USING THE CARD TO MAKE THE GRADE
AND INCREASE MATH CONFIDENCE
A Thesis
Presented to the faculty of the Department of Education
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF ARTS
in
EDUCATION
(Curriculum and Instruction)
by
Elizabeth Gutiérrez
SPRING
2013
© 2013
Elizabeth Gutiérrez
ALL RIGHTS RESERVED
ii
USING THE CARD TO MAKE THE GRADE
AND INCREASE MATH CONFIDENCE
A Thesis
by
Elizabeth Gutiérrez
Approved by:
__________________________________, Committee Chair
Rita M. Johnson, Ed.D.
__________________________________, Second Reader
Frank Lilly, Ph.D.
____________________________
Date
iii
Student: Elizabeth Gutiérrez
I certify that this student has met the requirements for format contained in the
University format manual, and that this thesis is suitable for shelving in the Library
and credit is to be awarded for the thesis.
, Department Chair
Susan Heredia, Ph.D.
Date
Department of Teacher Education
iv
Abstract
of
USING THE CARD TO MAKE THE GRADE
AND INCREASE MATH CONFIDENCE
by
Elizabeth Gutiérrez
Statement of the Problem
As stated in the Education Commission of the States (ECS) website, the
graduation requirements shown are the minimum course requirements, however, local
boards may adopt graduation requirements above and beyond those authorized by the
state. In the state of California, a student must complete two units of math while in
high school, grades 9-12, and must meet or exceed the Algebra 1 course. If the student
met the requirement of Algebra 1 before high school, two years of math are still
required, excluding Algebra 1 (ECS, 2007).
Algebra is very important for all children to take in their educational career and
is also a high school graduation requirement in California. The students are also
learning problem solving strategies and applying algebra to solving real world
situations. Researchers and teachers have investigated many strategies of effective
v
studying in an effort to improve student assessment scores. Research by Duncan
(2007) found that individual students will perform slightly better on exams if they
prepare reference notes to use for their exams. The effect of making and using a note
card during quizzes, however, has not been explored in high school Algebra 1 classes.
Sources of Data
Data was collected and pooled from students in two high school algebra 1
classes whom currently attend a suburban public high school located in Northern
California. This mixed methods study occurred over the course of one algebra 1 unit,
which consisted of two quizzes and one unit test. The experimental group consisted of
one algebra 1 high school class that was allowed to use one 3" x 5" note card on each
quiz, but not the test. The control group consisted of another algebra 1 high school
class that were not allowed any use of a note card.
There was a pre-test, two quizzes, end of unit test (post-test), and finally a
post-post-test that was given after a couple of weeks to see if they retained the
information. There was also a survey that included open-ended questions, provided
qualitative data, and was conducted using pre-test and post-test.
Conclusions Reached
Although the results of this research did suggest that the use of note cards was
significantly related to achievement for one specific subgroup and not the whole
sample, a teacher may want to consider adopting the idea of making and using note
cards on specific assessments at a students' early stage of education, so they become
vi
familiar with the process prior to high school. Students can be taught the strategy of
thinking about what is known and unknown and how using a note card to quantify
critical information may be helpful to learning the material.
, Committee Chair
Rita M. Johnson, Ed.D.
Date
vii
ACKOWLEDGEMENTS
I would like to express my special thanks of gratitude to the students studied in
this thesis whom were so patient during the research process and I came to learn about
so many new things because of them. I would also like to thank my parents for
believing in me and helping me through this crazy adventure with their wonderful
words of wisdom and prayers. Jason, my amazing fiancé, who helped me a great deal
on finishing this thesis with his knowledge, advice, support, and most importantly,
endless love. Dr. Rita Johnson supplied positive feedback and advice with solid
deadlines, allowing me to learn every step of the way.
THANKS AGAIN TO ALL WHO HELPED ME.
viii
TABLE OF CONTENTS
Page
Acknowledgements ................................................................................................... viii
List of Figures.............................................................................................................. xi
Chapter
1. INTRODUCTION .................................................................................................. 1
Purpose of the Study ......................................................................................... 3
Statement of the Problem ................................................................................. 4
Significance of the Study.................................................................................. 6
Methodology..................................................................................................... 7
Limitations ........................................................................................................ 9
Theoretical Basis of the Study.......................................................................... 9
Definition of Terms ........................................................................................ 10
Organization of the Thesis.............................................................................. 12
2. REVIEW OF LITERATURE ............................................................................... 13
Introduction .................................................................................................... 14
Math Confidence and Attitude ....................................................................... 16
Student Environment (Supportive Teacher to Parent Support) ...................... 22
Gender Differences ......................................................................................... 25
Learning Theories ........................................................................................... 28
Use of Note Cards on Assessments, Previous Findings ................................. 34
Conclusion ...................................................................................................... 36
3. METHODOLOGY ............................................................................................... 39
Introduction .................................................................................................... 39
Research Design ............................................................................................. 39
School Profile ................................................................................................. 40
Sample ............................................................................................................ 41
ix
Data Source .................................................................................................... 41
Data Analysis.................................................................................................. 42
4. RESULTS ............................................................................................................. 43
Analysis .......................................................................................................... 44
Summary......................................................................................................... 52
5. SUMMARY AND DISCUSSION ....................................................................... 55
Summary......................................................................................................... 55
Conclusions .................................................................................................... 57
Concerns ......................................................................................................... 59
Further Recommendations.............................................................................. 60
Appendix A. Letter to Parents .................................................................................. 62
Appendix B. Quizzes ................................................................................................ 64
Appendix C. Tests .................................................................................................... 69
Appendix D. Surveys................................................................................................ 74
Appendix E. Extras ................................................................................................... 77
References .................................................................................................................. 83
x
LIST OF FIGURES
Figures
Page
1.
The Information Processing Model ................................................................. 32
2.
Change From Pre-Test to Post-Test Scores (No Prior Algebra) ..................... 45
3.
Change From Pre-Test to Post-Test Scores (Prior Algebra) ........................... 46
4.
Change From Pre-Test to Post-Test Scores For Females (No
Prior Algebra) .............................................................................................. 47
5.
Change From Pre-Test to Post-Test Scores For Females (Prior
Algebra) ...................................................................................................... 48
6.
Change From Pre-Test to Post-Test Scores For Males (No Prior
Algebra) ...................................................................................................... 49
7.
Change From Pre-Test to Post-Test Scores For Males (Prior Algebra) .......... 50
8.
Change From Pre-Survey to Post-Survey Q2 in the Experimental
Group .......................................................................................................... 51
9.
Change From Quiz 1 to Quiz 2 Scores in the Experimental Group ................ 52
xi
1
Chapter 1
INTRODUCTION
Tommy is currently a Sophomore in High School and is enrolled in his third
year of algebra 1. He is frustrated and embarrassed going to his algebra 1 class every
day because it is the same material for the third year in a row and he hates being
surrounded by immature Freshmen. He is hoping that this year will be different
because he understands that he must pass algebra 1 to graduate and he must also earn
credits for three years of different math courses, as required by his high school. That
means that he must pass this year if he wants to take two other math courses his Junior
and Senior year. He could take summer school, but he always looks forward to the
summer off. What Tommy's teacher doesn't know is that he gets test anxiety and
seems to forget everything that he studied from the night before. However, Tommy's
studying techniques are to skim his notes and the book briefly without working out
any problems or writing anything down, and also has little parent influence at home.
He has no confidence going into the test nor during the test. He is hoping his new
algebra 1 teacher will finally pass him so he can just get algebra 1 out of the way.
The way Tommy feels is the way that many students taking algebra 1 in high
school feel. Yet, there are many new laws and requirements in the education system of
California that are putting more pressure on students taking mathematics in middle
school and high school. Mathematics achievement and math confidence holds back
many students from graduating high school. No Child Left Behind, (NCLB) a Federal
2
Act of 2001, requires states to reach the “proficient” level on performance assessments
and holds schools accountable that 95% of all students in every identified subgroup
reach this level (Matthews, 2004). Enforced by Governor Arnold Schwarzenegger and
supported by the California School Boards Association, all eighth graders in
California must take algebra in middle school. He states that when we set the bar high
for our students, they can do anything (Asimov, 2008). Also, the National Governor’s
Association (NGA), Common Core Standards Initiative 2010 and the Council of Chief
State School Officers (CCSSO), have created the Common Core Standards Initiative
(CCSI), in an effort to bring more alignment, rigor, and consistency to student
‘proficiency’ in elementary, middle, and high schools and to foster improvement in
college-and-career readiness across the nation (Rivers, 2011).
A few areas leading to low student success rates in passing algebra, as
specified in the literature, are test anxiety, student motivation, and student use of study
aids. Studies of the effectiveness of note cards during assessments cover a variety of
approaches in different areas of curriculum. There is comparatively little research
done in the past ten years that explores the effects that making and using note cards
have on the students’ performance on the exam and math confidence, specifically in
mathematics at the high school level. Whether the students are taking algebra for the first
time or repeating it, a teacher is always looking for different approaches for students to retain
and understand the mathematical concepts in algebra as well as lower test anxiety and help
students feel more confident during tests. This study will examine an approach to allow
3
students make and use a note card on certain assessments to help understand algebraic
concepts and processes and gain math confidence.
Purpose of the Study
The purpose of this experimental study is to test the usefulness and
effectiveness of making and using a note card on specific assessments to improve
algebraic understanding, increase test scores, and boost math confidence. The note
card is intended as a form of a study aid before each quiz given in the unit. The results
of this study will be added to the body of research on issues relating to the
effectiveness of study aids in an algebra 1 classroom and other mathematics
classrooms.
The null hypotheses are:
1. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the students that have not
repeated algebra 1 between the experimental and control groups.
2. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the students that have
repeated algebra 1 between the experimental and control groups.
3. There is no significant difference in the growth in the test scores from the
pre-test assessment to the post-test assessment for the females that have not
repeated algebra 1 between the experimental and control groups.
4
4. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the females that have
repeated algebra 1 between the experimental and control groups.
5. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the males that have not
repeated algebra 1 between the experimental and control groups.
6. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the males that have repeated
algebra 1 between the experimental and control groups.
7. There is no significant difference in the growth of math confidence from
the pre-test survey to the post-test survey amongst students in the
experimental group.
8. There is no significant difference in the growth in scores from the quiz 1 to
quiz 2 amongst the students in the experimental group.
Statement of the Problem
Algebra is a math course that most students take in middle school and is often
their first venture into abstract reasoning and multistep problem-solving. It is also a
high school graduation requirement in many states. In California, a student must
complete two units of math while in high school, grades 9-12, and must meet or
exceed the algebra 1 course. If the student met the requirement of algebra 1 before
high school, two years of additional mathematics are still required, excluding algebra
5
1 (Education Commission of the States [ECS], 2007). Not only is algebra 1 required
for all students in California, but the students are also learning problem solving
strategies and applying algebra to solving real world situations.
Algebra is thinking logically about numbers rather than computing with
numbers, and it is important that students understand core algebraic concepts in order
to become mathematically literate. Whether the students are taking algebra for the first
time or repeating it, a teacher is always looking for different approaches for students to
retain and understand the mathematical concepts in algebra. Understanding
mathematics, especially algebra, is important for everyone because of its usefulness in
the real-world.
As stated earlier, there are many new laws and requirements in the education
system of California that are putting more pressure on students taking mathematics in
middle school and high school. Mathematics achievement, fear of failing, test anxiety,
and math confidence holds back many students from graduating high school and
succeeding in many math courses. Reauthorization of the Elementary and Secondary
Education Act of 1964, the NCLB act calls for all students to be proficient in
mathematics in 2014. However, with the implementation of Common Core Standards
next year, the schools are able to ignore the NCLB act, but still have national
standardized tests that assume everyone is at the same level of math, which may not be
any better. In this specific high school, students have significant needs to perform
better in algebra.
