WHY DO STUDENTS FAIL AT ALGEBRA?
Elizabeth Islip
B.A., California State University, Sacramento, 1987
THESIS
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF ARTS
in
EDUCATION
(Curriculum and Instruction)
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
FALL
2009
WHY DO STUDENTS FAIL AT ALGEBRA?
A Thesis
by
Elizabeth Islip
Approved by:
, Committee Chair
Dr. Sherrie Carinci
, Second Reader
Dr. Elisa Michals
Date
ii
Student: Elizabeth Islip
I certify that this student has met the requirements for format contained in the
University format manual, and that this project is suitable for shelving in the Library
and credit is to be awarded for the thesis.
, Department Chair
Dr. Robert Pritchard
Department of Teacher Education
iii
__________________
Date
Abstract
of
WHY DO STUDENTS FAIL AT ALGEBRA?
by
Elizabeth Islip
Statement of Problem
Increasing the mathematical knowledge of students is a goal of the National
Council on Education. Business leaders are calling on public education to produce
students with a deeper, stronger background in mathematics and science. Nationwide,
Algebra 1 is an eighth grade subject, yet 50% of the students enrolling in a public
independent study high school had not completed the algebra requirement. Research
has shown teaching and learning strategies, teacher behavior, and student motivation
play an important part in education, yet little research was found in the literature
regarding teaching and learning strategies for mathematics. Understanding student
perception of the educational endeavor to learn algebra may help researchers and
teachers understand how to better facilitate learning.
Sources of Data
Data was collected and pooled from cumulative school histories, student
interviews, observation of students learning during algebra class, and surveys
iv
completed by the students. Data and insight was provided regarding the students’
perceptions, histories, and personal needs for learning mathematics.
Conclusions Reached
The level of intelligence was not the cause of the students’ deficiency in
earning algebra credits. Social skills, attendance, attention span, personal attention
needs, and family divorce prohibited them from succeeding to their intellectual best.
These students had unique, complicated issues. It would take a special teacher with
exceptional counseling skills, prolific background information, and abundant time for
researching and planning to tackle and help solve these students’ issues with school.
They were at a high risk for dropping out of high school. If intervention
strategies had been identified early, and systemic support followed through over
multiple school years, these students may not have been at risk at all. The cumulative
folders, a communication tool already in place in public schools, could be used as a
vehicle for transmitting teacher knowledge of the individual student’s optimum
learning environment, including counseling recommendations and follow up.
Accountability for every student’s educational success is the heart of No Child Left
Behind, and an individualized education plan for each at risk student may be ideal.
, Committee Chair
Dr. Sherrie Carinci
Date
v
ACKNOWLEDGEMENTS
My sincere appreciation goes to the students studied in this thesis. They were
the primary source of my learning. My three daughters provided me with the
motivation and encouragement for this endeavor. Patrick, my husband, gave his
enduring love and support for which I am grateful. Dr. Sherrie Carinci supplied
positive advice and steady deadlines, allowing me to learn every step of the way.
Thank you all.
vi
TABLE OF CONTENTS
Page
Acknowledgments ....................................................................................................... vi
List of Tables ............................................................................................................... ix
Chapter
1. INTRODUCTION .................................................................................................. 1
Purpose of the Study ......................................................................................... 2
Statement of the Problem ................................................................................. 3
Significance of the Study.................................................................................. 4
Methodology..................................................................................................... 5
Limitations ........................................................................................................ 6
Theoretical Basis of the Study.......................................................................... 6
Definition of Terms .......................................................................................... 8
Organization of the Thesis................................................................................ 9
Background of the Researcher........................................................................ 10
2. REVIEW OF LITERATURE ............................................................................... 11
Introduction .................................................................................................... 11
Student Motivation ......................................................................................... 13
Teacher Behavior............................................................................................ 23
Learning Strategies ......................................................................................... 29
Summary of Literature Review ...................................................................... 33
vii
3. METHODOLOGY ............................................................................................... 36
Introduction .................................................................................................... 36
Research Questions ........................................................................................ 37
Research Design and Data Collection ............................................................ 37
Participants ..................................................................................................... 39
Setting ............................................................................................................. 41
Procedure ........................................................................................................ 43
Summary......................................................................................................... 45
4. RESULTS AND ANALYSIS OF THE DATA ................................................... 47
Introduction .................................................................................................... 47
Results and Analysis of the Data .................................................................... 48
Summary......................................................................................................... 59
5. DISCUSSION, LIMITATIONS, RECOMMENDATIONS, AND
CONCLUSIONS ............................................................................................ 61
Discussion....................................................................................................... 61
Limitations ...................................................................................................... 67
Recommendations .......................................................................................... 68
Conclusions .................................................................................................... 71
Appendix A. Consent to Participate in Research ..................................................... 73
Appendix B. Student Journal Questions ................................................................... 75
References .................................................................................................................. 78
viii
LIST OF TABLES
Page
1.
Percentage of Students Enrolled in Algebra by Grade Level ........................... 4
2.
Student Information ........................................................................................ 40
3.
Student Perception of Previous Algebra Class ............................................... 50
4.
Notes from Observation 2008/2009 ............................................................... 52
5.
Results of Interviews, Quoted from Students ................................................. 55
6.
Analysis of Cumulative School History ......................................................... 58
ix
1
Chapter 1
INTRODUCTION
Jay checked out of the educational arena in the fifth grade. He was 11 years
old. In his subsequent seven years of public education, no teacher was able to engage
or motivate Jay. He scored proficient on state common assessments in math and
English, yet failed seventh grade. That was the last year he put effort into the state
common assessments. He left the traditional high school for an alternative education,
independent study program, after school officials in his 11th year noticed aberrant
deficiency in credit accumulations required for graduation. Jay had not sufficiently
learned algebra, yet had received three years of instruction in the subject.
Alternative education is an educational option some students choose after
emotional, social, behavioral, or academic disillusionment at the traditional school
setting. Many of the students enrolling at this rural, public, independent study high
school in northern California register without previously obtaining the required
algebra credit. Mathematical achievement holds back many students from graduating.
“No Child Left Behind,” (NCLB) a federal act of 2001, calls for all students to
increase their academic achievement on performance assessments each year. This has
led to a scramble by teachers, schools, and school districts for the most effective and
efficient instructional practices (Robelen, 2009). Three areas important to student
success, as indicated in the literature, are student motivation, student perception of
teacher behavior, and student use of learning strategies.
2
There is little evidence of effective teaching strategies for low achieving
mathematics students. Research is heavy in the reading interventions, but only one
study was identified from an exhaustive search by Vannest, Temple-Harvey, and
Mason (2008) for students with disabilities in mathematics intervention. Meanwhile,
most of the intervention is directed at providing students with additional classroom
time. Pointedly, in mathematics, knowledge varies considerably from student to
student causing student frustration and boredom when intervention instruction is
delivered in a one size fits all manner (Balfanz, Legters, & Jordan, 2004).
Purpose of the Study
The purpose of this study was to examine the factors in students’ educational
and social experience that disabled them from completion of algebra credits in prior
attempts. Research has reported the motivation, perception of teacher behavior, and
learning strategies students held while enrolled in algebra classes facilitated success or
failure of algebra class (Bandura & Locke, 2003; Hopkins, 2005; Pajares, 2002;
Pintrich & deGroot, 1990; Weinstein, Ridley, Dahl, & Weber, 1989). This study’s
purpose was to explore the variances behind students’ life and classroom experiences
and review patterns in these three areas which could be identified as disruptive to the
individual learning process.
This study solicits students’ retrospective perceptions of the conditions
surrounding their prior algebra classes to ascertain the elements that contributed to the
student not acquiring the knowledge or being able to demonstrate knowledge of
3
algebra on performance tests. In doing so, the students and teacher gained awareness
about themselves, their strengths, and where to look for support.
Specific questions included: Did students know what circumstances made it
easier for them to learn? What effect did teaching style, positive climate, collaborative
learning, peer pressure, and parental support have on student ability to learn? What
distracted the student from his/her ability to learn algebra concepts? What challenges
have students encountered as they attempted to complete algebra? What strategies
have students used to learn algebra?
Statement of the Problem
The public school system’s fundamental purpose is to facilitate student
learning. Imagine if Starbucks defined their purpose as delivering good coffee, and yet
only 66% of their coffee was received as “good” by the consumer. The United States
has a drop out rate purported to be 33%. Many disenfranchised students opt for
alternative education in their quest for a high school diploma, yet algebra class
remains a stumbling block for many. To be successful students must be motivated to
learn, have knowledge of learning strategies and be encouraged and supported in their
educational endeavor.
Little research has been done to ease the acquisition of mathematical skills for
low performing students (Ketterlin-Geller, Chard, & Fien, 2008). Startling statistics
show an urgent need to examine the factors related to repeated failure in algebra. Data
from the National Center for Education Statistics (NAEP, 2001), states 35% of 12th
graders still score below basic in mathematics. This level has not changed significantly
4
since 1992. This northern California County has a 35% failure rate for freshmen
algebra students.
Reauthorization of the Elementary and Secondary Education Act of 1965,
NCLB (2001), calls for all students to be proficient in mathematics in 2014. States are
declaring there is not enough funding to educate all students and the courts are
agreeing that present funding is inadequate (Hoff, 2003) Prevention of mathematical
aversion and planning for intervention requires the early identification and early
response to individual student needs. In this independent study school, students have
significant needs in algebra. The table below compares all students in the county with
students in this independent study high school.
