figure 2). 10% neither agreed nor disagreed, 34% agreed and 52% strongly agreed that they
could change a mixed number into an improper fraction (see appendix B, figure 2). 12% neither
agreed nor disagreed, 38% agreed, and 46% strongly agreed that they could change a fraction
into a decimal (see appendix B, figure 2). Finally, showing much improvement, 38% agreed and
48% strongly agreed that they could change a fraction into a percent (see appendix B, figure 2).
The general population delivered that 13.3% neither agreed nor disagreed, 36.7% agreed, and
50% strongly agreed that they could change an improper fraction into a mixed number (see
appendix B, figure 4). 13.4% neither agreed nor disagreed, 33.3% agreed, and 53.3% strongly
agreed that they could change a mixed number into an improper fraction (see appendix B, figure
17
4). With a success rate of 26.7% on a problem that required them to change a mixed number into
an improper fraction before they follow the rules for multiplication, how could such a large
percentage feel so sure that they could change a mixed into an improper (see appendix A, figure
2)?! Even more disappointing, 10% disagreed, 23.3% neither agreed nor disagreed, 26.7%
agreed, and 40% strongly agreed that they could change a fraction into a decimal (see appendix
B, figure 4). 10% strongly disagreed, 10% disagreed, 26.6% neither agreed nor disagreed,
26.7% agreed, and 26.7% strongly agreed that they could change a fraction into a percent (see
appendix B, figure 4).
Overall, in the model population, there was a cumulative growth of 7.5% from the results
in the pre test to the results in the post test. Frighteningly, the overall difference in results of the
pre test versus post test in the general population was a decrease of 50.9%. I dont quite
understand how there could be a decline in mathematical skills during a semester of mathematics
unless the students did not take the survey seriously. It is a grave travesty that the results were so
poor.
Pre Test Surveys-Algebra 2
All Algebra 2 classes were given the same two surveys. Twenty-four students in the
model population and one hundred fifty-one students in the general population completed the
surveys.
In the model population, 54.2% agreed and 33.3% strongly agreed that they could add
fractions (see appendix B, figure 5). Operations survey shows that 54.2% answered 98
75+
correctly and 58.3% answered 214
853 + correctly (see appendix A, figure 3). Just like the
18
geometry students, a higher percentage scored properly on the problem containing mixed
numbers. The general population conveyed that 51% agree and 36.4% strongly agreed they could
add fractions (see appendix B, figure 7). Accordingly, and happily, 76.2% responded correctly to
98
75+ , while 70.9% answered
214
853 + correctly (see appendix A, figure 4). This was the only
case where students scored lower on the mixed number addition problem, which is what I would
expect.
As for confidence with subtraction, 16.7% neither agreed nor disagreed, 54.2% agreed,
and 29.2% strongly agreed in the model population (see appendix B, figure 5). 62.5% of students
in the model population answered 31
76 correctly and 50% responded correctly to
654
219 (see
appendix A, figure 3). This level of students seemed to be more capable than the geometry
students in borrowing. In the general population, 49% agreed and 34.4% strongly agreed that
they could subtract fractions (see appendix B, figure 7). Backing up their confidence, 74.8%
scored correctly on 31
76 and 49% were correct when answering
654
219 (see appendix A,
figure 4). Again, this level scored better on the borrowing problem.
The model population was still a little shy when 16.7% neither agreed nor disagreed,
45.8% agreed, and 37.5% strongly agreed that they could multiply fractions (see appendix B,
figure 5). Similar to geometry data, 66.7% responded correctly to 31
94 and only 37.5%
properly answered multiplying with mixed numbers (see appendix A, figure 3). A more
confident general population boasted that 40.4% agree and 49.7% strongly agreed that they could
multiply fractions (see appendix B, figure 7). A nice strong 73.5% correctly answered 31
94 and
19
44.4% responded favorably to 11101
518 (see appendix A, figure 4). That last result was still low,
but higher than the model population or any of the geometry sections.