6
Many students are also "prepared" to fear and dislike mathematics at a very
young age which tends to lead to math anxiety. They may believe that the reason they
are in algebra 1 as a freshmen or sophomore, is because they are dumb or that they are
just terrible at math. Whether their parents were bad at it, or they just fear failing
because they have done it so much, which can lead to test anxiety. The fear may also
keep the learning from being successful, and retrieval is precarious. There is a need for
more research to see if making and using a note card on specific assessments increases
test scores and boosts math confidence.
Significance of the Study
Researchers and teachers have investigated and are still investigating different
teaching strategies of effective studying in an effort to improve student assessment
scores. The effects of making and using a note card during quizzes had not been
explored in the high school algebra 1 classes where this study was conducted. It is
hoped that the results of this study provide support for the use of a note card on
quizzes that will result in hopes of improved mathematical understanding and
learning, as measured by assessment tests, and boost math confidence. The algebra 1
teachers where I currently teach are also currently discussing whether the use of a note
card during quizzes and not on tests are beneficial to the students and should be
regularly used. Many of the science classes at the high school where I currently teach
already allow the use of a note card during their assessments, and have proven to
perform better than the previous year's scores. It is necessary to do research to see if
7
this strategy is effective. This research added to the body of research on usefulness and
effectiveness of note cards in any educational classroom and math confidence.
Methodology
This study will occur over the course of one Algebra 1 unit, which consists of
two quizzes (see Appendix B) and one chapter test. The experimental group consists
of one algebra 1 high school class that will be allowed to use one 3" x 5" note card on
each quiz, but not the test. The blank note cards will be given to the experimental
group two days before their quiz. The students will not be specifically told what to
write on the note card, just that they are able to write anything they think will help
them to answer the questions on the quiz correctly. They are required to turn the note
card in with their quiz. The control group consists of another algebra 1 high school
class that will not be allowed any use of a note card. The same material in both classes
will be covered throughout the lessons and taught in the same instruction with
different teaching strategies. Those strategies will include: direct instruction,
collaborative learning, group work, critical thinking activities, discussion strategies,
games, problem based learning, social networking tools, and many more.
There will be a pre-test, two quizzes, end of chapter test (post-test), and finally
a post-post-test that is given after a couple of weeks to see if they retained the
information. The pre-test, post-test, and post-post-test (see Appendix C) are similar
assessments with different numbers replaced within the problems. These assessments
will be given to both classes in order to compare the students’ assessment scores by
8
using a statistical test. The scores of all of the assessments (pre-test, two quizzes, posttest, and post-post-test) will be compared and placed on an excel spreadsheet. Student
names will be replaced with numbers and also have the demographics of gender,
grade, and how many times they have previously taken algebra 1. The statistical data
will be quantified with the use of a statistical T-test.
There will also be a survey that includes open-ended questions, provides
qualitative data, and measures using a thematic approach. The survey (see Appendix
D) will be conducted using pre-test and post-test. Open-ended questions will be asked
on the survey to measure math confidence towards the effectiveness of making and
using note cards on quizzes. Following collection of the information from the surveys,
the researcher will analyze data using both a quantitative and a qualitative approach.
The quantitative data will be analyzed using a statistical T-test and the qualitative data
will be examined and coded looking at recurrent themes.
The study being extended to cover all Algebra 1 units, is dependent on whether
the note cards show to be effective for the experimental group. If this occurs, the
experimental group and the control group will both be allowed to use a note card on
each quiz for the rest of the year, otherwise, the study will stop because it will have
shown that note cards are ineffective and not useful for those students.
The students in this study attend a suburban public high school located in
Northern California. They are of mixed ethnicity, but the majority of the students are
white/ Caucasian.
9
Limitations
There are several limitations to this study. One limitation is the number of
students who are enrolled in each class. The research is specific to both algebra 1
classes and curriculum. It may not be true for other curriculums, such as, English or
History. This may also not be applicable to upper division math courses, unless the
teacher felt the need for the use of note cards on assessments. Another limitation is
that the students may have forgotten to make their note card for their quiz because of
outside factors and came into class with a blank note card or no note card. This may
have also been due to the lack of motivation. All students are between 13 and 16 years
old. They did not represent a cross section of California.
Theoretical Basis of the Study
The theories of preparedness theory, stereotype threat, information processing
theory, and dual coding theory play major roles in this study. Individuals with a
phobia can be susceptible to anxiety induced reactions and feelings toward an object
or situation which in turn can "prepare" them to place major constraints on everyday
life (McNally, 1987). Stereotype Threat occurs when a group is positively or
negatively stereotyped, the group will live up or down to the expectation (Spencer,
Steele, & Quinn, 1999). Also, when females perform math, they risk being judged by
the negative stereotype that they have weaker math ability. According to Shah and
Miyake (1999), the information processing model (IPM) consists of three main
mechanisms: sensory memory, working memory, and long-term memory. The main
10
goal of the IPM put to use during any task or knowledge domain is to identify the
knowledge and processing activities that motivate performance. Lastly, dual coding
theory is about the nature of two systems of cognition: language and mental imagery.
The formation of mental images helps in learning when developing this theory
(Sadoski & Paivio, 2001).
Allowing the students to make and use a note card on quizzes, allows them a
different study strategy to gain math confidence and retain the information in a
different manner. Many students assume they are not going to succeed in mathematics
because of their phobia of math. Offering students the chance to try a new studying
technique will hopefully allow for a gain in math confidence that they are lacking and
raise their test scores in hopes of finally passing their algebra 1 class they have failed
multiple times. The note card can also act as another form of preparedness, counter
conditioning students to feel confident about their skills.
Definition of Terms
Algebra - Algebra is a general method of computation by certain signs and
symbols which have been contrived for this purpose, and found convenient. It is called
an universal arithmetic, and proceeds by operations and rules similar to those in
common arithmetic, founded upon the same principles (Maclaurin, 1748).
Chapter Quiz - An assessment consisting of 10 to 12 short answer problems
from that chapter.
11
Chapter Test (Pre-Test, Post-Test, Post-Post-Test) - An assessment consisting
of 25 to 30 short answer problems from that chapter.
Dual-Coding Theory - Explains human behavior and experience in terms of
dynamic associative processes that operate on a rich network of modality-specific
verbal and nonverbal (or imagery) representations (Clark & Paivio, 1991).
Information-Processing Theory - According to Shah and Miyake (1999), the
most researched and best conveyed model is the information processing model (IPM)
that was developed in the early 1950s. The IPM consists of three main mechanisms:
sensory memory, working memory, and long-term memory.
Note Card - A blank 3” by 5” index card on which students are free to write
anything they would like and use during a quiz.
Preparedness Theory - The preparedness theory of phobia holds that humans
are biologically prepared to learn to fear objects and situations that threatened the
survival of the species throughout its evolutionary history (Seligman, 1971). It is also
about the learning theory involved and then how people can become prepared to adapt
to situations and evolve. Preparedness may also mean when we are presented with
information before an event occurs, we are more prepared to adjust to that
information.
Stereotype Threat - Women risk being judged by the negative stereotype that
women have weaker math ability, which is known as stereotype threat; and it is
12
hypothesized that the apprehension it causes may disrupt women’s math performance
(Spencer et al., 1999).
Organization of the Thesis
This thesis follows the guidelines in the Graduate Student Handbook prepared
by the College of Education, Teacher Education Program and contains five chapters.
Chapter 1 gives a brief introduction into the study of making and using a note card to
raise assessment scores and increase math confidence, along with the purpose of the
study, statement of the problem, and significance of the study. Chapter 2 will be a
detailed review of research literature in an examination of the dimensions of the lack
of math confidence in mathematics, particularly in algebra 1, as well as the dimensions
of the need for a new learning style to boost math confidence and test scores. Chapter
3 discusses the methodology that was used for this study including the research
design, school profile, sample, data source, and data analysis. Chapter 4 is a discussion
of the quantitative findings that were compared and placed on an excel spreadsheet.
This chapter compared the findings between the experimental and control groups.
Chapter 5 is an interpretation of the data and detailed description of significant
findings. Areas of concern and future research are discussed. Following Chapter 5 is
the Appendix and a list of all references used in this study.
13
Chapter 2
REVIEW OF LITERATURE
This chapter of the review of literature is an examination of the dimensions of
the lack of math confidence in mathematics, particularly in algebra 1, as well as the
dimensions of the need for a new learning style to boost math confidence and test
scores. The first section offers a brief historical background on the history of
mathematics, the inclusion of mathematics into education, the inclusion of algebra into
schools, and the current requirements of algebra that affect every student. The next
section discusses the many reasons behind math confidence and the different attitudes
students have towards mathematics. It also examines math anxiety and test anxiety,
which both have negative influences on the success of a student.
The third section discusses how students' math success is greatly affected by
their environment; whether it is from the amount of support given by a teacher or
parent/guardian, or the amount of access to various external help that is readily
available. The fourth section discusses the research behind gender differences within
mathematics, along with looking into stereotype threat and how students are affected
by this. The fifth section examines the general learning theories and takes a closer look
at the learning theories that are specific to the use of note cards. The sixth section
provides background information on past studies that were done on the use of note
cards in the classroom. This chapter tells how it all pertains to the use of a note card in
hopes of increasing math confidence and raising student test scores.
14
Introduction
In the 1920s, the teaching of algebra and geometry in the United States was
regarded as an "intellectual luxury" because they were referred to as "useless"
subjects. Students were only taught basic math skills that had immediate practical
applications to their everyday life. After World War II, there was a big advocate for a
stronger math curriculum, which led to the space race of the 1950s. In the 21st
century, schools continue to work on increasing students' higher math skills so that the
United States can be competitive in a global economy (Martin, 2010).
Supported by the Elementary and Secondary Education Act of 1994, all
students are being held to the same academic standards, which is now being upheld
and continued by The No Child Left Behind Act of 2001 (NCLB). The legislation is
expanding the basis of the 1994 Act along with the federal role in public education.
They plan on doing this by requiring more school accountability, more rigorous
qualifications for teachers, and a greater emphasis on programs and strategies with
demonstrated effectiveness. All students must meet the state standards by 2014, close
the achievement gaps based on ethnicity, race, income, and language, and that they
achieve "proficiency" on state standards (Reeves, 2003). By mandating this, NCLB is
considered the most rigorous and demanding of standards-based strategies for
reforming schools (Simpson, LaCava, & Graner, 2004).
In 2004, algebra 1 became a high school graduation requirement in California;
however, students are encouraged to take it in eighth grade. In order to bring
15
California's eighth grade math testing into compliance with NCLB, the federal
government authorized the mandatory algebra 1 testing by eighth grade (Asimov,
2008). In 2008, there were approximately 500,000 eighth graders in California public
schools, of which 52% were taking Algebra 1. Those that were not enrolled in algebra
1 as an eighth grader, were required to take the California Standards Test (CST) for
pre-algebra (Asimov, 2008). Forty nine percent of the student population who took
algebra last year in middle school or high school, scored "Below Basic" and "Far
Below Basic" on their California Standards Test (CST) scores (CDE website, 2012).
The sad truth is that an eighth grade algebra teacher with a class full of 26 students,
usually has about two students that have difficulty understanding any math. These
students are best known as "misplaced" students, whom are so far behind that they are
required to learn a full year of algebra including the six years of math they have not
yet learned (Bracey, 2008).
Algebra teachers are being asked the impossible to tend to a student's needs,
when there may be anywhere from 26 to 36 students in a class, each at a different level
of math. The federal government keeps mandating more laws on math teachers
because of the need to be ranked higher in the world with our mathematical abilities,
without looking at the true needs of our students. The United States is solely ranking
the children and their schools due to the fact that we are heading down a fast-paced
track to improve our math scores. Mathematics is considered a core subject, but
students of all ages are more likely to tell you that the reason they are learning math is
16
to show they understand the material as opposed to whether they appreciate the beauty
of the subject or the way it piques their interest (Boaler, 2012).