Table 1
Percentage of Students Enrolled in Algebra by Grade Level
Grade level
Placer County 2008
8th
Freshmen
Sophomores
Juniors
Seniors
60%
42%
20%
11%
Not collected
Independent Study High
School 2009
(high school, grades 9-12)
96%
95%
45%
25%
Significance of the Study
Algebra is a gateway class to higher mathematical attainment. Students
furthering their mathematical achievement past algebra have a greater likelihood of
seeking careers in mathematics, thus entering the workforce in higher paying
5
professions (Kortering, deBettencourt, & Braziel, 2005, Ruffins, 2007). Careers in
mathematics and science are important to the United States as scientific and
technological changes rapidly occur.
This study asks students to analyze their previous failure in algebra classes.
Including students in the inquisition process gives value to their feelings and increases
motivation. Students rarely have opportunities to speak up and advocate for
themselves. Students as partners in their education can be an effective tool in the
endeavor to achieve more complicated mathematical knowledge (Myslinski, 2008).
Student input and autonomy may be a link to success at learning. The results of this
study will add to the body of research on issues relating to prevention of math aversion
and intervention activities.
Methodology
This study examined the comments, records and behavior of 12th grade
students deficient in algebra credits and enrolled in an independent study high school.
Qualitative case study applications (Bogdan & Biklen, 1998; Merriam, 2001;
Spradley, 1980) were used to examine the students’ educational endeavors in algebra.
In their final year of high school, students were enrolled in an algebra class that aimed
to address the individual learning needs of each student. Confidence building, peer
tutoring, social skills, learning strategies, formative assessments, open discussion, peer
support, and team and family values were part of the daily lessons in an attempt to
assist students in identifying problem areas and reaching academic goals. The analysis
tools used to identify students’ reasons for failing to meet educational goals came from
6
information in the literature review regarding internal and external motivation, the
impact of positive and negative teacher behavior, and the strength of learning
strategies. Teacher experience and educational background contributed to the analysis
of student comments and behavior.
Limitations
There are several limitations to this study. One limitation is the number of
students who participated in the research. Eight 12th graders were included in this
study. They retrospectively looked at their prior algebra classes, but memory is not
always clear. Another limitation of the study is the fact that these students are still
adolescents. They may not be able to discern what was happening in their life two or
three years ago. Accessing prior experiences with an objective eye may be difficult. In
addition, a new textbook was used by the eight students and teacher during small
group instruction at an independent study high school. Lastly, this study lacked a
multicultural perspective since the students were from a semi rural community, with
little ethnic diversity. Six of the students were White and two were half Hispanic and
half White. All students were between 17 and 18 years old. They did not represent a
cross section of California.
Theoretical Basis of the Study
The perception of the student as to whether the school, or teacher, or classroom
peers care about each other is foundational for learning (Noddings, 2005). Nel
Noddings expresses that educators pursue goals for their students yet may not take the
relationship aspect of caring into consideration. Teachers may view themselves as
7
caring for students but unless the individual student feels cared for, the relationship is
not present. The number of students in classrooms, the mountain of standards that
must be covered by teachers, the inability of students to follow their own curiosities
and interests, deter the ability for caring relationships to flourish. If students are
listened to and valued, they are more likely to positively accept the standards brought
before them as educational goals (Noddings).
Trusting that students can reflect on their own needs and strengths and
developing confidence in students to advocate for assistance is a step in caring for
students as individuals. Individual education plans for students in Special Education
describe student, as well as parent and teacher, goals. Special Education students also
are encouraged to advocate for their own individual needs. Yet, outside of Special
Education, this is not the case. Students are given little opportunity in high school to
express individual needs.
William Glasser, (1997) describes four needs inherent to every person: the
need to belong, the need for power, the need for freedom, and the need for fun.
Fulfilling any of these four driving forces at school makes school desirable.
Belonging, power, freedom, and fun can be obtained through peer relationships,
autonomy in the classroom, choosing learning strategies, and involvement.
Relationships with teachers, trust in teachers as well as with the self, are at the core of
student success (Glasser).
8
Definition of Terms
Algebra - the science which teaches how to determine unknown quantities by
means of those that are known (Katz, 2007).
Engagement - students’ use of metacognitive and self-regulatory strategies
(Turner et al, 1998).
Formative assessment - formative assessments gives insight on how well the
student is learning the incremental steps and procedures. It provides the educator
knowledge of whether to continue with new material, or reteach (Ketterlin-Geller, et
al, 2008).
Intervention classes - direct involvement of a student in curriculum for a
specific purpose, such as passage of the California Exit Exam or passage of Algebra 1.
In this independent study high school, it means more intense teacher participation.
Involvement - complex interaction of student cognition, motivation, and affect.
Focused concentration and comprehension (Turner et al, 1998)
Learning strategies - are behaviors or thoughts that facilitate learning
(Weinstein et al., 1989).
Self efficacy - student’s judgments of their capacity to accomplish a task or
succeed in an activity (Pajares, 2002).
Self-regulation - proactive efforts to learn, self motivational processes (goal
setting, self-efficacy perceptions, attributions, self-consequences) and metacognitive
learning processes (planning, monitoring, adapting) (Pintrich & deGroot, 1990;
Zimmerman, 1996). Forethought, performance, and self reflection of student learning.
9
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 the layout of the thesis including the statement of the problem and the
significance of the study.
Chapter 2 is an examination of current knowledge gained in the field of
motivation, teacher behavior, and learning strategies. Motivation is broken into its
sub-components of perception and self efficacy, intrinsic motivation, and learning and
performance goal orientation. Teacher behavior is examined regarding a positive
climate and student involvement. Scaffolding of instruction is also covered as well as
the importance of autonomy and development of student self regulation. The last
portion of the literature review is an abridgment of information regarding learning
strategies; cognitive, self regulation, and use of resources.
Chapter 3 discusses the methodology used for this study. Information
regarding the participants’ background and the school, as well as the research
questions and the surveys used are included. A qualitative study, this research drew
upon interviews, observations, and student’s school history.
Chapter 4 is a discussion of the qualitative findings in the surveys and data
collected. This chapter sought patterns in the stories of the students, and looked for
areas of overlap or themes.
Chapter 5 is an interpretation of the data and a reflection to see how findings
compare to the literature review. Areas of concern and future research are discussed.
10
Recommendations for prevention rather than intervention of low skilled students in
mathematics are included. Following Chapter 5 is the Appendix and a list of all
references used in this study.
Background of the Researcher
Elizabeth Islip is a graduate student pursuing her Masters in Education with a
major in Curriculum and Instruction at California State University, Sacramento.
Elizabeth graduated from California State University, Sacramento with a bachelor’s
degree in Business Education and received a single subject and a multiple subject
teaching credential. Additionally, she has earned a certificate in Drop out Prevention
and English as a Second Language.
Teaching at the fourth grade level in English Language Learners’ classrooms
and moving to seventh grade English and Social Studies, then to a high school
independent study, teaching ninth through twelfth graders, Elizabeth now specializes
in 12th grade curriculum and advising a leadership group.
Besides teaching, she coordinates a High Priority School Grant for the school.
This entails facilitating progressive student improvement on performance testing for
all sub-groups including students with Individual Education Plans in Special
Education. Currently she is a participant with Placer County in a research study
focused on teaching mathematics in schools using rigorous instruction in mathematics
strategies and positive classroom management and focus techniques.
11
Chapter 2
REVIEW OF LITERATURE
Introduction
Congressional bipartisan support for the No Child Left Behind Act of 2001
(NCLB) indicates the act is here to stay (Robelen, 2009). Reform in education began
with increased accountability for teachers and schools. The states increased standards
and expectations, and directed common performance assessments for all students. A
scramble for the most effective and efficient research based methods to increase
academic achievement ensued. The results of this reform movement are shown by the
Trends in International Mathematics and Science Study [TIMMS] 2007 (Robelen). In
1999, the United States ranked 19 out of 38 nations in math performance levels. In
2003, the United States moved to15th place, and in 2007 the United States ranked 9th
(Robelen). States are mandated to continue with the acceleration of academic
achievement for all students and prove this increase through performance assessments.
Sean Cavanagh (2008) reports that in spring of 2008, the state of California
mandated (raised from recommended) all eighth grade students be enrolled in Algebra
1 by the year 2011. Public uproar and expert opinion moved the state to rescind the
requirement. The push behind the national Algebra 1 mandate at the eighth grade level
is the achievement gap between students of color and low economic status, and white
students. The national No Child Left Behind Act requires all eighth graders to be
tested in algebra no matter if the course has been taken. California was out of
compliance with NCLB allowing eighth graders not enrolled in algebra to take the
12
performance assessment in general math (Cavanagh). Hence California’s endeavor to
mandate all eighth graders be enrolled in Algebra. California wants to maintain
national credibility and funding by complying with federal requirements on algebra
testing.
In unison, businesses are very concerned and are pushing for more students to
gain higher math skills (Business Higher Education Forum, 2006; Cavanagh, 2008).
Business leaders strongly communicate that California’s math and science education is
losing ground globally. Presently, only 23% of eighth graders pass the general math
skills test at a proficient level. Breaking out the numbers shows that 13 % of African
Americans and 16% of Hispanic students reached the proficient mark in general math
in 2006 (Cavanagh, 2008; Ketterlin-Geller et al., 2008). Algebra is a gateway class to
higher learning and crucial to success in math and science careers. Without positive
attitudes towards math, students opt out of taking higher level math classes and thwart
chances in a wide array of high paying nationally needed careers (Kortering et al.,
2005; Ruffins, 2007).