The last real measurable category was division. Following a familiar pattern, 16.7%
neither agreed nor disagreed, 41.7% agreed, and 29.2% strongly agreed in the model population
(see appendix B, figure 5). With the best result yet, 83.3% scored on 31
65 and 62.5% answered
32
726 correctly within the model population (see appendix A, figure 3). In the general
population, 11.3% neither agreed nor disagreed, 37.7% agreed, and 42.4% strongly agreed that
they could divide (see appendix B, figure 7). A less strong 72.8% correctly responded to 31
65
and 55% answered 32
726 correctly (see appendix A, figure 4).
Uniquely, in the model population, the percentages for converting improper fractions into
mixed numbers were identical to the percentages for changing mixed numbers into improper
fractions. 16.7% neither agreed nor disagreed, 29.2% agreed, and 50% strongly agreed (see
appendix B, figure 5). Not the same were the responses for the general population where 39.1%
agreed and 50.3% strongly agreed that they could convert improper into mixed while 40.4%
agreed and 40.4% strongly agreed that they could do the reverse (see appendix B, figure 7).
Again, since the problem 11101
518 required more knowledge than just conversions, it would not
be reliable for me to use those results to measure if students really could convert mixed numbers
into improper fractions.
Lastly, pre test results showed that in the model population, 37.5% neither agreed nor
disagreed, 29.2% agreed, and 25% strongly agreed that they could change a fraction into a
20
decimal (see appendix B, figure 5). With one more step in the process of changing a fraction into
a percent, a significant 25% disagreed, 37.5% neither agreed nor disagreed, 29.2% agreed, and
only 8.3% strongly agreed that they could convert fractions into percents (see appendix B, figure
5). The general population results revealed a very similar and sad outcome. 23.2% neither agreed
nor disagreed, 37.1% agreed, and 30.5% strongly agreed that they could change a fraction into a
decimal (see appendix B, figure 7). 16.6% disagreed, 32.5% neither agreed nor disagreed, 29.8%
agreed, and 19.2% strongly agreed that they could change a fraction into a percent (see appendix
B, figure 7). This outcome demonstrates that students really do not understand the connections
between fractions, decimals, and percents.
Post Test Surveys-Algebra 2
Twenty-three students in the model population completed the post test surveys. A large
sample of one hundred thirty-four completed the post survey for the general population, a
considerable seventeen fewer subjects than the pre test. This decrease in involvement is due to
the fact that some students take Algebra 2 in their senior year. These surveys were distributed
after the seniors had graduated.
In the model population, 47.8% agreed and 39.1% strongly agreed that they could add
fractions (see appendix B, figure 6). 56.5% correctly answered 98
75+ , representing 2.3% higher
than the pre test, and a mere 47.8% responded correctly to 214
853 + , showing a decline of 10.5%
(see appendix A, figure 3). In contrast, while 47% agreed and 39.5% strongly agreed that they
could add fractions (see appendix B, figure 8), 78.4% of the general population actually could
21
correctly respond to 98
75+ , which showed a growth of 2.2% (see appendix A, figure 4). Also
revealing a 6% growth, 76.9% properly responded to 214
853 + (see appendix A, figure 4).
When deliberating about subtraction, 13.1% of the model population neither agreed nor
disagreed, 39.1% agreed, and 39.1% strongly agreed that they could subtract fractions (see
appendix B, figure 6). When in reality, even though there was an increase of 2.7%, only 65.2%
correctly answered 31
76 , and significantly low, 34.8% scored on
654
219 (see appendix A,
figure 3). This result was 15.2% less than the pre test results. As for the general population, 47%
agreed and 39.5% strongly agreed that they could subtract fractions (see appendix B, figure 8).
As one of the most correct answers, 82.1% responded correctly to 31
76 , which demonstrated a
growth of 7.3% (see appendix A, figure 4). The borrowing problem, 654
219 , yielded an 11.4%
growth rate with 60.4% of students properly responding (see appendix A, figure 4).