Math Confidence and Attitude
Cornell (1999) made very broad and valid arguments towards the reasoning
behind students' dislike towards mathematics in his article “I Hate Math!: I Couldn't
Learn it, and I Can't Teach it!” In his study, he asked graduate students taking a math
instruction seminar for certification as elementary school teachers these questions: Did
you enjoy math in elementary school? Did you do well in math in elementary school?
Do you enjoy math now? Do you do well in math now? His research findings could be
generalized to seven main bases as to why many students in all levels of mathematics
may dislike the subject so much.
The first basis was a teachers' assumption of students' knowledge. Many
teachers assume that a student has mastered the concept or has little time to further
review the concept, and moves on. Time is of the essence when a teacher has
standards they must uphold, which causes them to rush through the material. The
second basis was the obscure vocabulary. Unique math vocabulary is not thoroughly
explained and may intimidate a student that may not be used to the math language,
such as, variables, specific notation, etc. The third basis was incomplete instruction.
Students may get frustrated by the lack of instruction in the mathematical procedures
or a specific teaching strategy that does not bode well with that student. Every student
learns in a different manner, therefore different teaching strategies should be enforced
17
in one's instruction. The fourth basis was too many skill-and-drill exercises. Students
may also get frustrated with the same form of assessments that a teacher may give,
such as a large amount of homework problems assigned every night. Tediousness
leads to boredom. Boredom then may lead to a dislike of a subject.
The fifth basis was a frustration at not being able to keep up with the class.
Trying to keep pace with the other students and the teacher may cause a loss of hope
in even passing the class. They may think there is no hope, therefore any point to do
well and be successful. The sixth basis was the math instruction had little tangible
relationships to the real world. Many students wonder why math is even useful and
where in the real world will they be able to use the concept being learned. If a teacher
does not make it relatable and important to the everyday life, then a student is quick to
lose interest. The last basis was math anxiety. According to Russell (2012) at
About.com, math anxiety is the fear that one won't be able to do the math, that it is too
hard, or the fear of failure which often stems from having a lack of confidence.
Math Anxiety
“Math anxiety is commonly defined as a feeling of tension, apprehension, or
fear that interferes with math performance” (Ashcraft, 2002). Erin Maloney and Sian
Beilock (2012), who is a post doctoral scholar and an associate professor of
psychology at the University of Chicago, respectively, found that when students are
put under math stress, they are unable to perform math problems successfully. The
stress is said to obstruct and block retrieval from their long-term memory, which is the
18
part of the brain that holds math facts. Maloney and Beilock also found that math
anxiety has an enormous impact on those students who have the potential to take
higher level math courses. Students with high math anxiety tend to avoid any type of
math as much as possible. They avoid taking math courses in high school and college,
as opposed to those who have low math anxiety. They also have negative attitudes
toward math and hold negative self-perceptions about their own math abilities
(Ashcraft, 2002).
Richardson and Suinn's (1972) identified math anxiety as having anxious
reactions to using math in ordinary life and academic situations and also designed a
98-item Mathematics Anxiety Rating Scale (MARS) that led to mixed results. A few
math anxiety studies done in elementary and secondary schools showed that math
anxiety scores increase across age (Brush, 1980; Meece, 1981). However, during the
high school and college years, female students report more anxiety about math than do
male students (Brush, 1980). Meece (1981) concluded that grade-level differences in
math anxiety were stronger and more prevalent than gender differences.
A study done by Wigfield and Meece (1988), showed that math worry was
highest in ninth graders, intermediate in seventh, eighth, tenth, eleventh, and twelfth
graders, and lowest in sixth graders. It was told in the study that the findings for the
sixth graders could be due to the less pressured environment of the elementary school
and that evaluation is not as emphasized as in junior high or high school. The findings
for the ninth graders could be due to the new environment of high school and their
19
first encounter of more difficult college-preparatory courses. Algebra 1 is a math
course that mainly contains ninth and tenth graders and is a high school graduation
requirement. The study also showed that the anxiety that students’ report represents
more than a lack of confidence in math; rather, it also centers on negative affective
reactions to math (Wigfield & Meece, 1988).
Many mathematics teachers believe that math anxiety originates primarily
from one’s own fears of failure and feeling of inadequacy. Math anxiety may not be
extreme or overwhelming, but it continues to develop and trouble most people
throughout their mathematical careers (Perry, 2004). Higbee and Thomas (1999)
found that there are correlations between math anxiety, test anxiety, and lack of
confidence in one’s ability to complete math tasks. This may also indicate that student
achievement is easily related to external factors, such as the teacher and their teaching
style, but also to internal factors, the student’s attitude toward mathematics. Harms
(2012) stated that many high-achieving students experience math anxiety at a young
age, which is a problem that can follow them throughout their lives. Worries about
math can disrupt working memory, which students could otherwise use to succeed.
Over a two year study, Meece, Wigfield, and Eccles (1990) implemented a
Student Attitude Questionnaire and examined math grades on a sample of seventh
through ninth graders to assess the influence of past math grades, math ability
perceptions, performance expectancies, and value perceptions on the level of math
anxiety. In this study, they focused on the interceding influence of math anxiety on
20
students' course enrollment plans and performance in mathematics. It was found that
students' current performance expectancies in math and the perceived importance of
mathematics had the strongest direct effects on their math anxiety. This goes to show
that math anxiety may stem from the unpleasant past experiences in with learning
mathematics, whatever they may be.
Test Anxiety
Anxiety is a phenomenon that human beings regularly come across within their
daily experience due to different factors. It is considered to be one of the most
common and invasive human emotions, with a large part of the world’s population
suffering from excessive and overbearing levels (Rachman, 2004).In the foundation of
test anxiety, Mandler and Sarason (1952) explained one's test performance on the
basis of two learned psychological drives: task-directed and anxiety. The task-directed
drive efforts to finish the task and thereby reduce the anxiety and the self-directed
task, anxiety drive, leads to irrelevant responses, which are manifested by feelings of
inadequacy, helplessness, heightened somatic reaction, anticipations of punishment or
loss of status and esteem, and implicit attempts to leave the testing situation. Later,
Liebert and Morris (1967) stated similar results that there are two components of test
anxiety: worry, which is the cognitive component, consisting of self-deprecatory
thoughts about one’s performance, and emotionality which is the affective component,
including feelings of nervousness, tension, and unpleasant physiological reactions to
21
testing situations. They stated that worry relates more strongly than emotionality to
poor test performance and interferes most with achievement performance.
People with high-test anxiety typically perform more poorly on tests than do
people with low-test anxiety, particularly when the tests are administered under
stressful, evaluative conditions around other people (Wine, 1971). As told by Soffer
(2008), test anxiety can produce a physiological hyperarousal, that interferes with
students’ mental processes and debilitates their ability to function during an exam, as
well as in the days and weeks leading up to a exam. However, Tobias (1985) declared
an alternative cause to test anxiety where the students with test anxiety receive lower
test scores due to inadequate study habits or poor test taking skills. The opposite is
true, that awareness of poor past performance is the cause of test anxiety. The main
reason for interest in researching test anxiety has been its relationship with
performance measures. Having high levels of test anxiety have been shown to be
negatively correlated with IQ, aptitude, academic achievement in reading, English,
math, natural sciences, foreign language, psychology, and mechanical knowledge,
problem solving, memory, and grades (Hembree, 1988). These effects were identified
in students ranging from third grade through graduate school.
Schwarzer and Jerusalem (1992) noted that individuals with high levels of test
anxiety are more likely to worry about the outcome of the test, compare their abilities
to others, or dwell on the notion that they are not fully prepared for the exam. Hill,
Wigfield, and Plass (1980), showed that when a student has time pressure on a test as
22
well as the level of the evaluative pressure that are placed on the students through the
instructions of the test, manipulated the degree to which test anxiety was related to test
performance. High-stake tests have always played an increasingly important role in
measuring student achievement and school effectiveness. Test anxiety has risen with
the use of tests in educational decision making (Von Der Embse, Barterian, & Segool,
2013). Overall, there are many factors that lead to test anxiety and lowering math
confidence that may or may not be easily avoided.
Student Environment (Supportive Teacher to Parent Support)
Educators and psychologists continue to express concern about the continuing
support of a teacher in a math classroom and the support a student gets at home.
Student and teacher relationships reflect a classroom’s ability to promote
development, and it is precisely in this way that relationships and interactions are the
key to understanding engagement (Pianta, Hamre, & Allen, 2012). In a longitudinal
study, it was found that student/teacher relationships deteriorated after the transition
from elementary school to junior high school (Feldlaufer, Midgley, & Eccles, 1988). It
was also said that students felt that the junior high math teachers cared less about
them, were less friendly, and graded them less fairly than the teachers they had for
elementary school. Other studies have shown that students who transition from
elementary school and progress to high school, report less favorable interpersonal
relationships with their teachers (Hawkins & Berndt, 1985; Hirsch & Rapkin, 1987;
O'Conner, 1978). Trickett and Moos (1974) found a strong relationship between
23
teacher support and high school students' academic interest and feelings of satisfaction
and security. The shift in student/teacher relationships from elementary school to high
school may contribute to a decline in student motivation and math confidence. This
may also indicate that student achievement is easily related to external factors, such as
the teacher and their teaching style, but also to internal factors, the student’s attitude
toward mathematics.
Adolescent girls have a greater need than boys for social connectedness and
may therefore be more sensitive to teacher support in the classroom (Veroff, 1983). A
study showed that the value of math increases for students who continue having more
supportive teachers and decrease for those that have the opposite pattern of change
(Feldlaufer et al., 1988). It has been documented that many adolescents do not actively
ask for help with their academic work when needed (Newman, 1990). Teachers not
only teach the material, but they also set up the norms and procedures in the classroom
that should lead to a warm and safe environment for all students to feel comfortable in.
Research in general education has documented a strong connection between a
child's success in school and parent involvement in school activities, such as
participating in general school-wide meetings, attending a school/class event, and
volunteering at school (Frew, Zhou, Duran, Kwok, & Benz, 2012). Epstein (1995)
states that there are six different ways that parents are able to support their children in
their education by: developing and using skills to support learning, communicating
about student progress, volunteering at school, helping with homework, becoming
24
involved in school-wide issues and decisions, and organizing community services that
will enhance the learning experience. Parental involvement has been shown to
influence achievement in mathematics, behavior problems, and the likelihood that a
child drops out of high school (Griffith, 1996; Miller & Kelley, 1991). This goes to
show that a parent's help with math homework plays an important role in students'
achievement. Parents who showed excessive support in and out of the classroom were
found to be too excessive by their grown children, but they reported better
psychological adjustment and life satisfaction than grown children who did not receive
intense support (Fingerman et al., 2012).
Although some parents may feel ill-equipped to help their children with their
math homework in high school, it is still very beneficial to help them get the help
necessary through tutoring or teacher help. Many research studies have examined that
high-socioeconomic status parents are more involved in helping with their child's
homework than are low-socioeconomic status parents and that more involvement leads
to improved academic performance (Astone & McLanahan, 1991; Lareau, 1987;
Muller, 1993; Revicki, 1981). However, it is also true that students whose parents are
more involved, regardless of their socioeconomic status, experienced higher
achievement than did those that were not (Clark, 1993). There is inconsistent support
concerning the correlation of parent education level with family involvement and
student achievement of the many studies that were found.
25
Gender Differences
Girls are said to earn higher grades than boys throughout elementary, middle,
and high school. However, boys tend to perform better on achievement or IQ tests.