Schools, teachers, and students have been trying to increase the academic level
of students in mathematics. Nationwide, eighth graders enrolled in Algebra 1 between
2003 and 2005 has increased 7% points, with California increasing 14% (Cavanaugh,
2008). NCLB mandates teachers use researched teaching methods yet there is
insufficient data to support researched intervention strategies for students with
emotional and behavioral disabilities, a subgroup that accounts for 91% of its students
behind academically (Vannest et al., 2008). Evidence based practices are effective in
13
classrooms; higher academic levels are achieved, students are more engaged in the
educational practice, less disruption occurs. Teaching strategies are powerful (Vannest
et al., 2008). The educational progress of all students is being monitored, tallied,
calculated and reported by school districts, states, and the federal government. The
professional teacher ensures all students are learning and uses instructional time to
most effectiveness. Asking students to become partners with teachers in their learning
is a strategy teachers and the state of California is employing (Myslinski, 2008).
Mathematics achievement has eluded some students with mathematic
difficulties (Mazzocco, 2005). Research in the field of best practices for students with
math difficulties is scant, but studies looking for areas to gain ground point in the
direction of student motivation, teacher behavior, and learning strategies (House &
Telese, 2008, Ketterlin-Geller et al., 2008; Pintrich & deGroot, 1990, Tanner & Jones,
2003).
Student Motivation
Motivation facilitates students to become cognitively engaged. “Motivation
refers to the incentive for goal directed behavior,” writes Dr. Susan Davis (2007), and
is developed through socialization. Interestingly, motivation can be attained in varying
ways and is adaptive. Motivation is dynamic and multidimensional. Motivation can
also be content specific (Linnenbrink & Pintrich, 2002). Different motivational beliefs
effect students’ ability to successfully complete algebra by promoting, sustaining or
facilitating learning. One motivational focus is student perception of ability to
complete the task; self efficacy, “Can I do this task?” (Pintrich & DeGroot, 1990). A
14
second motivating focus is task value belief, or intrinsic motivation; the individual’s
perception of the task’s importance, “Why am I doing this task?” The third focus is
goal orientation (Pintrich, 1999). “How do I feel about doing this task?” (Pintrich &
deGroot). Three general goal orientations are identified; mastery learning, extrinsic
motivation, and relative ability orientation.
Self-efficacy
Self-efficacy is the degree to which a student believes they can accomplish a
goal (Pintrich, 1999). Albert Bandura and Edwin Locke (2003), from Stanford
University and University of Maryland respectively, report “…perceived self efficacy
and personal goals enhance motivation and performance attainments”(p. 87). Bandura
and Locke understand “…efficacy contributes significantly to the level of motivation
and performance” (p. 87). Pajares (2002) believes self efficacy to be intuitive. Selfefficacy influences choices in three ways. Students
1. choose tasks they feel confident in, but avoid tasks they don’t feel
confident in,
2. choose how much effort, resiliency, and persistence will be expended on an
activity, and
3. feel an amount of stress and anxiety, from serenity to great apprehension
based on self efficacy.
Indirectly, teacher behavior and learning strategies have an effect on self-efficacy and
student motivation (Bandura & Locke). Bandura and Locke cite evidence from nine
large bodies of methodology and strategies, including work in laboratories and field
15
studies done by Sadri & Robertson in 1993, and investigations where Boyer (2000)
controlled efficacy beliefs experimentally. The studies conducted in the 1990s and
early 2000s have encompassed diverse populations, using different formats and
different instruments. Bandura and Locke powerfully state that self-efficacy beliefs
predict how a person will act, either positively or negatively. Self-efficacy determines
how a person will make decisions at important points, and also, how effectively a
person self motivates. Also, self-efficacy determines how a person perseveres in
difficulties. Bandura and Locke go on to state emotional well-being and vulnerability
is also dependent on self-efficacy. Howard Zimmerman (1996) states that “…efficacy
apparently is largely induced from contemporary classroom experiences” (p. 11). Selfefficacy is influential in the choices students make (Pajares, 2002). For example, when
a student is free to choose activities, they tend to choose activities they feel confident
in, and avoid activities where confidence is lower. Self-efficacy determines the
amount of effort, persistence, and resilience students expend. Whether serenity or
apprehension exists is due to self efficacy (Pajares).
House and Telese (2008) analyzed the Trends in International Mathematics
and Science Study 2003 (TIMSS 2003) containing data involving 12,000 thirteen year
old American and Japanese students. Japan’s adolescents scored above average on
international assessments; therefore it was of interest to study Japanese students’
perceptions, beliefs, and attitudes and compare them to American adolescents. The
analysts examined the relationship between mathematical ability belief, instructional
strategies, and algebra achievement.
16
In the TIMSS 2003 study, students were given 12 questions regarding their
beliefs and attitudes about mathematics. Examples of statements describing student
beliefs were, “I usually do well in mathematics,” and “I would like to get a job that
involved using mathematics.” Students responded by indicating disagree a lot,
disagree a little, agree a little, and agree a lot to the statements. Additionally, students
responded to 14 statements regarding instructional strategies used in the classroom.
For example, “We explain our answers,” “We work together in small groups,” and
“We use calculators.” Students indicated never, some lessons, about half the lessons,
or every or almost every lesson to indicate their experience with instructional
strategies in the classroom (House & Telese, 2008).
Each student also took a performance assessment for the TIMSS 2003, named
the International Mathematics Assessment. The assessment had relatively few test
questions for each content area, therefore the analysts generalized five plausible score
values for each question and took the average for each student on the algebra
assessment. They believed this method to be consistent with statistical procedures
(House & Telese, 2008). In this study, the House and Telese used quantitative
methods to determine the relationship between mathematics beliefs and attitudes about
mathematics, and algebra test scores. A multiple regression procedure was used to
determine the relative contribution of self beliefs and classroom instructional strategy
to explanation of algebra test scores (House & Telese).
House and Telese (2008) found for both Japanese and American students,
when a student perceives they are capable of learning mathematics more easily than
17
their peers, their achievement in mathematics is higher and their selection of careers
requires more mathematics education. Additionally, the student relates being more
interested in careers in science and engineering (p. 109). Beliefs, attitudes, and self
confidence towards academic achievement influence math achievement (p. 102). The
relationship between higher mathematic achievement on test scores and student belief
in ability was positive, as well as the converse. Predominantly, students not
performing well on test scores related negative self appraisals. Interesting was the
finding that students earning high and low test scores communicated that they enjoyed
learning mathematics, needed mathematics for careers they choose, and believed
mathematics helped in day to day life (House & Telese, 2008).
Pintrich and deGroot (1990) developed and used a 56 item self reporting
questionnaire to look at correlations between motivation beliefs and the use of
cognitive learning strategies. The questionnaire entitled Motivated Strategies for
Learning Questionnaire (MSLQ) is a four part survey. The questionnaire is divided
into self-efficacy, intrinsic value, text anxiety, and cognitive strategy (Pintrich &
deGroot). This questionnaire was given to 173 seventh graders in eight science and
seven English classes. Student performance was measured from classroom quizzes and
tests, seatwork, homework, and reports, averaged over the semester to garner a
summary of all tasks.
The results of this study indicated self regulation, self efficacy and test anxiety
were the best predictors of student performance. The findings indicated cognitive
learning strategies and self regulation are related to high performance. Pintrich and
18
deGroot (1990) found students with high efficacy rates reported more use of cognitive
learning strategies such as rehearsal, elaboration, and organization. This indicated self
efficacy is facilitative to acquiring cognitive learning strategies (Pintrich & deGroot).
Another finding in this study by Pintrich and deGroot (1990) was the
relationship between intrinsic value and performance. Students who valued schoolwork for its importance and interest are intrinsically motivated. Good grades were not
the goal for these students, interest in learning was the goal. Performance and intrinsic
value were not directly related. Pintrich and DeGroot explained that the study
suggested intrinsic value was used by students to choose to become cognitively
engaged.
Intrinsic Motivation
Intrinsic motivation is the desire to learn for learning sake, for the pleasure of
learning (Linnenbrink & Pintrich, 2002). Interest is a factor in intrinsic motivation and
is multifaceted (Linnenbrink & Pintrich). Personal interest is somewhat stable and can
be measured by asking a student what they like or do not like. Situational interest can
be short or long term and has to do with the classroom climate or the context and
environment. Linnenbrink states in her description of situational interest, “For the
most part, the researchers who have studied situational interest have been reading
researchers who have focused on how different aspects of text can generate and
sustain interest” (p. 319).
19
Learning and Performance Goal Orientation
Dr. Carol Dweck is a professor at Columbia University and a prominent
researcher in the field of motivation. During an interview with Gary Hopkins, editor
from Education World (2005) Dr. Dweck explained the importance of intelligence as a
potential to develop using mastery learning. Mastery learning is learning focused on
strategies, challenge, and persistence at learning. Mastery learning is not worrying
about incompetence, or intelligence level, or how easy the learning is, nor about the
grade one will receive from the teacher. “Sustained effort over time is the key to
outstanding achievement,” states Dr. Dweck (Hopkins, p. 3). In describing mastery
learning, she explains how learning goals differ from performance goals. Learning
goals are the focus of learning, where a student will put effort. Performance goals are
assessment results. Assessment results show a student’s current level of performance.
Intellectual skills are accumulated through effort in challenges, reading, and education
(Hopkins). As students are taught intelligence is not static, their view of personal
ability changes and they are more willing to face challenges (Hopkins).
Dweck (Hopkins, 2005) explains many top athletes are mastery oriented. They
do not take a bad game or score as information that they are less talented then
opponents, but as learning focuses. Before the next event, the athlete will view taped
performances and practice relentlessly on newly indentified skills and improvement.
Formative testing is similar in this respect. It points the student and teacher to
skills needing focus and effort. Giving students the information on how and where to
focus and improve their learning is empowering (Tanner & Jones, 2003). Summative
20
tests are used at the end of learning, before going to another topic. Summative tests
give a picture of how well the student learned the information and point to high or low
ability and may be a reason students become disenfranchised with education during
middle school (Tanner & Jones). Developing learners who are in charge of their own
learning should be the educator’s goal (Tanner & Jones). Using formative assessments
is a means for students to take control.