Multiplication appeared to break even in the model population. 13% neither agreed nor
disagreed, 43.5% agreed, and 43.5% strongly agreed that they could multiply fractions (see
appendix B, figure 6). 69.6% correctly answered 31
94 , representing a 2.9% increase, while only
34.8% correctly responded to 11101
518 , showing a deficit of 2.7% (see appendix A, figure 3).
The general population demonstrated remarkable improvement in the operations survey. 42.5%
agreed and 48.5% strongly agreed that they could multiply fractions (see appendix B, figure 8).
22
With an improvement of 10.8%, 84.3% answered 31
94 correctly, and with an increase of 3.4%,
47.8% scored on 11101
518 (see appendix A, figure 4).
Division, for the model population, remained a challenge. 13% disagreed, 17.4% neither
agreed nor disagreed, 26.1% agreed, and 43.5% strongly agreed that they could divide fractions
(see appendix B, figure 6). Although 78.3% correctly answered 31
65 , it was a 5% deficit from
the pre test (see appendix A, figure 3). A score of 52.2% on 32
726 was also lower by 10.3%
(see appendix A, figure 3). In the general population, 12% neither agreed nor disagreed, 45.5%
agreed, and 36.6% strongly agreed that they could divide properly (see appendix B, figure 8).
This group actually scored lower than the model population in this area. 77.6% of the general
population answered 31
65 correctly, representing an increase of 4.8% from the pre test, but .7%
lower than the model population (see appendix A, figure 4). 46.3% responded correctly to
32
726 , 8.7% lower than the pre test and 5.9% lower than the models post test (see appendix A,
figure 4).
The model population seemed to believe that 43.5% agreed and 47.8% strongly agreed
that they could change an improper fraction into a mixed number (see appendix B, figure 6).
17.4% neither agreed nor disagreed, 39.1% agreed, and 39.1% strongly agreed that they could
change a mixed number into an improper fraction (see appendix B, figure 6). On confidence in
changing a fraction into a decimal, 13% neither agreed nor disagreed, 43.5% agreed, and 30.4%
strongly agreed (see appendix B, figure 6). Ironically enough, since one would ordinarily change
a fraction into a decimal in order to convert into a percent, it was surprising to see that 17.4%
23
neither agreed nor disagreed, 34.8% agreed, and a high 34.8% strongly agreed that they could
change a fraction into a percent (see appendix B, figure 6). According to the general population,
40.3% agreed and 49.3% strongly agreed that they could change an improper fraction into a
mixed number (see appendix B, figure 8). 36.6% agreed and 48.5% strongly agreed that they
could change a mixed number into an improper fraction (see appendix B, figure 8).
Unfortunately, students appeared less confident when asked questions regarding decimals and
percents. 10.4% disagreed, 15.7% neither agreed nor disagreed, 43.3% agreed, and 27.6%
strongly agreed that they could convert a fraction into a decimal (see appendix B, figure 8).
17.2% disagreed, 20.9% neither agreed nor disagreed, 35.8% agreed, and 22.4% strongly agreed
that they could change a fraction into a percent (see appendix B, figure 8).
As a whole, the general population of Algebra 2 students had greater growth than the
model population with an increase of 37.2% from pre test to post test. The model population had
a deficit of 35.8% when the results from the pre test and post test were tallied. The percentages
of the general population should have actually been more accurate than those of the model
population because it was a much larger sample. Again, how could there be a decline in
mathematical skills at the completion of a math course, especially when I observed an incredible
teacher continuously reviewing different methods of adding, subtracting, multiplying, and
dividing fractions on a daily basis?
The Action Plan
I cannot understand why fractions are so difficult for most students. They are given exact
methods for adding, subtracting, multiplying, dividing, and even changing a fraction into a
decimal, why doesnt the information stick? After chatting with my daycare provider one
24
morning, she revealed to me that her elementary age students are great with their multiplication
facts and division skills. She told me that they do lack in the area of fractions. She said that her
son knew that 41 could also be represented by .25, but he didnt know why or how to get .25. Do
the students really not understand the concept of a fraction as being part of a whole? Apparently
not, after reviewing the results from my data collection.