These gender differences favoring boys on assessment tests and favoring girls' grade
point averages throughout school continue on to this day (Duckworth & Seligman,
2006). In Halpern's studies (1986), she stated that gender differences begin to emerge
in the adolescent stages, between 13 and 16 years of age. Although there are some
variations, many are in agreement that gender differences in math performance have
existed in the past and are still present. One may typically hear that males simply
outperform females on mathematics tests. Hyde, Fennema, & Lamon (1990) firmly
conclude from their research that the overall differences in mathematics performance
are not evident in early childhood, but appear in adolescence and usually favor boys in
tasks involving high cognitive complexity and favor girls in tasks of less complexity.
On the mathematics section of the Scholastic Aptitude Test (SAT), a recent
study on gender differences found that males performed reasonably better on
geometry/arithmetic items, as compared to female students, who performed reasonably
better on miscellaneous and arithmetic/algebra items (Harris & Carlton, 1993). Tate
(1997) reported in his research that there are changes in mathematics achievement in
the United States which were reviewed by national trend studies, advanced placement
tests, and college admissions exams. The research determined trends in mathematics
achievement of various social groups corresponding to gender, race, ethnicity, class,
26
and language proficiency. The findings signified that the males tended to outperform
females on standardized measures by a small margin and generally not significant. It
also suggested that male and female students perform similarly on assessments of
basic mathematical skills; and the differences were more distinct on assessments
entailing complex mathematical reasoning, where the males outperformed the females.
Both Hyde and Tate reached similar conclusions on the areas of outperformance by
each gender, which was shown by their research. Much of the research that was found
on gender differences in mathematics reiterated the same information as said above. It
may also be concluded that gender differences in mathematics performance are small.
However, during the high school and college years, female students report
more anxiety about math than do male students (Brush, 1980). Girls reported
experiencing significantly more negative affect about math than did boys in all grade
levels. This means, as math courses get harder, girls will be more likely to stop taking
math when they have that option. No gender differences were observed on the worry
scale (Wigfield & Meece, 1988). Data gathered by Veroff (1983) suggests that
adolescent girls have a greater need than boys for affiliation and social connectedness
and may therefore be more sensitive to teacher support or the lack of it in the
classroom. However, a longitudinal study of gender difference in math performance in
the United States, like many other studies, suggested that both male and female
students demonstrated the same growth trend in mathematics performance (Ding,
Song, & Richardson, 2006).
27
Stereotype Threat
Few researchers will deny that there are gender differences in math
achievement, but the main question that should be asked is what factors contribute to
these differences, especially given that it will be impossible to close the gender gap
without understanding these factors (Stoet & Geary, 2012). Women risk being judged
by the negative stereotype that women have weaker math ability, which is known as
stereotype threat; and it is hypothesized that the apprehension it causes may disrupt
women’s math performance (Spencer et al., 1999). These beliefs about gender
differences, mainly in math abilities, still affect females today. Many studies show that
stereotype threat can direct adolescent girls to choose easier problems to solve, lower
their performance expectations, and diminish mathematics as a career choice (Aronson
& Good, 2002; Davies, Spencer, Quinn, & Gerhardstein, 2002; Stangor, Carr, &
Kiang, 1998). Therefore, negative stereotypes about women can hinder their
performance and depress their self-assessments of ability.
In a significant study, Simon and Feather (1973) found that the performance
outcomes in female and male students were attributed to different measures. When
female students failed, they were more likely to attribute poor performance to the
difficulty in a test or bad luck, and less likely to attribute it to lack of effort. However,
when both female and male students succeeded, they stressed internal explanations
rather than external for their good performance. Put another way, both female and
male students are likely to internalize their successes, but females externalize their
28
failures more than males. Jamieson and Harkins (2012) stated that motivation was
found to play an important role in the effect of stereotype threat on females’ math
performance. Threatened females were motivated to refute the validity of the negative
stereotype, but performed more poorly because they withdrew effort (Jamieson &
Harkins, 2012).
A study done by Ernest (1976), surveyed teachers and found that 63% felt boys
were naturally better at math than girls. No teacher believed that girls were naturally
better. It was also reported that parents felt math to be more easier for their sons that
their daughters (Yee & Eccles, 1988). With teacher and parental support for gender
stereotypes of differences in math abilities, it can be believed that adolescent girls and
young woman have less math confidence, take fewer advanced math courses, and have
a more negative attitude toward math (Catsambis, 1994; Hyde et al., 1990). Women's
aspirations and career choices can easily be influenced by the combination of these
effects, which can steer them away from a degree or career in mathematics. If you tell
a females that they commonly score lower on math tests and have less math
confidence than males, then they will start believing it and adhere to the gender
stereotype.
Learning Theories
There is not said to be one general learning theory or learning style, for
everyone learns and thinks differently. Learning theories have been developed over
the past century and a half, in which some were developed as a negative effect to
29
earlier theories, and others built upon initial theories. Cooper (2009) discusses four
fundamental perspectives for learning theories: behaviorist, cognitive, humanistic, and
social learning; along with general learning theories of memory and intelligence, and
instructional theories. A general description of each perspective is discussed in further
detail below, which was taken from the website titled "Theories of Learning in
Educational Psychology" composed by Sunny Cooper (2009).
In modern Western society, the earliest development of a learning theory was
behaviorism which was founded by John B. Watson in the early 20th Century. The
work of Thorndike, Tolman, Hull, Skinner, and others, followed over the next few
decades. From the behaviorist perspective, the focus on behavior is observed rather
than on internal cognitive processes, the environment is the sculptor of learning and
behavior, and the principles of being contiguous and reinforcement are important to
explaining the learning process.
The behaviorist point of view was challenged in the mid twentieth century by
the Gestalt psychologists who criticized behaviorism and felt it was too dependent on
external behaviors to explain learning. The work of Wertheimer, Kohler, Koffka, and
Lewin followed in cognitive theories of learning, which are concerned with processes
that occur inside the brain and nervous system as a person learns. They believe that
people actively process information and learning takes place through the efforts of the
learner, including: inputting, organizing, storing, retrieving, and finding relationships
between information.
30
Freud, Maslow, and Rogers contributed most to the humanistic perspective.
The humanists discarded the ideas of behaviorism and said that motivation, choice,
and responsibility are influences of learning. The concepts that found their way into
the humanistic learning theories are the subconscious mind, anxiety, repression,
defense mechanisms, drives, and transference. Humanistic theories believe that life's
experiences are the central arena for learning.
Social learning theories built upon behaviorism in the 1940s, signifying that
when imitative responses were reinforced, they led to the observed learning and
behavioral changes. In the 1960s, the work of Bandura, Vygotsky, and Brown seceded
from the behaviorist views. The primary mechanism of social learning theories is
interactions between people. Learning is based on observation of others in a social
setting.
Every student is capable of learning, they just happen to learn in different
ways. Theorists will continue to identify new learning theories because no matter a
person's race, ethnicity, sex, gender, or age, every student's ability to learn depends on
multiple factors. The theories of preparedness theory, information processing theory,
and dual coding theory are specific to the use of note cards and play major roles in this
study.
Preparedness Theory
Many students are "prepared" to fear and dislike mathematics at a very young
age which tends to lead to math anxiety. They may believe that the reason they are in
31
algebra 1 as a freshmen or sophomore, is because they are dumb or that they are just
terrible at math. Whether their parents were bad at it, or they just fear failing because
they have done it so much, this leads to test anxiety and forgetting everything they had
studied the night before. Students are just "conditioned" to choke up and assume they
are going to fail math. Preparedness is due to evolution. It is the way evolution has
"prepared" us to learn about things that are threatening. Seligman's (1971) theory of
preparedness supports the theory of classical conditioning, which emphasized the fact
that phobias are not random. Many people are afraid of thunder or spiders because
their ancestors were afraid of them. Individuals with a phobia can be susceptible to
anxiety induced reactions and feelings toward an object or situation which in turn can
place major constraints on everyday life. Phobias are described as a conditioned
reaction that is specific, persistent, intense, and irrational with a compelling need to
avoid the object or situation (McNally, 1987).
Preparedness theory (Schwartz, 1974) presumes that traditional theories of
learning cannot be consistently applied to all stimuli interacting with all organisms.
Seligman (1971) hypothesized that the strength of human phobic reactions, like a fear
for mathematics, is due to the high degree of preparedness of certain stimuli. It is also
about the learning theory involved and then how people can become prepared to adapt
to situations and evolve. Preparedness may also mean when we are presented with
information before an event occurs, we are more prepared to adjust to that
32
information. Many students are unaware that they have "prepared" themselves to have
a phobia of mathematics as a result of our evolutionary history.
Information Processing Theory
According to Shah and Miyake (1999), the most researched and best conveyed
model is the information processing model (IPM) that was developed in the early
1950s. The IPM consists of three main mechanisms: sensory memory, working
memory, and long-term memory. Refer to the diagram in Figure 1 showing the flow of
information between the three main mechanisms. The input of information into the
sensory memory only registers for a few seconds unless the person pays attention or
recognizes a pattern in the information, then will it be processed into the working or
short-term memory. The use of rehearsal helps the person to store the information in
the long-term memory. Through encoding and recovery, the person allows the
information back into the working memory and uses it to process and create responses
on new input that has come to the person’s attention (Reiser & Dempsey, 2007).
Figure 1. The Information Processing Model. (from Schraw & McCrudden, 2003)
33
Sensory memory and working memory allow people to manage limited
amounts of incoming information during initial processing, while long-term memory
acts as a permanent storage for knowledge. The IPM is used as a metaphor for
successful learning and provides a well-articulated means for describing the main
cognitive structures (memory systems) and processes (strategies) in the learning cycle
(Schraw & McCrudden, 2003). The information processing theory is the analysis of
the way a human being learns new information. There is an unchanging pattern of
events that take place in any situation, and by knowing this pattern we can help people
learn new things faster (Thadani, 2010). The main goal of the IPM put to use during
any task or knowledge domain is to identify the knowledge and processing activities
that motivate performance. Knowledge is thought of in terms of what is known, how it
is organized, represented, stored, and how it is used (Pellegrino & Goldman, 1987).
Making and using a note card to be used on an assessment, can be used as a method or
technique to enhance the processing of information into the long-term memory.
Dual Coding Theory
Dual coding theory is about the nature of two systems of cognition: language
and mental imagery. The formation of mental images helps in learning when
developing this theory (Sadoski & Paivio, 2001). According to the dual coding theory,
students actively process information through the selection, organization, and
representation of information in condensed form during the making of a note card
(Dorsel & Cundiff, 1979; Hindman, 1980). The dual coding theory also affirms that
34
preparing a note card to be used on an assessment is an efficient way to code material
for memory. Since students are given a limited space to write on, they must
summarize, or code, the material that they are going to be tested on. Studies suggest
that coding helps students memorize and understand information (Wickelgren, 1975).
The making and use of a note card may be used as a form of a study aid, which may
enhance learning, allowing students to perform better on assessments.
Use of Note Cards on Assessments, Previous Findings
Studies of the effectiveness of the use of note cards during assessments cover a
variety of approaches in different areas of curriculum. There is comparatively little
research done in the past ten years that goes into the effects that making and using
note cards on assessments have on the students’ performance on the exam, specifically
in mathematics at the high school level. One research study, in particular, investigates
the effects of using note cards in mathematical exams at the University level by only
conducting a survey on those students that were allowed to use cheat sheets (Butler &
Crouch, 2011). The students surveyed said that they found their note card useful, but
each had a different definition as to what useful was, depending on the expectations.
Two studies (Drake, Freed, & Hunter, 1998; Erbe, 2007) stated that student
learning is greatly enhanced by allowing students to prepare a note card for the exam
to help structure the study time and reduce test anxiety. When people experience
extreme distress and anxiety in testing situations that impairs learning and hurts test
performance, it is known as a psychological condition, test anxiety (Cherry, 2012).
35
Whether the students are taking Algebra for the first time or repeating it, a teacher is
always looking for different approaches for students to retain and understand the
mathematical concepts in Algebra as well as lower test anxiety and feel more
confident during tests.