Students facing a middle school algebra class after elementary arithmetic may
find it challenging. Because many middle school students have not been faced with
challenging mathematics classes in previous school years, they question whether
intelligence is limiting them in algebra. Dweck (Hopkins, 2005) makes the point that if
students view intelligence as fixed, the student may feel there is no need to further
attend to learning the difficult algebra concepts. Also, students may be afraid to try the
more difficult algebra concepts, for fear of proving their intelligence is low. This
causes a student to withdraw the value of education. Dweck’s research has shown no
relation between success in education and the ability to confront challenges, meaning
that if a child is successful in school that does not mean they will be good at
confronting challenges. Inversely, neither is there a relationship between the student
struggling in school and their ability to confront challenges (Dweck, as cited in
Hopkins).
Praising students for effort is much more powerful than praise or reward for
performance (Dweck, as cited in Hopkins, 2005). Linnenbrink and Pintrich (2002)
take this idea one step further and state praise should be for effort of specific learning
21
strategies. Reward for performance begets students choosing only winning situations
without the question of losing, thereby eliminating challenges. Dweck (Hopkins,
2005) makes clear that studies with geniuses show them exhibiting tremendous,
sustained effort. That is what makes them geniuses. The greatest knowledge students
may seize is the importance of continual learning, facing challenges, and confronting
obstacles.
By feeling successful as learners, students are able to set goals. It is by setting
goals that students become metacognitive concerning their learning (Winstead, 2004).
“Success breeds success” is an old axiom, stated again by Lisa Winstead, a researcher
and professor at the University of the Pacific. She writes that academic success is a
prerequisite to goal setting. Also, for students to succeed at goals, they must realize
that errors are part of the learning process. When students understand errors are a
natural, inevitable step in learning, students have less tendency to give up. Mastery
learning supports students’ self efficacy and lessens negative thoughts, thereby
allowing more achievement (Linnenbrink & Pintrich, 2002).
William Glasser’s (1997) position on choice theory states that students’ need
for belonging, power, freedom and fun must be met in the classroom. If a teacher uses
extrinsic motivation, for example in the form of candy or bonus points on summative
tests, this is destructive to caring relationships. Extrinsic motivation is coercive
(Glasser). The ultimate goal of schools is for the child to be motivated to choose
education as a means of satisfying the four needs, thus intrinsically motivating. The
22
student learns, because their goal is to satisfy the need for power, freedom, fun, and
belonging and the learning environment satisfies these goals.
Extrinsic Motivation
Extrinsic motivation is the desire to learn so the student can achieve something
else, or as a means to an end (Linnenbrink & Pintrich, 2002). For middle school
students, extrinsic motivation is not positively tied to performance (Pintrich, 1999).
Engagement in this type of motivation, leads to cognitive thought on other matters
besides the task (Linnenbrink & Pintrich). Examples of extrinsic motivation as
described in the literature are the need to “be better than” another student, or for the
need to fit in “the higher level” math group, or to achieve high performance on
assessments (Linnenbrink & Pintrich; Pintrich). Students using relative ability
orientation have a desire to get a good grade for their parents, or to work hard for the
teacher. It appears this type of motivation, extrinsic, is not long lived for students in
middle school. Yet for college level students, extrinsic motivation and relative ability
orientation is positively tied. Pintrich’s writing regarding this difference between
middle school and college students is that the choice college students have regarding
classes, effort, and time may account for the positive correlation between extrinsic
motivation and performance. Pintrich also postulates the college student may
understand the need for good grades in certain classes will allow her to get where she
wants to be in another area.
The United States Department of Education’s practice guide Reducing
Behavior Problems in the Elementary School Classroom (Epstein, Atkins, Cullinan,
23
Kutash, & Weaver, 2008) negates Glasser’s (1997) theory, citing research stating
positive rewards encourage appropriate behavior that leads to engagement, student
success, and ultimately positive student academic perception. The stated manner for
teachers’ use of positive rewards is to employ small rewards frequently, close to the
student’s use of appropriate behavior, using rewards that make the student feel good,
and then to gradually reduce and eliminate rewards. This use of extrinsic motivation
appears to apply a tactic of behavior modification that may additionally provide
modeling for the non motivated students. This practice guide from the United States
Department of Education (Epstein et al., p. 67) acknowledges the controversy with
extrinsic motivation. Therefore researchers for the United States Department of
Education examined 128 studies, concluding that extrinsic motivation, or external
reinforcement, enhanced student interest and time on task if used in appropriate
manners as stated above.
Integral to student motivation and the school experience is teacher behavior.
How a teacher motivates, how a teacher rewards behavior and what a teacher says can
have a profound effect on a child’s view of learning. Michael Fullan and Andy
Hargreaves (1996) write in the book What’s Worth Fighting for in Your School? that
teachers make over 100 decisions, on the spur of the moment, in every given school
day.
Teacher Behavior
The foundation of public education rests upon the student recognition of a
caring teacher (Noddings, 2005). The responsibility to establish and maintain this
24
personal, caring relationship with children lies with the teacher. In addition, the
professional teacher makes sure all students learn in an effective manner, according to
the needs and attitudes of the child (Vannest, Temple-Harvey, & Mason, 2008). It is
well documented in the literature, that positive climate and student involvement is
essential to effective learning (Epstein et al., 2008; Shipero,1993).
Student Involvement
Studying student involvement in mathematics, Julianne Turner et al. (1998)
define involvement as a “…psychological state that is concerned with the quality of
experience during learning” (p. 731). This state of mind is different than engagement
and interest, sitting between them on the continuum line of experiences. Engagement
is “doing” and interest is being intrinsically motivated “to do”. Involvement requires
deep concentration and an in-depth understanding of the task and goal at hand. Turner
et al. define characteristics of student involvement as student concentration is pointed,
time passes quickly, students are emotionally invested, and a persistence to continue
with the task is evident. Student perception of involvement incorporates challenges
and student skills that are both high and balanced (Turner et al.).
Turner et al. (1998) observed seven classroom teachers’ instructional practices
in a total of 34 different class sessions. They also studied classroom processes related
to motivation. The researchers observed and audio taped class sessions, categorizing
the class’ discussion. They asked 42 fifth and sixth grade students, an equal number of
boys and girls, representing high to low mathematical abilities to generalize the class
25
experience using a 13 statement questionnaire with a Likert scale. Statements regarded
involvement, challenge, and skill level (Turner et al.).
Findings from this study showed that teachers in high involvement classrooms
monitored and supported students’ skills, persistence, interest, and emotional stamina.
Additionally, teachers upholding high involvement in classrooms gave students
control by praising autonomy. The third noticeable difference between high and low
involvement classrooms in this study was the enthusiasm for mathematics among
teachers. Students in three of the seven classrooms were designated high involvement
students by the tendency to work fruitfully and positively. Using motivational and
emotional supports, teachers enable students through confidence building by the use of
scaffolding instruction (Turner et al., 1998).
Turner et al. (1998) defines scaffolding as assisted instruction. Scaffolding
instruction in the literature is also called the Zone of Proximal Development
formulated by Vygotsky. Stephen Krashen calls scaffolding instruction i+1 (Winstead,
2004). Teachers with high student involvement used scaffolding to maintain student
focus and concentration. Teachers used feedback regarding goals and adjusted the
student assignment level, which allowed new understanding for the student.
Scaffolding instruction utilized diagnostic and formative assessment allowing the
student to begin instruction at a comfortable academic level. Thus they provided the
student a level of success and challenge (Linnenbrink & Pintrich, 2002; Turner et
al.,1998). As the students progressed through the curriculum, teachers using
scaffolding asked students to verbalize or write their conceptual understanding. In
26
addition, teachers also allowed students to use their own strategies for computation
and exploration. This autonomy allowed for the transfer of learning to the student
(Turner et al.). Assessment was used to ensure small bits of context had been learned
by the student. If the formative assessment showed the student was unsuccessful, the
teacher intervened and supported the student. The student felt positive and was
motivated to continue (Winstead). High involvement classrooms in the Turner study
had teachers that discussed the exploration of learning strategies and self evaluation.
Scaffolding supported students cognitively as they demonstrated understanding.
Turner et al (1998) identified and discussed the importance of teacher
behaviors in high and low involvement classrooms, but did not delve into the
measurement of student performance as it related to high or low involvement
classrooms. Neither did the researchers address the differing skill level of the 42
students answering the questionnaire. A question for further research is how did the
students of differing skill level perform on summative tests, and did the lower
performing students view their involvement in the classroom in the same way as their
peers.
Learning Environment
Van Grinsven and Tillema (2006) report in the article, “Learning Opportunities
to Support Student Self –Regulation: Comparing Different Instructional Formats,” on
the need for teachers to provide environments for students to learn and participate in
organizing their own learning and knowledge acquisition. This study in vocational
programs, involving 623 sixteen to eighteen year old students, categorized factors of
27
different types of learning environments. Students then reported on motivation and use
of self regulation strategies through responses to questionnaires including Pintrich &
De Groot’s (1990) Motivated Strategies for Learning Questionnaire (MSLQ). The
central question of the research was to determine the best learning environment to
enhance student motivation and self regulated learning strategies.