Since this topic appears to be of great concern to most mathematics teachers, I feel as
though I need to further my research. I should redistribute the surveys next year to see if there is
a change. Maybe I did not have a large enough sample. An idea that I could develop is to use
daily repetition in my courses as well and see if there is more of a leveling relationship that
makes some classes more successful than others.
Another method for validity is for me to go to the particular classes and tell the students
myself the purpose of the research. I tried to relay the importance of my data collection to each
of the teachers, but I dont really know how serious they understood it to be. I could also ask
teachers if I could take the students into the auditorium where they could spread out and see how
important the data is.
Some teachers asked me, after the fact, whether students could use calculators to answer
the survey. Some teachers allowed their students use of a calculator while others did not. This
variant should have been addressed prior to distributing surveys. I would choose that students not
use calculators so that I could follow work and know that they didnt just punch numbers in
without any thought. Some students know that there is a key that allows you to enter mixed
numbers as well as proper fractions in order to add, subtract, multiply, divide, and other
operations.
25
I thought that most of Christine Jeffersons students would answer six to eight problems
correct. I do not understand how there was not a greater increase in ability. I can only assume
that students want to impress their teachers at the beginning of a semester while at the end, they
are just thinking about summer.
I also plan on discussing these results with Christine so that I can receive some feedback
that I could use for my next data collection. I would like to work with Christine and develop a
plan that we both could use to better develop the students fraction skills. It also makes me think
that we might possibly be too late at this point to really make a difference. I suppose that is what
my future research will tell me. One trial may not allow me to get at the real conclusion. I know
that I would like to time several trials to find my best speed for a race.
Now that I have brainstormed ideas to continue my research, I would like to begin next
semester by drafting a new survey, with Christine, just requiring students to add, subtract,
multiply, and divide fractions. I feel as though the Likert Scale was not very useful in
determining whether we actually improved students understanding of operations involving
fractions. We can create daily mini lesson plans that revolve around using fractions, not just drill.
We will schedule time to implement said plan in the spring semester. I think that we should use
that same plan for the following fall semester to see if the time of the year makes a difference.
After we have collected and analyzed all of this data, we can create a report to present to
the elementary grades at a professional development day. We can urge the elementary teachers to
join us in acquiring data from the lower grades. Working with the elementary teachers, we can
create yet another plan from which younger students can benefit. This way, we would be able to
compare future results with present results and discover if this idea does, in fact, make students
more successful.
26
References
Fleming, N.D. (1995). I'm different; not dumb. Modes of presentation (VARK) in the tertiary
classroom, in Zelmer,A., (ed.) Research and Development in Higher Education, Proceedings of
the 1995 Annual Conference of the Higher Education and Research Development Society of
Australasia (HERDSA), 18, 308 313.
Millsaps, G. M., & Reed, M. K. (1998). Curricula for teaching about fractions. ERIC
Clearinghouse for Science, Mathematics and Environmental Education. Online at
http://www.ericdigests.org/2000-2/fractions.htm
Ofen-Noy N., Dudai Y. and Karni A. (2003). Skill learning in mirror reading: How repetition
determines acquisition. Cognitive Brain Research, 17, 507-521
Pagni, D. (2004). Fractions and Decimals. Australian Mathematics Teacher, 60 (4), 28-30.