Many studies (Duncan, 2007; Dickson & Miller, 2005; Larwin, Larwin, &
Gorman, 2012; Whitley, 1996) tested the use of note cards in information systems
courses, statistics courses, child development courses, and psychology courses by
using a quasi-experiment and testing multiple sections through various testing
techniques (use of note card, make a note card but not use it, to not use a note card,
etc.). Another study (Wachsman, 2002) also used the quasi-experiment method among
four Economics classes and used the T-test to analyze their data. Following their
experiment, they gave the same students a survey of their opinions on how useful they
found note cards to be.
One study (Dickson & Bauer, 2008) was implemented in a college Psychology
course where fifty-three students were given a pre-test without the use of a note card
and also a post-test with the use of the note card. Both of the tests were the same and it
was shown that the post-test results were higher than the pre-test because of the use of
the note card. A second study (Theophilides & Koustelini, 2000) was given to 276
Education majors where they were given a closed-book exam and a few weeks later,
they were given an open-book exam. Even though they were given two different
exams, the T-test results showed that they performed much better on the open-book
36
exam. The students were also given a questionnaire after and stated that they felt more
confident and found the open-book exam easier.
The 11 studies shown above had different sample groups for testing their
hypotheses, some followed by surveys, however, each agreed that preparing and using
a note card can potentially affect students’ test performance in a positive manner.
Nevertheless, the organization of making a note card to be used on an assessment is
important on preparing and studying for an exam. As told by Kathleen Thayer a
professor at Purdue University, note cards are an efficient means of organizing
information that can be written in a straightforward manner and the information is
learned by making them. Making and using a note card to use on an assessment is just
one metacognitive strategy used to improve the understanding of the material and will
be tested within the study.
Conclusion
This chapter of the review of literature examined the dimensions of the lack of
math confidence in mathematics, particularly in algebra 1, as well as the dimensions of
the need for a new learning style to boost math confidence and test scores. In the state
of California, an increasing number of eighth graders have taken algebra courses since
2003 (Liang, Heckman, & Abedi, 2012). In a study done by Liang et al. (2012),
students’ California Standards Test (CST) results in grades seven through eleven were
examined and revealed that ninth grade students have a 69% greater chance of
succeeding in algebra if they passed the CST for General Mathematics in eighth grade
37
compared to those who failed the CST for algebra 1. Mathematics achievement and
lack of math confidence holds back many students from graduating high school and
being able to pass algebra 1. Without looking at the true needs of students, the federal
government is more concerned about mandating more laws on math teachers and the
curriculum in order for students to be ranked higher in the world in mathematical
abilities.
Whether the students are taking algebra for the first time or repeating it, a
teacher is always looking for different approaches for students to retain and
understand the mathematical concepts in algebra. Unfortunately for many students,
math anxiety and test anxiety do not allow for an easy opportunity to pass their algebra
1 course. As stated by Maloney and Beilock (2012), low math ability is not the entire
explanation for why math anxiety and poor math performance occur in tandem; when
faced with a math task, math anxious individuals tend to worry about the situation and
its consequences. Attention to student motivation, student environment, and a need for
different study strategies are necessary to help boost math confidence and raise test
scores.
Little research has been done in the past ten years that examines the making
and using of note cards on assessments improve students’ performance on an exam,
specifically in mathematics at the high school level. Allowing students to make and
use a note card on quizzes, allows them a different study strategy which may help
them gain math confidence and retain the information in a different manner. Many
38
students assume they are not going to succeed in mathematics because of their phobia
of math. Offering students the chance to try a new studying technique will hopefully
allow for a gain in math confidence that they are lacking and raise their test scores in
hopes of finally passing their algebra 1 class they have failed multiple times.
39
Chapter 3
METHODOLOGY
Introduction
The purpose of this experimental study was to test the usefulness and
effectiveness of making and using a note card on specific assessments to improve
algebraic understanding, increase test scores, and boost math confidence. The control
group (Group 1) consisted of a heterogeneous group of high school students in one
algebra 1 class and the experimental group (Group 2) consisted of another
heterogeneous group of high school students in another algebra 1 class at the same
high school. The same material in both classes were covered throughout the lessons
and taught in the same instruction with different teaching strategies. Those strategies
included: direct instruction, collaborative learning, group work, critical thinking
activities, discussion strategies, games, problem based learning, social networking
tools, and many more.
Research Design
This mixed methods study occurred over the course of one algebra 1 unit,
which consisted of two quizzes and one unit test. The experimental group consisted of
one algebra 1 high school class that was allowed to use one 3" x 5" note card on each
quiz, but not the test. The blank note cards were given to the experimental group two
days before each quiz. The students were not specifically told what to write on the
note card, just that they were able to write anything they thought would help them to
40
answer the questions on the quiz correctly. They were required to turn the note card in
with their quiz. The control group consisted of another algebra 1 high school class that
were not allowed any use of a note card.
There was a pre-test, two quizzes, end of unit test (post-test), and finally a
post-post-test that was given after a couple of weeks to see if they retained the
information. The pre-test, post-test, and post-post-test were similar assessments with
different numbers replaced within the problems. There was also a survey that included
open-ended questions, provided qualitative data, and was measured using a thematic
approach. The survey was conducted using pre-test and post-test. The pre-test survey
was given before the first lesson of the chapter was taught and the post-test survey was
given after the end of chapter test was turned in. Open-ended questions were asked on
the survey to measure math confidence towards the effectiveness of making and using
note cards on quizzes.
The study being extended to cover all algebra 1 units, was dependent on
whether the note cards showed to be effective for the experimental group. If this
occurred, the experimental group and the control group would both be allowed to use a
note card on each quiz for the rest of the year, otherwise, the study would stop because
it would have shown that note cards were ineffective and not useful for those students.
School Profile
The students in this study attend a suburban public high school located in
Northern California. The school is part of a large, pre-school through 12th grade
41
district. They are of mixed ethnicity, but the majority of the students were white/
Caucasian.
Sample
Group 1 consisted of 33 students who were enrolled in a high school algebra 1
course for the 2012-2013 school year. There were sixteen females and seventeen
males in the class. Twenty four of the students were Freshmen, eight were
Sophomores, and one was a Junior. Group 2 consisted of 35 students who were also
enrolled in a high school algebra 1 course for the 2012-2013 school year. There were
15 females and 20 males in the class. Twenty seven of the students were Freshmen,
seven were Sophomores, and one was a Junior. The control group (Group1) and the
experimental group (Group 2) were both taught by the same teacher. Group 2 was
chosen to be the experimental group at random.
Data Source
The quantitative data was analyzed using a statistical t-test and the qualitative
data was examined by looking at recurrent themes and coded by using a statistical ttest. The scores of all of the assessments (pre-test, two quizzes, post-test, and postpost-test) from both classes were compared and placed on an excel spreadsheet.
Student names were replaced with numbers and also had the demographics of gender,
grade, and how many times they had previously taken algebra 1. Each question on
both the pre-test and post-test surveys were coded specifically and placed on the same
excel spreadsheet.
42
Data Analysis
All data was entered into an excel spreadsheet. A statistical t-test for
independent means was used to analyze the data utilizing the Statistical Package for
Social Sciences (SPSS) program. Data from both main groups and sub groups were
analyzed.
43
Chapter 4
RESULTS
This experimental study tested the usefulness and effectiveness of making and
using a note card on specific assessments to improve algebraic understanding, increase
test scores, and boost math confidence. The null hypotheses were:
1. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the students that have not
repeated algebra 1 between the experimental and control groups.
2. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the students that have
repeated algebra 1 between the experimental and control groups.
3. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the females that have not
repeated algebra 1 between the experimental and control groups.
4. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the females that have
repeated algebra 1 between the experimental and control groups.
5. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the males that have not
repeated algebra 1 between the experimental and control groups.
44
6. There is no significant difference in the growth in test scores from the pretest assessment to the post-test assessment for the males that have repeated
algebra 1 between the experimental and control groups.
7. There is no significant difference in the growth of math confidence from
the pre-test survey to the post-test survey amongst students in the
experimental group.
8. There is no significant difference in the growth in scores from the quiz 1 to
quiz 2 amongst the students in the experimental group.
The control group, Group 1, was allowed the use of a note card on the quizzes
and not on the test. The experimental group, Group 2, was not allowed the use of a
note card. The independent variable was the note card. The dependent variable was the
test/survey score.
Analysis
Null Hypothesis 1: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the students that
have not repeated algebra 1 between the experimental and control groups.
To test the first hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the students that have not repeated algebra 1
for Group 1 (M = 40.4091, SD = 18.804) and Group 2 (M = 50.6250, SD = 24.745)
was not statistically significant, t(17) = -1.025, p = (0.320). See Figure 2.
45
Change from pre-test to post-test (no prior
algebra)
60
50
50.625
40.4091
40
30
20
10
0
Control
Experimental
Figure 2. Change From Pre-Test to Post-Test Scores (No Prior Algebra).
Null Hypothesis 2: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the students that
have repeated algebra 1 between the experimental and control groups.
To test the second hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the students that have repeated algebra 1 for
Group 1 (M = 79.5000, SD = 8.0436) and Group 2 (M = 68.3913, SD = 13.3714) was
statistically significant, t(42) = 3.299, p = (0.002). See Figure 3.
46
Change from pre-test to post-test
(prior algebra)
100
80
79.5
68.3913
60
40
20
0
Control
Experimental
Figure 3. Change From Pre-Test to Post-Test Scores (Prior Algebra).
Null Hypothesis 3: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the females that
have not repeated algebra 1 between the experimental and control groups.
To test the third hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the females that have not repeated algebra 1
for Group 1 (M = 34.3333, SD = 3.055) and Group 2 (M = 37.100, SD = 20.870) was
not statistically significant, t(6) = -0.221, p = (0.832). See Figure 4.
47
Change from pre-test to post-test for Females
(no prior algebra)
50
40
37.1
34.3333
30
20
10
0
Control
Experimental
Figure 4. Change From Pre-Test to Post-Test Scores For Females (No Prior Algebra).
Null Hypothesis 4: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the females that
have repeated algebra 1 between the experimental and control groups.
To test the fourth hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the females that have repeated algebra 1 for
Group 1 (M = 78.625, SD = 9.158) and Group 2 (M = 73.000, SD = 12.559) was not
statistically significant, t(20) = 1.214, p = (0.239). See Figure 5.
48
Change from pre-test to post-test for Females
(prior algebra)
100
80
78.625
73
60
40
20
0
Control
Experimental
Figure 5. Change From Pre-Test to Post-Test Scores For Females (Prior Algebra).
Null Hypothesis 5: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the males that have
not repeated algebra 1 between the experimental and control groups.
To test the fifth hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the males that have not repeated algebra 1
for Group 1 (M = 42.6875, SD = 21.9250) and Group 2 (M = 73.1667, SD = 7.2514)
was statistically significant, t(9) = -2.293, p = (0.048). See Figure 6.
49
Change from pre-test to post-test for Males
(no prior algebra)
100
73.1667
80
60
42.6875
40
20
0
Control
Experimental
Figure 6. Change From Pre-Test to Post-Test Scores For Males (No Prior Algebra).
Null Hypothesis 6: There is no significant difference in the growth in test
scores from the pre-test assessment to the post-test assessment for the males that have
repeated algebra 1 between the experimental and control groups.
To test the sixth hypothesis, a t-test for independent means was done. The
difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the males that have repeated algebra 1 for
Group 1 (M = 80.6667, SD = 6.614) and Group 2 (M = 64.8462, SD = 13.349) was
statistically significant, t(20) = 3.271, p = (0.004). See Figure 7.
50
Change from pre-test to post-test for Males
(prior algebra)
100
80.6667
80
64.8462
60
40
20
0
Group 1
Group 2
Figure 7. Change From Pre-Test to Post-Test Scores For Males (Prior Algebra).