Van Grinsven and Tillema (2006) set up five learning environments
specifically
1. the traditional education mode for whole class instruction of teacher chosen
subjects with little or no student choice, or attention to learning strategies,
and graded on product only,
2. an open learning center, where students worked independently on teacher
chosen subjects with a high degree of autonomy but no attention to
learning strategies, graded on product and process,
3. thematic, independent group work, with high student autonomy and again,
no attention to learning strategies, graded on product and process,
4. thematic, project oriented learning in small groups of students, with no or
little choice, yet high attention to learning strategies, graded on product and
process, and lastly
5. a full project-based learning (PBL) program with small groups of students
working thematically with a high degree of autonomy , with explicit
attention to learning strategies, graded on product and process.
28
Van Grinsven and Tillema (2006) found student motivation had the strongest
influence and largest impact on use of self regulated learning strategies. Motivation
was greater in environments where autonomy rather than teacher control was the
norm. Another finding of the study regarded student perception of the learning
environment. If students perceived the environment as promoting self regulated
learning, the data indicated students participated and put more effort into learning.
Overall, this study found traditional instructional environments to be in need of
reform for students in vocational education and about to enter the workforce (Van
Grinsven & Tillema, 2006). It also pointed out students felt most supported by
teachers in an environment where learning strategies were taught. The difficulties
arose with the five different environments when assessing individual effort in a group
project and in the design of tasks for self regulated learning (Van Grinsven &
Tillema).
Questions that remain after synthesizing this study involve the lack of data
analyzed for gender, and the lack of information on the type of subjects and themes
offered in the differing vocational education environments. The researchers also did
not indicate the performance level of students in the five differing environments,
although they did indicate it was problematic to assess group projects per individual.
No Child Left Behind mandates researched teaching methods be used in
classrooms yet little research has been done in mathematics classrooms with at risk
students. Tom Lester (2009), a consultant for “Math Matters”, and mathematics
teacher has developed a program with promise for struggling students. Through
29
positive reinforcement of behavior and answers, deep respect for students, clear
communication of expectations and mode of response, carefully crafted specific
questioning, and cognitive use of “wait time”, teachers were able to involve students
in conceptual understanding and intellectual conversations with peers in a positive
climate.
Reaching the students who have self efficacy, parental support, are organized
and goal motivated are the easy to teach. What about the students that are struggling
with parents, finances, peers, and homework? At risk students need to be in
classrooms where teachers use classroom management techniques that increase
student involvement, improve student motivation, and focus attention to rigorous
mathematics achievement.
Learning Strategies
“Self regulated learning is neither easy nor automatic,” Paul R. Pintrich (1999)
claims (p. 467). Self regulated learning involves more time, more effort, and more
engagement and importantly, must be promoted. Self regulated learning utilizes
strategies to regulate cognition and metacognition (Pintrich; Weinstein et al., 1989).
Self regulated learning strategies are grouped into cognitive strategies, self regulation
strategies, and use of resource strategies (Pintrich).
Undoubtedly, student motivation and student self regulation are entwined
(Pintrich & deGroot, 1990). Tanner and Jones (2003) found a strong correlation
between students’ self efficacy scores and their belief in self regulated learning
strategies. In a study of 47 middle school mathematic students, statements regarding
30
self-efficacy, metacognition, and self regulated learning strategies were investigated
regarding student attitudes towards beliefs as learners of mathematics. Even though
the students believed, for the most part, that self regulated learning would be helpful,
few applied the learning strategies. Furthermore, with this group of students, data
revealed that student knowledge of their personal strengths and weaknesses was not
evident. Nor was, in 50% of the students, knowledge of effective learning strategies.
Also, a large minority expressed beliefs that mathematic knowledge was not in their
control. In this large minority, students believed mathematical knowledge was fixed.
The necessity to explicitly teach self regulation learning strategies is evidenced by this
study (Tanner & Jones).
Many educators believe learning strategies are acquired as students progress
through school. But unless students are explicitly taught learning skills, they may not
incorporate them into regular use (Weinstein et al., 1989). To suppose that students
innately discover learning strategies is in error (Weinstein et al.). Modeling and
meaningful practice must be incorporated into the curriculum.
Cognitive Strategies
In the Trends in International Mathematics and Science Study 2003 (TIMSS)
researchers, Daniel House and James Telese (2008) acquired student responses to
statements regarding learning strategies and measurement of perceptions of self
aptitude and belief. Responses to these statements were analyzed with the students’
answers on the TIMSS International Mathematics assessment. From this analysis,
House and Telese found more active learning was conducive to higher performance on
31
assessments. For example affirmative responses to the statements, “We work on
problems on our own,” “We interpret data in tables, charts, or graphs,” “We explain
our answers,” “We decide on our own procedures for solving complex problems”
(House & Telese, 2008, p. 104) resulted in a tendency for higher performance test
scores in algebra.
Contrary to earlier studies, students responding that they engaged frequently in
cooperative learning produced lower performance scores than did students working
problems out on their own (House & Telese, 2008), pointing to further research
needed in the area of cooperative learning and mathematics. In addition, House and
Telese described a need for reflection after active learning sessions, as well as hands
on projects, and student self management to improve positive attitudes towards
mathematics.
Richard Lesh and Richard Lehrer (2006) make a point on cognitive strategies
for learning mathematics that
Models are a type of knowledge that is useful for developing, describing,
explaining, predicting, and controlling complex systems; and “survival of the
useful” is the main criteria which determines the acceptance or rejection of
models (as well as the underlying conceptual systems that they embody). (p.
19)
To be accepted, models must be powerful, reusable, and sharable with others (Lesh &
Lehrer). Lesh and Lehrer explain that the highest demanded people in the work world
can make sense of complex systems, communicate in teams, adapt quickly and readily,
32
work with many layers of people, and develop and share tools. Expression is the most
sought after skill, not computation. Lesh and Lehrer believe mathematics teachers and
students should spend time investigating, revising, inventing and developing models.
Barry Zimmerman’s research (1996) involves observations of social learning.
He has found social models powerful in knowledge acquisition strategies.
Interestingly, his research has found self regulation to be developed over four phases.
First is observation, then imitation, next is practice of self control, and finally adapting
self regulatory skills according to one’s needs. Self regulation begins socially, then
moves to the individual. Zimmerman and Manuel Martinez-Pons’(1988) studies have
sought information on how students achieve high academic success. Findings indicate
students achieving high academic success approach learning in a strategic fashion,
they self monitor their learning (self regulation), and sustain learning through use of
self efficacy (Zimmerman & Martinez-Pons, 1988).
Self Regulation
Zimmerman and Martinez-Pons (1988) developed 14 categories of self
regulatory processes. Using responses to a questionnaire, the researchers were able to
accurately identify high achieving academic students from regular achieving students.
The process of self regulation increases students’ academic motivation and self
efficacy (Zimmerman,1996). Intervention in self regulation can enhance perceptions
and decision making. When students recognize these enhancements, students may
accept responsibility for their learning. Self regulation of learning strategies promises
33
to improve the academic performance of many youth, yet it is still a hidden factor in
education (Zimmerman, 1996).
Use of Resources
In a study to provide insight into the needs of students with learning
disabilities in algebra, researchers Larry Kortering, Laurie deBettencourt, and Pat
Braziel (2005) found that students enjoyed working with their peers and asked for
more support and assistance from caring teachers. Peer Assisted Learning (PALS) is a
learning strategy used with high success in language arts and reading classrooms, and
in 2nd through 6th grade mathematics instruction. Students are paired with other
students of differing academic ability and coached in providing response, inquisition,
and reward. At the high school level, intervention strategies that show promise
include work at problem solving that has value after high school (Kortering,
deBettencourt, & Braziel).
Vannest et al. (2008) searched for articles on mathematics intervention for
students with behavioral and emotional disabilities and found most would not meet
NCLB’s standard of scientifically based research. They did identify learning strategies
that showed promise of academic performance improvement. These three strategies
include visual and manipulative organizers, various academic strategies such as “Say
it before you do it,” and permanent model, and lastly, musical mnemonic technique.
Summary of Literature Review
All students deserve to have a quality education that meets their needs.
Education in mathematics achievement for low performing students is an area with
34
scant research and attention. Few researched based studies are available to teachers
regarding mathematic intervention programs. Strategies that have been employed in
classrooms for students who are behind academically center around more time-on-task
(Balfanz et al., 2004) or remediation classes in mathematics are limited to narrow test
preparation and life skills. Views on learning mathematics are wide ranging, from a set
of procedures that must be memorized, to a conceptual view of integrating algebra and
geometry to make sense of the world (Balfanz et al., 2004). What is needed is an early
identification system using diagnostic and formative assessments, and required
intervention using scaffolding instruction (Balfanz et al., 2004). Self efficacy beliefs in
students must be regularly scrutinized by teachers to keep students from becoming
discouraged. All students must be enrolled in classes with rigorous instructional
programs that sanction the self efficacy of students. It is imperative that low
performing students are taught explicit learning strategies and have caring, supportive
teachers, with high expectations for behavior and achievement.
Synthesis of studies indicates a significant need for more research in the area
of mathematical achievement. Attention to student motivation, teacher behavior, and
explicit teaching of learning strategies needs to occur in a timely manner for all
students.
In addition to school reform, students as partners in identifying their specific
learning needs may impede disenfranchisement from the educational realm.
Linnenbrink and Pintrich (2002) tells us the student’s active regulation of motivation,
learning strategies and connected behavior in the academic content plays a big role in
35
achievement. By engaging students in the work of monitoring their academic
achievement, a feeling of empowerment, autonomy, and intrinsic interest may come
about.
Passage of performance assessments in algebra is required for a high school
diploma and passage of higher level mathematics classes is required for higher paying
jobs. Positive self efficacy in mathematics is attainable for all students. Let us do as
the law states, and leave no child behind.