27
APPENDIX A:
Operations Survey Results
28
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
0102030405060708090
Percent
Survey Questions
Geometry Model Population
Pre TestPost Test
Figure 1:
Q1 % Q2 % Q3 % Q4 % Q5 % Q6 % Q7 % Q8 % TOTALPre 32 69.6 35 76.1 35 76.1 17 37 37 80.4 17 37 36 78.3 23 50 46 Post 36 72 36 72 34 68 22 44 41 82 26 52 36 72 25 50 50
Figure 2:
Q1 % Q2 % Q3 % Q4 % Q5 % Q6 % Q7 % Q8 % TOTALPre 20 54.1 18 48.6 21 56.8 14 37.8 18 48.6 14 37.8 21 56.8 15 40.5 37 Post 21 70 18 60 14 46.7 10 33.3 11 36.7 8 26.7 11 36.7 6 20 30
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
010203040
5060
70
Percent
Survey Questions
Geometry General Population
Pre Test
Post Test
29
Figure 3:
Q1 % Q2 % Q3 % Q4 % Q5 % Q6 % Q7 % Q8 % TOTALPre 13 54.2 15 62.5 14 58.3 12 50 16 66.7 9 37.5 20 83.3 15 62.5 24 Post 13 56.5 15 65.2 11 47.8 8 34.8 16 69.6 8 34.8 18 78.3 12 52.2 23
Figure 4:
Q1 % Q2 % Q3 % Q4 % Q5 % Q6 % Q7 % Q8 % TOTALPre 115 76.2 113 74.8 107 70.9 74 49 111 73.5 67 44.4 110 72.8 83 55 151 Post 105 78.4 110 82.1 103 76.9 81 60.4 113 84.3 64 47.8 104 77.6 62 46.3 134
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
0102030405060708090
Percent
Survey Questions
Algebra 2 Model Population
Pre TestPost Test
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
0102030405060708090
Percent
Survey Questions
Algebra 2 General Population
Pre TestPost Test
30
APPENDIX B:
Likert Scale on Pre and Post Tests
31
Figure 1 Geometry Model Population Pre Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 0 0 2 4.3 3 6.5 19 41.3 22 47.8 Q2 0 0 4 8.7 1 2.3 22 47.8 19 41.3 Q3 0 0 2 4.3 3 6.5 19 41.3 22 47.8 Q4 0 0 2 4.3 8 17.4 15 32.6 21 45.7 Q5 1 2.3 2 4.3 3 6.5 14 30.4 26 56.5 Q6 0 0 1 2.3 4 8.7 16 34.8 25 54.3 Q7 2 4.3 4 8.7 15 32.6 19 41.3 6 13 Q8 3 6.5 10 21.7 8 17.4 19 41.3 6 13 Figure 2 Geometry Model Population Post Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 1 2 0 0 1 2 15 30 33 66 Q2 1 2 0 0 1 2 18 36 30 60 Q3 1 2 0 0 1 2 18 36 30 60 Q4 1 2 0 0 6 12 16 32 27 54 Q5 1 2 1 2 3 6 18 36 27 54 Q6 1 2 1 2 5 10 17 34 26 52 Q7 1 2 1 2 6 12 19 38 23 46 Q8 1 2 1 2 4 8 19 38 24 48 Figure 3 Geometry General Population Pre Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 0 0 4 10.8 7 18.9 16 43.2 10 27 Q2 0 0 4 10.8 8 21.6 16 43.2 9 24.3 Q3 0 0 4 10.8 9 24.3 19 51.4 5 13.5 Q4 1 2.7 5 13.5 11 29.7 15 40.5 5 13.5 Q5 0 0 2 5.4 5 13.5 16 43.2 14 37.8 Q6 1 2.7 2 5.4 7 18.9 15 40.5 12 32.4 Q7 1 2.7 4 10.8 9 24.3 13 35.1 10 27 Q8 5 13.5 7 18.9 10 27 9 24.3 6 16.2
32