Null Hypothesis 7: There is no significant difference in the growth of math
confidence from the pre-test survey to the post-test survey amongst students in the
experimental group.
To test the seventh hypothesis, a t-test for independent means was done. The
difference between the Survey Q2 means in the growth of math confidence from the
pre-test survey to the post-test survey amongst students in the experimental group for
Pre-Survey Q2 (M = 1.88, SD = 0.332) and Post-Survey Q2 (M = 1.80, SD = 0.408)
was not statistically significant, t(24) = 1.000, p = (0.327). See Figure 8.
51
Change from pre-survey to post-survey
Q2 in the experimental group
2.5
2
1.88
1.8
Pre-Survey Q2
Post-Survey Q2
1.5
1
0.5
0
Figure 8. Change From Pre-Survey to Post-Survey Q2 in the Experimental Group.
Null Hypothesis 8: There is no significant difference in the growth in scores
from the quiz 1 to quiz 2 amongst the students in the experimental group.
To test the eighth hypothesis, a t-test for independent means was done. The
difference between the growth in scores from the quiz 1 to quiz 2 amongst the students
in the experimental group for quiz 1 (M = 19.14, SD = 9.391) and quiz 2 (M = 20.23,
SD = 8.889) was not statistically significant, t(34) = -0.867, p = (0.392). See Figure 9.
52
Change from quiz 1 to quiz 2 scores in
the experimental group
25
20
19.14
20.23
Quiz 1
Quiz 2
15
10
5
0
Figure 9. Change From Quiz 1 to Quiz 2 Scores in the Experimental Group.
Summary
This study examined the usefulness and effectiveness of making and using a
note card on specific assessments to improve algebraic understanding, increase test
scores, and boost math confidence, comparing two high school algebra 1 classes at an
urban high school.
After testing the first hypothesis, the difference between the group means in
the growth in test scores from the pre-test assessment to the post-test assessment for
the students that have not repeated algebra 1 was not statistically significant. After
testing the second hypothesis, the difference between the group means in the growth in
test scores from the pre-test assessment to the post-test assessment for the students that
have repeated algebra 1 was statistically significant. After testing the third hypothesis,
the difference between the group means in the growth in test scores from the pre-test
assessment to the post-test assessment for the females that have not repeated algebra 1
53
was not statistically significant. After testing the fourth hypothesis, the difference
between the group means in the growth in test scores from the pre-test assessment to
the post-test assessment for the females that have repeated algebra 1 was not
statistically significant. After testing the fifth hypothesis, the difference between the
group means in the growth in test scores from the pre-test assessment to the post-test
assessment for the males that have not repeated algebra 1 was statistically significant.
After testing the sixth hypothesis, the difference between the group means in the
growth in test scores from the pre-test assessment to the post-test assessment for the
males that have repeated algebra 1 was statistically significant. After testing the
seventh hypothesis, the difference between the Survey Q2 means in the growth of
math confidence from the pre-test survey to the post-test survey amongst students in
the experimental group was not statistically significant. Lastly, after testing the eighth
hypothesis, a the difference between the growth in scores from the quiz 1 to quiz 2
amongst the students in the experimental group was not statistically significant.
The results of the t-tests conducted on null hypotheses 1, 3, and 4 showed that
there is no significant difference in student achievement scores between the control
and experimental groups with the use of a note card for the quizzes but not for the pretests and post-tests. The result of the t-test conducted on null hypothesis 7 showed that
there is no significant difference in math confidence between the control and
experimental groups before and after the unit was taught. The result of the t-test
conducted on null hypothesis 8 showed there is no significant difference in student
54
achievement scores amongst the students in the experimental group from the quiz 1 to
quiz 2 with the use of a note card. This researchers' opinion would believe that with
proper introduction and practice on how to correctly make and use a note card to use
on the quizzes, the data would have been different.
The result of the t-test conducted on null hypothesis 2 showed that there is
significant difference in student achievement scores for the students that have repeated
algebra 1 with the control group higher. However, the result of the t-test conducted on
null hypothesis 5 showed that there is significant difference in student achievement
scores for the males that have not repeated algebra 1 with the experimental group
higher. Anecdotally, non-repeating algebra students commented to the researcher that
the use of the note cards boosted their math confidence and they felt they had
performed better. Lastly, the result of the t-test conducted on null hypothesis 6 showed
that there is significant difference in student achievement scores for the males that
have repeated algebra 1 with the control group higher. This researcher’s opinion
would believe that control group was all around stronger in their math ability than the
experimental group due to many outside factors.
55
Chapter 5
SUMMARY AND DISCUSSION
Summary
Algebra is a way of thinking. Students who learn and understand algebra can
use the understanding to apply it to any real-world problem that they run into. Most
people do not think algebra is used very often, but it is actually used every day in
different purposes, and many times without even knowing it. In today's world of high
stakes testing, students must become proficient in their knowledge of algebra to
succeed and even graduate from high school. Having the ability to do algebra will help
students excel into the area of expertise that they decide to specialize in and keep
many options open for their future careers.
Although basic math skills and understanding fundamental algebra are
important for everyday life, many people describe feeling anxious when faced with the
idea of doing any math. Math anxiety and test anxiety have been associated with
negative feelings surrounding math and unfavorable outcomes on both math
performance and the confidence to learn math (Bekdemir, 2010). This hinders many
students' abilities to perform well on math assessments, which lowers their math
confidence and likelihood to pass their algebra course. Paying close attention to
student motivation, student environment, and a need for different study strategies are
necessary to help increase math confidence and learning, which in turn will increase
test scores.
56
The theories of preparedness theory, stereotype threat, information processing
theory, and dual coding theory play major roles in this study. Students may have
already "prepared" themselves to believe they are terrible at math because of their
disappointment in math at an early age or because their parents were scared of it also,
therefore, math is meant to be impossible. Women also risk being judged by the
negative stereotype that women have weaker math ability, which is known as
stereotype threat and may disrupt women’s math performance (Spencer et al., 1999).
Furthermore, the main goal of the information processing model is to explain visibly
that the knowledge is thought of in terms of what is known, how it is organized,
represented, stored, and how it is used (Pellegrino & Goldman, 1987). Lastly, those
students that were allowed use of a note card, were given a limited space to write on,
where they must summarize, or code, the material that they are going to be tested on.
It is important to understand the theoretical basis for this study to appreciate the
usefulness of it.
Different teaching strategies of effective studying in an effort to improve
student assessment scores are constantly being investigated. Yet, depending on the
teacher, students, environment, and district, certain teaching techniques will only work
for certain students. The effects of making and using a note card during quizzes had
not been explored in the high school algebra 1 classes where this study was conducted.
However, introducing a new teaching technique will allow for another useful strategy
that may be applied in the classroom.
57
Conclusions
In this study, high school students at a suburban public high school located in
Northern California were studied. This study occurred over the course of one algebra 1
unit, which consisted of two quizzes and one chapter test, lasting approximately two
and a half weeks. Students in the experimental group consisted of one algebra 1 high
school class that were allowed to use one 3" x 5" note card on each quiz, but not the
test. Students in the control group consisted of another algebra 1 high school class that
were not allowed any use of a note card. The same material in both classes were
covered throughout the lessons and taught in the same instruction with different
teaching strategies.
Data analysis showed the difference in the test scores between the two groups
for the students that had not repeated algebra 1 were not statistically significant.
Therefore, the data suggests that the first null hypothesis cannot be rejected.
Data analysis showed the gains in the test scores of the control group for the
students that had repeated algebra 1 were statistically significant. The use of a note
card for the experimental group did not positively affect those specific students.
Therefore, the data suggests that the second null hypothesis can be rejected.
Data analysis showed the difference in the test scores between the two groups
for the females that had not repeated algebra 1 were not statistically significant.
Therefore, the data suggests that the third null hypothesis cannot be rejected.
58
Data analysis showed the difference in the test scores between the two groups
for the females that had repeated algebra 1 were not statistically significant. Therefore,
the data suggests that the fourth null hypothesis cannot be rejected.
Data analysis showed the gains in the test scores of the experimental group for
the males that had not repeated algebra 1 were statistically significant. The use of a
note card for the male students in the experimental group did positively affect those
specific students. Therefore, the data suggests that the fifth null hypothesis can be
rejected.
Data analysis showed the gains in the test scores of the control group for the
males that had repeated algebra 1 were statistically significant. The use of a note card
for the experimental group did not positively affect those specific students. Therefore,
the data suggest that the sixth null hypothesis can be rejected.
Data analysis showed the difference in the boost of math confidence in the
experimental group was not statistically significant. Therefore, the data suggests that
the seventh null hypothesis cannot be rejected.
Data analysis showed the difference in the quiz scores in the experimental
group was not statistically significant. Therefore, the data suggests that the eighth null
hypothesis cannot be rejected.
All in all, the use of note cards only presented useful and significant gains for
the males in the experimental group who had not repeated algebra 1.
59
Concerns
There are a number of variables and factors in this study that affect the ability
to translate these results into other classrooms, schools, or circumstances. The small
sample size in both the experimental group and the control group make the study less
likely to be translated. The location (a suburban public high school located in Northern
California), makes the study less likely to yield the same results in other locales. The
numerous differences among individual students may have lead to differences in test
scores and math confidence. Students may have been absent from the class during the
time of instruction, assessments, or surveys were given. There are many outside
factors that could have affected students negatively in the study: socio-economic
factors (different backgrounds), parents' education (outside help), school environment
and resources, safety and bullying, language barriers, teachers/administration, student
motivation, etc.
Additionally, even though many students are repeating algebra 1, it is difficult
to know how well prepared they were coming into the assessments due to their past
history. Students in both classes come from a variety of educational backgrounds and
are functioning at different academic levels. The control group was also taught earlier
during the day, while the experimental group was taught after lunch, which could also
have a huge impact on student motivation and willingness to learn. Lastly, for the
experimental group, with proper introduction and practice on how to correctly make
and use a note card to use on the quizzes, the data may have been different.
60
Further Recommendations
Although the results of this research did suggest that the use of note cards
presented useful and significant for the males in the experimental group that had not
repeated algebra 1, further research is needed to prove the generalizability of this
study and to refine study strategies. This particular study covered a span of one unit,
which lasted about two and a half weeks. It would be interesting to start the use of
note cards on quizzes at the beginning of the year and last a whole semester or even a
year. Researchers may also choose a different sample of students from different
backgrounds, grade levels, or math course.
Additionally, a school may want to consider adopting the idea of making and
using note cards on specific assessments at a students' early stage of education, so they
are familiar with the importance of how beneficial the strategy of thinking about what
is known and unknown and how to use a note card may be when used appropriately.
Continued research on this subject could lead to students going into their exams better
prepared and more confident. It is extremely important that researchers apply the latest
and finest teaching strategies that best suit their students. During this time of public
schools adopting Common Core Standards that will be changing the curriculum
completely, educators need to creatively find ways to boost student learning. A good
math teacher should motivate the math and engage the students. They should convey
the beauty of the subject. Every great teacher has his or her own teaching style and
61
philosophy but should always know their audience and apply the teaching strategies
that best fit their classroom.
Bob Moses (2001), founder of the Algebra Project, a nonprofit organization
that anticipates quality public school education for every child in America by the use
of mathematics as an organizing tool, noted that "the most urgent social issue affecting
poor people and people of color is economic access... [and that] depends crucially on
math and science literacy" (p. 5). As stated so perfectly by Donovan (2008), "math
literacy education must be offered to everyone, child or adult, cognitively gifted,
average, or slow. Given its critical importance in college-level mathematics, algebra 1,
in particular, must be made accessible."
