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.

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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

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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

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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

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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.

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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

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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,

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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),

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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

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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

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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).

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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

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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