Figure 4 Geometry General Population Post Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 0 0 1 3.3 5 16.7 14 46.7 10 33.3 Q2 0 0 1 3.3 7 23.3 12 40 10 33.3 Q3 0 0 2 6.7 4 13.3 13 43.3 11 36.7 Q4 1 3.3 3 10 8 26.7 10 33.3 8 26.7 Q5 0 0 0 0 4 13.3 11 36.7 15 50 Q6 0 0 0 0 4 13.3 10 33.3 16 53.3 Q7 0 0 3 10 7 23.3 8 26.7 12 40 Q8 3 10 3 10 8 26.7 8 26.7 8 26.7 Figure 5 Algebra 2 Model Population Pre Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 0 0 0 0 3 12.5 13 54.2 8 33.3 Q2 0 0 0 0 4 16.7 13 54.2 7 29.2 Q3 0 0 0 0 4 16.7 11 45.8 9 37.5 Q4 0 0 3 12.5 4 16.7 10 41.7 7 29.2 Q5 0 0 1 4.2 4 16.7 7 29.2 12 50 Q6 0 0 1 4.2 4 16.7 7 29.2 12 50 Q7 0 0 2 8.3 9 37.5 7 29.2 6 25 Q8 0 0 6 25 9 37.5 7 29.2 2 8.3 Figure 6 Algebra 2 Model Population Post Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 0 0 1 4.3 2 8.7 11 47.8 9 39.1 Q2 0 0 2 8.7 3 13 9 39.1 9 39.1 Q3 0 0 0 0 3 13 10 43.5 10 43.5 Q4 0 0 3 13 4 17.4 6 26.1 10 43.5 Q5 1 4.3 0 0 1 4.3 10 43.5 11 47.8 Q6 1 4.3 0 0 4 17.4 9 39.1 9 39.1 Q7 1 4.3 2 8.7 3 13 10 43.5 7 30.4 Q8 1 4.3 2 8.7 4 17.4 8 34.8 8 34.8
33
Figure 7 Algebra 2 General Population Pre Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 3 2 5 3.3 11 7.3 77 51 55 36.4 Q2 3 2 7 4.6 15 9.9 74 49 52 34.4 Q3 2 1.3 5 3.3 8 5.3 61 40.4 75 49.7 Q4 4 2.6 9 6 17 11.3 57 37.7 64 42.4 Q5 2 1.3 5 3.3 9 6 59 39.1 76 50.3 Q6 2 1.3 8 5.3 19 12.6 61 40.4 61 40.4 Q7 1 0.7 13 8.6 35 23.2 56 37.1 46 30.5 Q8 3 2 25 16.6 49 32.5 45 29.8 29 19.2 Figure 8 Algebra 2 General Population Post Test
Strongly Disagree % Disagree %
Neither Agree Nor Disagree % Agree %
Strongly Agree %
Q1 2 1.5 4 3 8 6 65 48.5 55 41 Q2 2 1.5 6 4.5 10 7.5 63 47 53 39.5 Q3 1 0.8 5 3.7 6 4.5 57 42.5 65 48.5 Q4 3 2.2 5 3.7 16 12 61 45.5 49 36.6 Q5 2 1.5 7 5.2 5 3.7 54 40.3 66 49.3 Q6 2 1.5 11 8.2 7 5.2 49 35.6 65 48.5 Q7 4 3 14 10.4 21 15.7 58 43.3 37 27.6 Q8 5 3.7 23 17.2 28 20.9 48 35.8 30 22.4
34
APPENDIX C:
Operations Survey
35
FRACTION SURVEY 1
Please complete the following problems to the best of your ability. This data collection is for a graduate research project. Do not write your name on this sheet. Please show as much work as you can. Thank you.
1. 98
75+
2. 31
76
3. 214
853 +
4. 654
219
5. 31
94
6. 11101
518
7. 31
65
8. 32
726
36
APPENDIX D:
Likert Survey
37
Fraction Survey 1 Please do not write your name on this survey. Just respond to the best of your ability. This is not for a grade in your class, but it will be used in a graduate research project. 1. I can add fractions. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 2. I can subtract fractions. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 3. I can multiply fractions. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 4. I can divide fractions. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 5. I can change an improper fraction into a mixed number. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 6. I can change a mixed number into an improper fraction. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 7. I can change a fraction into a decimal. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree 8. I can change a fraction into a percent. Strongly Disagree Neither Agree Agree Strongly Disagree Nor Disagree Agree