62
APPENDIX A
Letter to Parents
63
Dear Parents/Guardians,
I am currently in pursuit of my Masters degree in Education with a focus in
Curriculum & Instruction from California State University, Sacramento. I will be conducting
an experimental study for my thesis. The purpose of this study is to test the usefulness and
effectiveness of making and using a note card to improve test scores and increase math
confidence.
Whether the students are repeating Algebra 1 or taking it for the first time, a teacher is
always looking for different approaches for these students to retain and understand the
Algebraic concepts. This study will provide data to test my hypothesis, which is that the use
of note cards on quizzes will help students gain math confidence and raise their test scores.
The study will occur over the course of one Algebra 1 unit, which includes two
quizzes and one chapter test. The experimental group will consist of one Algebra 1 high
school class that may use a note card on the quizzes, but not the test. The control group will
consist of another Algebra 1 high school class that will not be allowed any use of a note card.
The subjects’ rights to privacy and safety will be protected by assuring the anonymity of all
participants. There will be no names used in any part of this research. All student materials
will be provided to the students.
A survey will also be given to those students that were allowed the use of a note card
with questions based on their experience. The study may be extended to cover two Algebra 1
units, which is dependent on whether the note cards were proven to be effective for the
experimental group. If this occurs, the experimental group and the control group will both use
a note card on all quizzes for the rest of the semester, otherwise, the study will stop because it
would have been shown that note cards are ineffective and not useful.
Deciding which class would be the experimental group and which class would be the
control group was done at random by one of my aides. He chose the number five, which meant
that my fifth period Algebra 1 class would be the experimental group. This research has been
deemed “no risk,” which means there will be no harm or discomfort inflicted on my subjects.
The students in both classes are still learning the California State standards at the same pace
and they will take the same assessments and have the same information introduced on a daily
basis.
If you have any questions and/or concerns about my research, please feel free to
contact me at (***) ***-**** ext. **** or by e-mail at ********@*****.***. Thank you!
Sincerely,
Elizabeth Gutiérrez
64
APPENDIX B
Quizzes
65
66
67
68
69
APPENDIX C
Tests
70
71
72
73
74
APPENDIX D
Surveys
75
76
77
APPENDIX E
Extras
78
Purpose of the Study
A few reasons why many students dislike math are because they think they are
horrible at the subject, they find no usefulness in the subject, or they have no
confidence and fear the subject. For example, the preparedness theory of phobia holds
that humans are biologically prepared to learn to fear objects and situations that
threatened the survival of the species throughout its evolutionary history (Seligman,
1971). In a sense, many students fear mathematics because they have failed before on
exams and in the class, therefore they are afraid to fail again, thus, they develop
anxiety. When we are prepared for something, we no longer fear it, thus adapt and
evolve. That is what the note cards hope to do, prepare students not to fail.
Dual coding theory is about the nature of two great symbolic systems of
cognition: language and mental imagery. They state that the formation of mental
images aids in learning when developing this theory (Sadoski & Paivio, 2001).
Therefore, the organization of making a note card is important on preparing and
studying for any assessment because the students are reading through their notes or
book and deciding what they feel is important to write down on their note card. It is a
form of studying by acknowledging and remembering the mental images of the taught
material that the student does know and does not know which they will then find
important to write down. As told by Kathleen Thayer, the Director of Academic
Success Center at Purdue University, index cards are an efficient means of organizing
information that can be written in a straightforward manner and the information is
79
learned by making them (Thayer, 2009). Constructing and organizing a note card was
just one strategy used to improve the understanding of the material and the
effectiveness of making and using them on quizzes was tested within the study.
Specific questions explored in this study include: Did making and using a note
card during assessments improve or hinder student assessment scores? Was there a
significant difference in the unit test scores between students who were allowed to
make and use a note card during quizzes than those who were not? Was there an effect
on student test scores and the motivation of making a note card that was differentiated
by the gender and the number of times that student had repeated Algebra 1? Was there
a significant difference in the students' math confidence before and after the use of the
note card?
Significance of the Study
There are many reasons why the use of a note card on quizzes may be
beneficial to students and improve students’ performance on tests and boost math
confidence. Many teachers may be unaware of the effects and usefulness of note cards
on assessments influencing student performance and math confidence. They may
prefer to test students on their ability to analyze rather than memorize formulas.
Allowing students to prepare their note card is a form of review that they may have
never done before, as in Tommy's case. The results of this study add to the body of
research on issues relating to the effectiveness of note cards on assessments in an
80
algebra 1 classroom and other mathematics classrooms. It will also add to the body of
research on issues relating to boosting math confidence in the classroom.
Theoretical Basis of the Story
Many students are "prepared" to fear and dislike mathematics at a very young
age which tends to lead to math anxiety and little math confidence. Seligman's (1971)
theory of preparedness supports the theory of classical conditioning, which
emphasized the fact that phobias are not random. Many students that are unaware that
they have "prepared" themselves to have a phobia of mathematics are a result of their
history and how they were brought up to like the subject. Females tend to internalize
failure, while males externalize it. Gender stereotypes affect the learner by lack of
interest, less connection to the curriculum, and lack of confidence in the subject.
Allowing all students to make and use note cards on their quizzes will give hope to
both females and males that they are capable of succeeding in mathematics, which will
boost math confidence. Knowledge is thought of in terms of what is known, how it is
organized, represented, stored, and how it is used (Pellegrino & Goldman, 1987).
Making and using a note card to be used on an assessment, can be used as a method or
technique to enhance the processing of information into the long-term memory.
According to the dual coding theory, students actively process information through the
selection, organization, and representation of information in condensed form during
the making of a note card (Dorsel & Cundiff, 1979; Hindman, 1980). The dual coding
theory also affirms that preparing a note card to be used on an assessment is an
81
efficient way to code material for memory. This also leads to taking ownership for the
information, more self-regulation, and more motivation.
Background of the Researcher
Elizabeth Gutierrez is a graduate student pursuing her Masters in Education
with a major in Curriculum and Instruction at California State University, Sacramento.
She attended Sierra Community College for two years before she transferred to
California Polytechnic State University, San Luis Obispo, where she graduated with a
bachelor's degree in Mathematics, a minor in Spanish, and her single subject teaching
credential in Mathematics.
Elizabeth's father used to play math games during dinner at the kitchen table
when she was a little girl, which led to her love for mathematics. It was not until she
was tutoring her peers in mathematics at Sierra College that she realized she wanted to
become a teacher and spread her knowledge and love for the subject.
Elizabeth is currently in her fourth year teaching Algebra 1 and Algebra 2 at a
suburban high school and loving every minute of it. She is the adviser for the Class of
2014 where she helps the student government class plan the homecoming float,
dances, design their class shirts, etc. She is also an adviser for the Kick Off Mentors,
which is a club that consists of over 100 upper-classmen that plan the Freshmen
Orientation at the beginning of the school year and monthly meetings to mentor the
Freshmen throughout the year.
82
Elizabeth is very much involved in many committees at her school for her
mathematics department to slowly implement the Common Core Standards into the
curriculum. She hopes to one day become the Division Leader of the department,
Math Coach of the District, and Curriculum Coach of the District. Her love for
mathematics and helping students strive to do well in mathematics will never go away.
83
REFERENCES
Aronson, J., & Good, C. (2002). The development and consequences of stereotype
vulnerability in adolescents. In F. Pajares & T. Urdan (Eds.), Academic
motivation of adolescents (pp. 299-330). Greenwich, CT: Information Age
Publishing.
Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive
consequences. Current Directions in Psychological Science, 11(5), 181-185.
Asimov, N. (2008, July 10). Algebra 1 to be required for all 8th-graders. The San
Francisco Chronicle, A-1. Retrieved from http://www.sfgate.com
Astone, N. M., & McLanahan, S. S. (1991). Family structure, parental practices and
high school completion. American Sociological Review, 56, 309-320.
Bekdemir, M. (2010). The pre-service teachers' mathematics anxiety related to depth
of negative experiences in mathematics classroom while they were students.
Educational Studies in Mathematics, 75, 311–328.
Boaler, J. (2012, July 3). Timed tests and the development of math anxiety. Education
Week. Retrieved from
http://www.edweek.org/ew/articles/2012/07/03/36boaler.h31.html?tkn=VOQF
K9Wqv%2FQyFHqgUJA20N6MixLTNTr2KA%2Fy&cmp=clp-edweek
Bracey, G. W. (2008). The algebra hoax. Phi Delta Kappan, 90(4), 306-307.
Brush, L. (1980). Encouraging girls in math. Cambridge, MA: Abt.
84
Butler, D., & Crouch, N. (2011). Student experiences of making and using cheat sheets
in mathematical exams. Paper presented at the AAMT-MERGA 23rd
Conference of the Australian Association of Mathematics Teachers
Conference, and 34th Mathematics Education Research Group of Australasia
Conference, Alice Springs, NT. Retrieved from
http://www.merga.net.au/documents/RP_BUTLER&CROUCH_MERGA34AAMT.pdf
California Department of Education. (2012, August 31). 2012 STAR results mark ninth
straight year of rising English and math scores. Retrieved from
http://www.cde.ca.gov/nr/ne/yr12/yr12rel79.asp
Catsambis, S. (1994). The path to math: Gender and racial-ethnic differences in
mathematics participation from middle school to high school. Sociology of
Education, 67, 199-215.
Cherry, K. (2012). What is test anxiety? A brief overview of test anxiety. About.com.
Retrieved from http://psychology.about.com/od/mentalhealth/a/testanxiety.htm
Clark, R. E., & Paivio, A. (1991). Dual coding theory and education. Educational
Psychology Review, 3, 149-210.
Clark, R. M. (1993). Homework-focused parenting practices that positively affect
student achievement. In N. E. Chavkin (Ed.), Families and schools in a
pluralistic society (pp. 85-105). Albany: State University of New York Press.
85
Cooper, S. (2009). Theories of learning in educational psychology. Retrieved from
http://www.lifecircles-inc.com/Learningtheories/learningmap.html
Cornell, C. (1999). I hate math! I couldn't learn it, and I can't teach it! Childhood
education, 75, 225-230.
Davies, P. G., Spencer, S. J., Quinn, D. M., & Gerhardstein, R. (2002). Consuming
images: How television commercials that elicit stereotype threat can restrain
women academically and professionally. Personality and Social Psychology
Bulletin, 28(12), 1615-1628.
Dickson, K. L., & Bauer, J. J. (2008). Do students learn course material during crib
sheet construction? Teaching of Psychology, 35(2), 117-120.
Dickson, K. L., & Miller, M. D. (2005). Authorized crib cards do not improve exam
performance. Teaching of Psychology, 32(4), 230-233.
Ding, C., Song, K., & Richardson, L. I. (2006). Do mathematical gender differences
continue? A longitudinal study of gender difference and excellence in
mathematics performance in the US. Educational Studies, 40(3), 279-295.
Donovan, T. (2008). The importance of algebra for everyone. WORLD, 35.
Dorsel, T. N., & Cundiff, G. W. (1979). The cheat-sheet: Efficient coding device or
indispensable crutch? Journal of Experimental Education, 48, 39-42.
Drake, V. K., Freed, P., & Hunter, J. M. (1998). Crib sheets or security blankets?
Issues in Mental Health Nursing, 19, 291–300.
86
Duckworth, A. L., & Seligman, M. E. (2006). Self-discipline gives girls the edge:
Gender in self-discipline, grades, and achievement test scores. Journal of
educational psychology, 98(1), 198.
Duncan, D. G. (2007). Student performance shows slight improvement when open
notes are used during information systems exams. Journal of Information
Technology Education, 6, 361-370.
Education Commission of the States. (2007, March 23). Standard high school
graduation requirements (50-state). [Data file]. Retrieved from
http://mb2.ecs.org/reports/Report.aspx?id=735
Epstein, J. L. (1995). School/family/community partnerships: Caring for the children
we share. Phi Delta Kappan, 76, 701-712.