36
Chapter 3
METHODOLOGY
Introduction
The purpose of this study was to determine factors in students’ lives that
perpetrated their continued failure in passing algebra. The study incorporated a
qualitative approach with questionnaires, interviews, examination of cumulative
school history folders, and observations. These methods provided a student
perspective of the factors that may have affected their mathematical achievement
(Merriam, 2001). Issues of self efficacy and motivation, teacher behavior, and learning
strategies are components in a successful academic program and were examined in
this study. Other social strains may have applied undue pressure on the student and
their success in algebra and were examined with this study. Students responded to
questions regarding their history in algebra classes. These responses were analyzed
based on vocabulary and intent, in order to perceive a picture of the students’ hurdles
in the algebra classroom (Merriam, 2001).
Questionnaires were given to students regarding their knowledge and use of
learning strategies, parent and peer support, and perception of teacher behavior.
Observations of student self efficacy and motivation were made by the algebra
teacher/ researcher. Student cumulative school histories were scrutinized. “This
interpretive approach stresses that it is the subjective experience of the individual that
37
is important and that it is individual perception that bestows meaning, rather than there
being any external objective meaning, “ states Middlewood, Coleman, and Lumby
(1999, p. 10).
Research Questions
Questions that plagued the researcher were: Why do 95% of sophomores, 50%
of juniors and 25% of seniors in this independent high school still need algebra credit?
What factors disabled students from passing algebra? What academic, social, and
emotional needs did the students have at the time of their initial attempt in algebra
class? What student needs for learning algebra were not being addressed?
Research Design and Data Collection
To answer the above questions, qualitative approaches were used in the study
(Bogdan & Biklen, 1998; Lyons & LaBoskey, 2002; Merriam, 2001; Middlewood et
al., 1999. Student stories, observation of the student learning process, interviews, and
combing student cumulative school history folders provided the researcher with each
student’s description of their mathematical learning. Categorizing the data in three
areas, motivation, teacher behavior, and learning strategies, helped to understand the
reasons behind students’ unsuccessful venture in previous algebra classes.
Because the participants learned algebra from both peer and teacher instruction
during this study, the teacher/researcher was a participant observer (Spradley, 1980).
Notes of student behavior and student comments were taken while in their current
algebra class, as well as after class, during less formal meetings. Interviews with
students were audio taped, then transcribed and categorized by the observer. Students
38
were interviewed using open-ended questions relating to motivation, teacher behavior,
learning strategies, and parent and peer support. The observation process was cyclical
(Spradley, 1980) and continued from August, 2008 to May, 2009.
From the student’s perspective, evaluation of their earlier algebra classes
regarding a positive classroom climate relating to behavior control, scaffolding of
instruction, student autonomy and involvement was examined. Student perception of
teacher support and caring was analyzed and documented. Learning strategies
explicitly taught by previous algebra teachers and used by the students were observed
by the researcher. In addition, student perceptions regarding learning strategies were
gathered from a questionnaire. Cognitive strategies, self regulation strategies, and use
of resources were observed. This study took into account algebra instruction from the
consumer’s point of view (Kortering et al., 2005).
The interview was voice recorded and transcribed with student understanding
that all names would be changed to protect the students’ identity. The students
preferred talking rather than writing when answering questions for this study. The
interview questions asked the students to relate their experiences in their first algebra
classes. Students were asked to assess whether the teacher behavior suited their
learning needs, and if it seemed the teacher cared about their academic success.
Engagement is an important component of academic achievement and students talked
about their involvement in the algebra classroom. Family support and peer
relationships were assessed by the students as being helpful or not. The observation
notes obtained by the teacher/researcher were grouped into three areas, motivation,
39
teacher behavior, and learning strategies to attempt to reduce the multiple variables
into a coherent scenario.
Participants
The participants in this study were two males and six females enrolled in an
independent study, alternative education high school in the northern California
foothills. The students were 12th graders. None of them had finished the 10 credit
algebra requirement, but some students had earned a few credits. Six students were
white, and two students had one Hispanic parent and one White parent. One
participant dropped out of high school on her 18th birthday during the second semester.
A second female participant stopped attending school in the second semester, but
attended an occupational training program through the county. These eight students
were classified as being at extremely high risk of dropping out of high school based on
credit accumulation and their lack of progression during the previous year at this
independent study. Following is a table of participating student data.
40
Table 2
Student Information
Student
male or
female
Parent
Education/
divorce
Socioeconomic
Ethnicity
Reading
grade
level
Kyle M
Graduate
level/
divorced
Graduate
level/
divorced
Vocational/
divorced
Middle
White
Middle
High
school/
neither
parent
works
(medical)
College/
divorced
Vocational/
divorced
High
school/
alcoholism
& violence
High
school/
divorced
Emily
F
John
M
Jocelyn
F
Elaine
F
Sharon
F
Amy
F
Tabatha
F
Designated
in need of
special
education
in high
school
Transportation
issues to
participate in 4
times per week
direct
instruction in
algebra
12+
Credits
needed
in
algebra/
(total
credits
needed
should
be 30)
9.5/ 40
No
White
12+
10/42
No
Low
Hispanic/
white
12+
10/50
No
Middle
Hispanic/
white
7
10/104/
No
Rode public
transportation
6:30-3:30
Walked to
friends, waited
to catch a ride
Walked to
public
transportation
None
Middle
White
12+
5/95
No
None
Low
White
12+
5/50
No
None
Low
White
12+
7/40
No
None
Low
White
7
5/30
No
None
All students in the classroom agreed to participate in this research regarding
their past experiences in algebra. They knew they were to be part of a study for the
teacher’s Masters project. Discussion frequently revolved around learning strategies,
group work, multiple ways to solve problems, motivation, and teacher behavior.
41
Two of the participants lived with both parents; the others were in single parent
or blended families. All of the students were presently in the regular education
program. Six students were proficient readers, performing at the 12th grade reading
level as indicated on the Star Diagnostic Reading Assessment. Two students scored in
the 7th grade level on this same diagnostic reading assessment. None of the eight
students had an identified reading disability. None of the students were designated in
need of special education during high school, although two students had previous
experiences in special education. Three students had transportation issues and had to
leave their homes at 6:30am, transferring on busses, or catching a ride at 7:30 with
friends that may or may not be going to school. Two students were alone in the
mornings at their homes and agreed to get themselves to school for the algebra class.
Three students had mothers or a grandparent at home in the morning to support them.
This information is included to help the reader understand the self motivation required
to attend a non standard, not required, four day a week algebra class at an independent
study high school.
Setting
The setting for this study was a newly planned classroom in an independent
study high school of 150 students. The classroom was designed for a small group, and
the class was implemented specifically for 12th grade students not yet completing the
ten credit algebra requirement for high school graduation. This was the first attempt
for the school to offer daily, rather than the traditional weekly classes. The participants
agreed to come to school four mornings per week for the first semester to receive and
42
participate in small group instruction in algebra. Independent study students do not
typically meet four days per week, and transportation is an issue.
During the one and a half hours of algebra instruction per day, the teacher first
began working with student perception of ability; dispelling the myth that intelligence
is fixed. Confidence building in mathematical ability was the top priority. Class started
with mental math problems within the students’ skill level. The classroom climate was
relaxed. Students brought their breakfast, drank tea; one student brought her lap dog to
class each day. The class celebrated when they finished algebra chapters and when
they all understood a difficult concept; celebrations occurred frequently. The students
participated in team building activities together. The class had momentum because the
students knew this was their ticket to a diploma and after 12 years in education, all
eight students wanted a diploma. The small class allowed for student autonomy and
demanded student interaction. Encouragement by the teacher and students for peer
tutoring was accepted by all.
The psychological setting for this algebra class was that all students would be
involved in learning algebra in a positive classroom with high regard for fellow
students and the teacher. It was verbalized and reinforced that no student would be left
behind. Expectation of the teacher and students was that all students would finish ten
credits in algebra before the first semester finished and that all students would
continue with additional required credit accumulation and graduate from high school.
Previously, the algebra study in the students’ 11th grade year at this
independent study high school was to be done at home via an algebra text, with the
43
student showing learning on summative assessments during the designated weekly
appointment with the teacher. This independent text book model of instruction was
unsuccessful for the students in this study. One student had tried the computer as an
instructional mode, using Plato. He was unsuccessful in passing the algebra course
using this method.
The students and teacher participating in this study were also participating in a
federal grant, titled Rigorous Instruction in Mathematics Strategies facilitated through
the local county office of education. This grant was to study the effect of rigorous
mathematics preparation by the teacher and a teacher delivery system that promoted
self efficacy, student autonomy, positive teacher behavior, and the use of self
regulation. The guidelines for classroom management were from Math Matters, a
program for staff development in teaching mathematics. Applied coaching,
observation and intensive instruction in teaching mathematics was provided to the
teacher.
The independent study high school was located in a rural community of 40,000
people and one of six schools in the high school district. Total enrollment reached
approximately 160 students in the spring, with 50% of students enrolling in this school
having not completed their algebra credit.
Procedure
Prior to questioning students regarding their commitment to learn algebra,
letters of consent (see Appendix A) were distributed to students and parents. During
the daily algebra class, students were given the opportunity to gain confidence in
44
mathematical ability with the scaffolding of curriculum. Concurrently, relationships
were stressed and several team building activities ensued. Mathematical rigor within
the students’ ability was expected and accepted by the students.
One aspect of the class that was different from most classes was the
understanding that students and teacher agreed not to leave any student behind. This
translated to reteaching many topics. If a student was absent, the other students and the
teacher would tutor the student what they had missed prior to going on with the new
lesson. These instructional practices may seem common to the reader, but the eight
students had varying school attendance problems, motivation issues, low self efficacy,
family support issues, and repeated negative experiences with school staff. They also
had a large variation in mathematical knowledge. The reteaching had two positive
results, it directly taught students what they had missed and it reviewed and reinforced
curriculum to the students who were in attendance. It also built relationships.