Erbe, B. (2007). Reducing test anxiety while increasing learning: The cheat sheet.
College Teaching, 55(3), 96-98.
Ernest, J. (1976). Mathematics and sex. Berkeley: University of California Press.
Feldlaufer, H., Midgley, C., & Eccles, J. S. (1988). Student, teacher, and observer
perceptions of the classroom environment before and after the transition to
junior high school. Journal of Early Adolescence, 8, 133-156.
Fingerman, K. L., Cheng, Y. P., Wesselmann, E. D., Zarit, S., Furstenberg, F., &
Birditt, K. S. (2012). Helicopter parents and landing pad kids: Intense parental
support of grown children. Journal of Marriage and Family, 74(4), 880-896.
87
Frew, L. A., Zhou, Q., Duran, J., Kwok, O., & Benz, M. R. (2012, January). Effect of
school-initiated parent outreach activities on parent involvement in school
events. Journal of Disability Policy Studies. Retrieved from
http://dps.sagepub.com.proxy.lib.csus.edu/content/early/2012/01/16/10442073
11427163.full.pdf+html
Griffith, J. (1996). Relation of parental involvement, empowerment, and school traits
to student academic performance. Journal of Educational Research, 90, 33-41.
Halpern, D. F. (1986). Sex differences in cognitive abilities. Hillsdale, NJ: Erlbaum.
Harms, W. (2012). Math anxiety causes trouble for students as early as first grade.
University of Chicago News. Retrieved from
http://news.uchicago.edu/article/2012/09/12/math-anxiety-causes-troublestudents-early-first-grade
Harris, A. M., & Carlton, S. T. (1993). Patterns of gender differences on mathematics
items on the Scholastic Aptitude Test. Applied Measurement in Education,
6(2), 137-151.
Hawkins, J. A., & Berndt, T. J. (1985). Adjustment following the transition to junior
high school. Paper presented at the biennial meeting of the Society for
Research in Child Development, Toronto.
Hembree, R. (1988). Correlates, causes, effects, and treatment of test anxiety. Review
of Educational Research, 58(1), 47-77.
88
Higbee, J. L., & Thomas, P. V. (1999). Affective and cognitive factors related to
mathematics achievement. Journal of Developmental Education, 23, 8-24.
Hill, K. T., Wigfield, A., & Plass, J. A. (1980, April). Effects of different kinds of
optimizing instructions on seventh- and eighth-grade children's achievement
test performance. Paper presented at the annual meeting of the American
Educational Research Association, Boston.
Hindman, C. D. (1980). Crib notes in the classroom: Cheaters never win. Teaching of
Psychology, 7, 166–168.
Hirsch, B. J., & Rapkin, B. D. (1987). The transition to junior high school: A
longitudinal study of self-esteem, psychological symptomatology, school life,
and social support. Child Development, 58, 1235-1243.
Hyde, J. S., Fennema, E., & Lamon, S. J. (1990). Gender differences in mathematics
performance: A meta-analysis. Psychological Bulletin, 107, 139-155.
Jamieson, J. P., & Harkins, S. G. (2012). Distinguishing between the effects of
stereotype priming and stereotype threat on math performance. Group
Processes & Intergroup Relations, 15(3), 291-304.
Lareau, A. (1987). Social class differences in family-school relationships: The
importance of cultural capital. Sociology of Education, 60, 73-85.
Larwin, K. H., Larwin, D. A., & Gorman, J. (2012). Mnemonic aids during tests:
Worthless frivolity or effective tool in statistics education? Retrieved from
http://www.aabri.com/manuscripts/111047.pdf
89
Liang, J. H., Heckman, P. E., & Abedi, J. (2012). What do the California standards
test results reveal about the movement toward eighth-grade algebra for all?
Educational Evaluation and Policy Analysis, 34(3), 328-343.
Liebert, R. M., & Morris, L. W. (1967). Cognitive and emotional components of test
anxiety: A distinction and some initial data. Psychological Reports, 20, 975978.
Maclaurin, C. (1748). A treatise of algebra in three parts. London: A. Millar.
Maloney, E. A., & Beilock, S. L. (2012). Math anxiety: Who has it, why it develops,
and how to guard against it. Trends in Cognitive Sciences. Retrieved from
http://hpl.uchicago.edu/Publications/TiCS%20Final_Maloney&Beilock_2012.
pdf
Mandler, G., & Sarason, S. (1952). A study of anxiety and learning. Journal of
Abnormal and Social Psychology, 47, 166-173.
Martin, M. (2010, September 28). Why is math so important for kids? Retrieved from
file:///F:/EDTE/Thesis%20%20ideas/history%20of%20algebra/Why%20Is%20Math%20So%20Important
%20For%20Kids%20-%20martin.htm
Matthews, D. (2004). No Child Left Behind: The challenge of implementation.
Spectrum: Journal of State Government, 77(2), 11-15. Retrieved from
http://web.ebscohost.com.leo.lib.unomaha.edu/ehost
90
McNally, R. J. (1987). Preparedness and phobias: A review. Psychological Bulletin,
101, 283-303.
Meece, J. (1981). Individual differences in the affective reactions of middle and high
school students to mathematics: A social cognitive perspective. Unpublished
doctoral dissertation, University of Michigan.
Meece, J. L., Wigfield, A., & Eccles, J. S. (1990). Predictors of math anxiety and its
influence on young adolescents' course enrollment intentions and performance
in mathematics. Journal of educational psychology, 82(1), 60.
Miller, D. L., & Kelley, M. L. (1991). Interventions for improving homework
performance: A critical review. School Psychology Quarterly, 6, 174-185.
Moses, R. P. (2001). Radical equations: Math literacy and civil rights. Boston:
Beacon Press.
Muller, C. (1993). Parent ties to the school and community and student academic
performance. Paper presented at the conference on Sociology of Education,
Asilomar, CA.
Newman, R. S. (1990). Children's help-seeking in the classroom: The role of
motivational factors and attitudes. Journal of Educational Psychology, 82, 7180.
O'Connor, J. L. (1978). Perceptions of self, ideal self, and teacher feelings in
preadolescent children. Elementary School Guidance and Counseling, 13, 8892.
91
Pellegrino, J. W., & Goldman, S. R. (1987). Information processing and elementary
mathematics. Journal of Learning Disabilities, 20(1), 23-32.
Perry, A. (2004). Decreasing math anxiety in college students. College Student
Journal, 38, 321-324.
Pianta, R. C., Hamre, B. K., & Allen, J. P. (2012). Teacher-student relationships and
engagement: Conceptualizing, measuring, and improving the capacity of
classroom interactions. Handbook of research on student engagement, 365386.
Rachman, S. (2004). Anxiety (2nd ed). New York: Routledge.
Reeves, C. (2003). Implementing the No Child Left Behind Act: Implications for rural
schools and districts. Naperville, IL: North Central Regional Educational
Laboratory.
Reiser, R. A., & Dempsey, J. V. (2007). Trends and issues in instructional design (2nd
ed.). Upper Saddle River, NJ: Pearson Education, Inc.
Revicki, D. A. (1981). The relationship among socioeconomic status, home
environment, parent involvement, child self-concept, and child achievement.
(ERIC Document Reproduction Service No. ED 206 645).
Richardson, F. C., & Suinn, R. M. (1972). The mathematics anxiety rating scale:
Psychometric data. Journal of Counseling Psychology, 79, 551-554.
92
Rivers, J. L. (2011, August 12). Common core standards and assessments. League of
Women Voters. Retrieved from http://www.lwv.org/content/common-corestandards-and- assessments.
Russell, D. (2012, September 24). What is math anxiety? About.com. Retrieved from
file:///F:/EDTE/Thesis%20-%20ideas/test-anxiety_files/anxiety%20%20russell.htm
Sadoski, M., & Paivio, A. (2001). Imagery and text: A dual coding theory of reading
and writing. Mahwah, NJ: Erlbaum.
Schraw, G., & McCrudden, M. (2003). Information processing theory. Retrieved from
http://www.education.com/reference/article/information-processing-theory/
Schwartz, B. (1974). On going back to nature. Journal of the Experimental Analysis of
Behavior, 21, 183-198.
Schwarzer, R., & Jerusalem, M. (1992). Advances in anxiety theory: A cognitive
process approach. Advances in test anxiety research, 7, 2–31. Lisse, The
Netherlands: Swets & Zeitlinger.
Seligman, M. E. P. (1971). Phobias and preparedness. Behavior Therapy, 2, 307-320.
Shah, P., & Miyake, A. (1999). Models of working memory. In P. Shah & A. Miyake
(Eds.), Models of working memory: Mechanisms of active maintenance and
executive control (pp. 1–25). Cambridge, England: Cambridge University
Press.
93
Simon, J. G., & Feather, N. T. (1973). Causal attributions for success and failure at
university examinations. Journal of Educational Psychology, 64, 46-56.
Simpson, R., LaCava, P., & Graner, P. (2004). The No Child Left Behind Act
(NCLB): Challenges and implications for educators. Intervention in School
and Clinic, 40(20), 67-75.
Soffer, M. E. (2008). Elementary students’ test anxiety in relation to the Florida
Comprehensive Assessment Test (FCAT). Unpublished master’s thesis, Florida
State University, Tallahassee, Florida.
Spencer, S. J., Steele, C. M., & Quinn, D. M. (1999). Stereotype threat and women’s
math performance. Journal of Experimental Social Psychology, 35, 4-28.
Stangor, C., Carr, C., & Kiang, L. (1998). Activating stereotypes undermines task
performance expectations. Journal of Personality and Social Psychology,
75(5), 1191.
Stoet, G., & Geary, D. C. (2012). Can stereotype threat explain the gender gap in
mathematics performance and achievement? Review of General Psychology,
16(1), 93.
Tate, W. F. (1997). Race-ethnicity, SES, gender, and language proficiency trends in
mathematics achievement: An update. Journal for Research in Mathematics
Education, 652-679.
94
Thadani, R. (2010). Information processing theory. Retrieved from
file:///F:/EDTE/Thesis%20-%20ideas/history%20of%20algebra/informationprocessing-theory%20-%20thadani.html
Thayer, K. (n.d.). Making and using study guides-aids to preparing for an exam.
Retrieved from
http://www.purdue.edu/studentsuccess/academic/asc/documents/study_guides.
pdf.
Theophilides, C., & Koutselini, M. (2000). Study behavior in the closed-book and the
open-book examination: A comparative analysis. Educational Research and
Evaluation, 6(4), 379- 393.
Tobias, S. (1985). Test anxiety: Interference, defective skills, and cognitive capacity.
Educational Psychologist, 20, 135-142.
Trickett, E. J., & Moos, R. H. (1974). Personal correlates of contrasting environments:
Student satisfactions in high school classrooms. American Journal of
Community Psychology, 2, 1-12.
Veroff, J. (1983). Contextual determinants of personality. Personality and Social
Psychology Bulletin, 9, 331-343.
Von der Embse, N., Barterian, J., & Segool, N. (2013). Test anxiety interventions for
children and adolescents: A systematic review of treatment studies from 2000–
2010. Psychology in the Schools, 50(1), 57-71.
95
Wachsman, Y. (2002). Should cheat sheets be used as study aids in economics tests?
Economics Bulletin, 1(1), 1.
Whitley, B. E. (1996). Does" cheating" help? The effect of using authorized crib notes
during examinations. College Student Journal, 30, 489-493.
Wickelgren, W. (1975). Coding and dynamics of memory. Final report. Washington,
DC: National Institute of Education, Office of Research Grants.
Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school
students. Journal of Educational Psychology, 80(2), 210.
Wine, J. (1971). Test anxiety and direction of attention. Psychological Bulletin, 76(2),
92.
Yee, D. K., & Eccles, J. S. (1988). Parent perceptions and attributions for children's
math achievement. Sex Roles, 19, 317-333.