The new text was set up sequentially with a few practice questions at the
beginning of each section showing the skills needed to be successful in the new unit.
Formative assessments were scheduled in the text frequently. These formative
assessments were useful in identifying content knowledge for each of the students.
The class did not progress forward until all students passed the formative assessments.
Students who understood the material taught students needing additional help. The
summative assessments at the end of each chapter were done as a class, a community
test. The teacher/researcher required that all students were to have the same answers
on each test item. This gave the teacher/researcher the information on persistence,
45
cooperative effort, and insight into individual student needs. Students were
encouraged to find and explain to the whole class their own way of construing
mathematical problems thus giving students autonomy, involvement, mathematical
vocabulary, self regulating behavior, and goal attainment. Completions of algebra
problems were also done openly, with multiple ways of working problems shown.
Students were encouraged to demonstrate methods that worked well for them in the
past to fellow students and teacher by posting information on the walls for others to
see and follow in future sessions.
Student data was collected in multiple ways. Observation notes were taken
throughout the semester by the teacher. Written questionnaires where completed by
the students. Face to face interviews using a voice recorder to document comments
were used so the students could document their perceptions without writing. The
teacher-student interviews were conducted after students had successfully earned most
of their algebra credits.
Cumulative folders of students’ school history were carefully examined by the
teacher after the school year had ended. Information in each folder was put in
chronological order, then information was entered on a matrix to facilitate
understanding. Gaps in student’s school histories were evident in several folders.
Summary
There are multiple possible causes for student’s failure in previous algebra
classes. This study sought an exploratory and interpretive view (Leedy & Ormrod,
2004). The students participating in the study did so voluntarily, without risk. They,
46
also, were interested in finding an explanation for their non success in previous
algebra courses.
The eight students were 12th graders, embarking upon their last year of high
school. The students were enrolled in an independent study high school, through the
public school system. Additionally, these students had been enrolled in this
independent study high school in the prior year of education for varying lengths of
time. They had entered their 12th year of school without earning the total ten credits of
algebra, an 8th grade class, required by the public school district for graduation.
The eight students enrolled in a daily algebra class designed to be conducted in
a positive, confidence building atmosphere. The students discussed reasons why
success had eluded them in prior mathematics classrooms, and responded to interview
questions, written questionnaires, and examined feelings retrospectively to try and pin
point what would help other students in the future to facilitate learning algebra. The
researcher collected comments spoken and written by students and categorized them
into groups to indentify reasons for previous failure in learning algebra. Additionally,
students’ cumulative school history folders were combed by the teacher/researcher for
information regarding student’s ability to learn algebra.
47
Chapter 4
RESULTS AND ANALYSIS OF THE DATA
Introduction
Algebra is an 8th grade subject, yet 25% of the 12th grade students in this
independent study high school had still not completed the course. Eight of these 12th
grade students agreed to attend algebra class four days a week, to facilitate earning the
needed algebra credit. It is characteristic of independent study students to meet with a
teacher/advisor only once per week, so meeting four times a week was not the typical
arrangement for independent study.
For 12 years these students had been enrolled in public education, yet now at
the end of the journey, were faced with an ominous task for the coveted high school
diploma. They must earn the required credits in algebra. As the year started it became
evident the students wanted to learn algebra; they showed up and were eager and
ready. Thus questions arose. Why had they not passed the algebra class earlier? What
got in the way? Was it student motivation? Teacher behavior? Did students know how
to learn math? To try and answer these questions, the teacher/researcher observed,
surveyed, interviewed, studied student’s cumulative school histories, and taught these
students algebra from August to June. The teacher/researcher participated in class
conversations, queried student learning needs, and facilitated the acquisition of algebra
knowledge.
48
Results and Analysis of the Data
Understanding why students fail at learning algebra is a complex subject. This
research attempts to translate the multiple realities of several students’ school lives as
it relates to learning mathematics. As Middlewood, Coleman, and Lumby (1999),
explain “People act on the basis of the sense that they individually make of a situation,
rather than acting directly in response to external stimuli” (p. 11).
Based on a review of the literature stating that student self efficacy, teacher
behavior, and use of learning strategies are essential elements of student success, the
student comments and teacher observations were grouped accordingly. Not all
students responded to data questions. Student attendance throughout the year was
sporadic. Students dealt with life pressures as well as academic pressure to graduate
from high school. The survey was presented and completed during algebra class time,
in early November, 2008. Observations were made during the algebra class and at any
other time contact was made with students and family. For example, the
teacher/researcher made many calls home. Also, students may not have attended the
formal algebra class on a certain day, but may have come to school after class was
over. Interviews took place when the students were available and willing after the first
semester. After January of 2009, students came in frequently during the week for
individual algebra tutoring, but no longer met in a class setting. The four methods used
in collecting data are presented in the chronological order of collection during the
year.
49
Responses to Journal Questions
The journal questions sought student perceptions regarding self efficacy,
teacher behavior, and learning strategies and environment in their previous algebra
classes. The journal is shown as Appendix A. In response to journal questions
regarding perceived motivation, students used negative terms “Gave up”, “Didn’t
care”, “Felt like an outcast”. Student perception of teacher behavior was also mostly
negative “She could not control her students.”, “She was mean.” “Classroom was
overfilled with disrespectful, ignorant, students that had no desire to learn.” This
negative perception was expected as the students had failed their algebra classes. One
student did not blame the teacher or the other students, but took the blame wholly on
himself. “I didn’t care.” And “I didn’t want to do the work so I didn’t.” Use of
learning strategies included their perceived maturity level and family support. One
student felt his family blamed him for not doing the work, another student’s mother
confronted the teacher, and then the student said the mother “Gave up.” The intent was
to understand how students experienced math class. Following is a table with journal
questions and student responses.
50
Table 3
Student Perception of Previous Algebra Class
Grade
student
first took
algebra
Self efficacy
(fractions?
compared to
others?)
Teacher behavior
(room setting?
classroom control?
teaching strategies?)
Learning strategies
(maturity level?
family support?)
Kyle
7th grade
algebra
class
“I was not
committed to
learning.” ”I did
excellent with
fractions but failed
because of not
showing my work.”
”I tried to learn at
first, then gave up.”
“I never got it
before.” “too much
homework …gave
up didn’t care
anymore.”
“There were 36
students, no help to
slow students.” “She
could not control her
students.”
“I was very mature
compared to my
classmates.” My
family “blamed me
for not doing my
work.”
“She was mean,
didn’t help us if we
didn’t get it. She
kept going on so we
got left behind and
failed.” “Big class,
no classroom
control”
John, 9th
grade
algebra
class
“I didn’t go that
often and quit.” “I
didn’t want to do
work so I didn’t.” “I
did ok with
fractions.”
“The teacher had
classroom control,
sometimes we were
in rows or
sometimes groups.”
Elaine,
8th grade
algebra
class
“Hard time with
fractions, no one
took the time to
explain them to
me.” “…felt very
uneducated.”
“I felt like an out
cast at the school,
so in all of my
classes but art I
struggled.”
“The class was
overfilled with
noisy, disrespectful,
ignorant, students
that had no desire to
learn.” “He had zero
control over the
class, he allowed the
students to harass
him…”
“Mom confronted
the teacher, teacher
didn’t care, mom
gave up, mom did as
much as possible.”
”Teacher gave us
questionnaires about
learning styles but
then didn’t change
anything!”
“My mom tried
really hard to get me
to pass but I didn’t
care enough” “My
mom yelled at me,
even sat in my
classroom with me,
but nothing helped.”
”Then, no body
believed that I was
so miserable, now
my dad realized that
I have been
struggling and I’m
growing and
learning.”
Emily,
8th grade
algebra
class
Learning
strategies
students
choose as
helpful
Using short
cuts, short
and more
frequent
tests.
Working
together in
a small
class, being
able to talk
in class
Group
work
51
Observations of Students
Observation by the teacher/researcher occurred usually during algebra class.
The students started algebra class on time using mental math problems. They then
worked two previous day’s problems on the white board to refresh or remind ourselves
how far we had come. Next on the agenda was the current day’s lesson. Through out
the algebra class, students tutored each other. About five students were present on any
given day due to absences and tardiness, therefore continual reteaching was necessary
and accepted. The teacher fixed the students cocoa, one student brought her lap dog
every day to class. Another student baked birthday cakes and we celebrated
frequently. Initially the time frame of the algebra class was August to December, but
in January, students were two chapters from finishing. These were the most difficult
chapters. The class size in January was usually at three students because one student
dropped out and two attended morning vocational classes. Attendance was poor. In the
end, students acquired the last 1 ½ credits one on one with the teacher. Through out
the school year, the teacher/researcher took notes of observed student behavior. Those
notes follow.
52
Table 4
Notes from Observation 2008/2009
8/2008
to
5/2009
Kyle
Self efficacy
Teacher behavior
Unrealistic
high sense of
skill level,
could
compromise
with teacher,
enjoyed the
computer.
Rarely
completed
work
assigned but
proved he
could
understand
concept.
Teacher/researcher
allowed student to
become student
teacher, teacher
did not call
attention to
deficits but
supplied
clarification and
computation
corrections.
Emily
Low sense of
self efficacy,
somewhat of
a
perfectionist.
Pencil hold
abnormal,
very stubborn
and
aggressive
when pushed.
Motivated to
graduate.
Learning
strategies used
Student had
conceptual
knowledge but
had
computation
errors, worked
well with
others, liked
being unique
with strategies.
Student needed
the confidence
of being a
smart math
student,
therefore was
frequently
asked his
opinion and to
explain his
reasoning.
Student attitude
Talking it out,
helped keep class
working it out
moving, enjoyed
on board,
positive peer
additional time
outlook, would not until problem
allow one on one
understood,
help
less problems.
Student needed
to be class
leader, she was
instrumental in
prodding the
class forward.
Relied on
calculator.
Algebra credits
earned/Graduation
10/Graduated
10/Graduated
during summer
school
53
John
He knew he
could learn
algebra. He
was
methodical
and thorough.
Attendance
problem.
Often had
head down on
desk.
Appeared
depressed.
Very quiet
student, teacher
had to initiate all
communication
and tutoring,
teacher provided
positive
experiences
(picture in school
newspaper)
Jocelyn
Low sense of
skill level,
would come
to school if
her mom paid
her. Often
promised to
do school
work, rarely
did.
Developed
understanding
that she could
learn algebra,
tried when in
class.
Attendance
problem, very
social and
self centered.
Did academic
work at home
occasionally.
Frequent calls
home to keep
parent informed of
absentee would
help get Jackie to
school
Elaine
Teacher set firm
limits for rude
behavior, used
inclusion,
acceptance of
student point of
view, pairing with
others
Less problems,
all in class
work, small
class so student
could get
additional
attention,
encouragement,
multiple
opportunities.
Student had
few needs. He
would come to
class if teacher
called mother.
Worked
slowly,
encouragement,
explanation of
alternative
methods of
receiving
diploma
Working with
other students,
allowing
students to
teach, extra
time, belief in
no child would
be left behind.
Student needed
friendship,
therefore
frequently
attempted to
work with
others
10/Mother kicked
student out of
house for smoking
pot. He is poised
to graduate from
Adult School in
December 09,
needing 1 night
class.
10/Is poised to
graduate from
Adult School in
December 09,
needing 2 night
classes. Student
moved out of the
home in April,
returned home in
late May.
10/Is poised to
graduate from
Adult School in
Sept, by using
GED credits.
Student lived with
father, but father
was frequently out
of town.
54
Sharon
Amy
Good self
efficacy level,
when in class
completed
assignments
accurately.
Attendance
problem
Low
academic self
esteem,
perfectionist,
short
concentration
time, social,
self centered.
Often
promised to
do school
work, rarely
did.
Tabatha Extreme math
anxiety, no
confidence.
Short
attention span
Praise, inclusion,
identification of
Student writing
career goal,
on board and
pairing with others explanation to
other students,
Acceptance and
attention, student
allowed to bring
lap dog to class
every day
Allowed to
work as a
group,
participated
when ready,
positive set up,
success
oriented
Teacher identified
and explained
anxiety, worked
one on one
focus on
wrestling with
the math
problems, not
giving up,
reducing
anxiety
8.5/Dropped out
on 18th birthday.
Student was
straight forward in
her lack of desire
for a HS diploma.
Student explained
only reason mom
wanted her to
finish HS was for
the child support
money.
9/ Is poised to
graduate from
adult School in
Dec, needing 1
night class, but
researcher
doubtful student
will follow
through.
10/Graduated,
student passed the
California Exit
Exam in May
after 5 tries.
55
Student Interviews
Interviews were conducted after the algebra class had finished, in February.
Students were interviewed individually without peer interaction using an audio
recorder. The teacher/researcher guided the interview with these three specific
questions. Why do you think you did not pass your previous algebra classes? Was the
teacher and classroom helpful to you as a learner? What strategies would have helped
you learn? During the interviews students were encouraged to explain whatever came
to their mind regarding previous algebra classes. As reported below, the students told
their own stories.
Table 5
Results of Interviews, Quoted from Students
Kyle
Self efficacy
I have always been
strong in math, due to
the programming. I have
had to do quantum
physics, either its so
simple or I am too
distracted. In my first
algebra class I thought I
would have no
problems, but if I had to
write down all the steps,
its long, and I got pissed
off.
Teacher behavior
I was in special
ed. It was a small
class. It went
pretty well; when
I did the test I
knew the answer
to every question.
It was a friendly
classroom but we
had disagreements
between kids,
things didn’t work
with the students.
The teacher gave
a lot of
homework; it was
insane, like 200
problems a night.
Learning strategies
My parents knew I was
smart in math; they looked
at the Mensa test.
Technically I have an IQ
50 points above brilliant.
Its just the fact of my
laziness and the horrible
way they taught it. I have
always had trouble
focusing in here and there.
I want to take geometry
where the teacher teaches
with you not at you.
Students should be able to
take short cuts, let students
prove they know the steps.
56
John
Jocelyn
Elaine
I hate school. When I
was little and everyone
was raising their hand, I
just sat there quiet, (I’ve
had) years of practice. I
am normal in math, my
dad is good in math, my
mom is normal. I
probably got my ability
from my dad.
I didn’t do good in large
classes, not every kid
gets help from the
teacher. I started
kindergarten a year
early. I have always had
a hard time in school. I
would cry, get upset, my
parents couldn’t help
like the teacher. I would
talk myself out of doing
math, Oh this is way too
hard. I couldn’t do
homework by myself.
My mom tried
everything, she even
poured water on my
head to get me up to go
to school.
No one told me credits
were important. I
thought they would just
pass me from year to
year. I rejected school.
The environment got in
the way of learning. I
didn’t ask questions,
because I thought I
would have to work
harder, stay under the
radar, you know.
They all tried, I
just didn’t want to
do homework. I
wanted to play
and run around.
If I just could do work in
class, I would be fine. But
they all wanted me to do
extra stuff. My friends
weren’t good students
either. I was the worst one
when I was younger. I was
the instigator.
She liked to put
people on the
spot, I would
throw a fit and
walk out. Tutor
helped but that
didn’t start until
the end of the
school year. I
never saw a
counselor except
to help me choose
classes.
I must be shown and
talked to at the same time,
need to be shown and do
it. I got worse in high
school. The big
classrooms were too big. I
like one on one help.
When I went to
the private school,
there were only 8
students and the
teacher had time
to help me learn. I
earned 5 credits
there.
If a student needs help (a
teacher should) offer it. If
you offer enough times, a
kid will take that offer. Be
easy; give them time to
catch on. Math was not my
hardest class. My dad
didn’t have time to keep
track of me. He doesn’t
understand girl’s
emotions. He always said
things would get better.
They didn’t.
57
Tabatha I stopped doing math
because I couldn’t get it.
I hate math, too much
thinking. I don’t like
numbers. My mom
didn’t like math either, it
just wasn’t her subject.
When people get around
me I just rush or guess, I
can’t focus if I am not
one on one.
Its not that I can’t stick
with it, its more. I can do
math, I am not stupid, I
know how to do it.
“When I see a test, I
focus more on “I need to
do this.” And get all
mixed up. I think of
more, like, how can I get
this done in a hurry. If I
am the last one done, oh
that’s it. And I say, oh
crap this is going in the
grade book.
Snider Middle
School, I hate that
school, it was big.
We tried everything,
Sylvan Learning Center,
tutoring, Score -another
tutoring place, my friends
have helped me since 5th
grade. I can learn but it
doesn’t stick. When I get a
paper it just goes whew. I
say ‘I know this, and then
I look at it and say ok, A’
(make a guess).
I think I needed to learn
math continually, and how
to break it down. I like
working one on one on the
white board.
Cumulative School Histories
Students’ cumulative school histories contained varying amounts of
information. Two students’ cumulative folders did not contain the number of student
absences. Written comments from each grade level teacher were generally about
attitude and suggestions for attitude improvement. Most cumulative school histories
did not contain information regarding interventions in math, nor suggestions for
specific learning strategies. The histories did contain some parental support
information, such as divorce and communications with parents. Scores on state
58
standardized tests were always included, and the researcher compared these to class
calculated grades. Following is an orderly assimilation of data found.
Table 6
Analysis of Cumulative School History
K-12
school
history
Kyle
Emily
John
Student
attendance
and
behavior
Teacher
comments
positive or
negative
regarding
student)
Good
Positive
attendance, remarks, yet
strong self Sharing &
image.
getting along
Student
with others a
took
problem
medication
for ADD
in 5th - 7th.
Attendance Positive,
problem,
student rec’vd
felt
“o”s in
inferior in citizenship
math
until middle
school, yet 4th
“Bossy, Her
way only”.
Learning strategies:
state assessments and
grades
Other themes
Special Ed 5th grade,
autistic, Proficient in
ELA, average in math,
grades were all over the
place over the 12 years.
1. Parents
divorce in k
2. began
academic
decline in 9th
3. attended 7
schools
Special ed in 4th
processing disorder
writing, when student
changes schools in 8th ,
mother does not
disclose sp/ed need
1. Parents
divorce in 4th
2. multiple
PhysEd
failure
3. began
academic
decline in 4th
Attendance
poor,
Behavior
trouble
from k.
Student
took
medication
for ADD
5th - 7th.
Consistently scores
high on state
assessments, yet middle
school. teachers want
him retained
1. lack of effort,
energy in 3rd
2. meets father
in 3rd
3. Mother
kicked out at
18 years old
Early teachers
want him to
listen,
behavior
“Roller
Coaster”
59
Jocelyn
Elaine
Sharon
Amy
Attendance
problem,
Behavior
trouble
from k
Attendance
problem,
Behavior
problems
from k
Teachers have
difficulty with
Jackie’s
disrespect
Proficient in ELA, basic
in math, low grades in
math
1. A’s in PhyEd.
Disrespect, no
close contact
with teacher
Low scores especially
in math, yet “very
capable”
1. Parents
divorce early,
father custody
2. Suspended in
5th
3. attended 13
schools
Attendance
problem,
excellent
behavior
until
divorce,
then
behavior
problem
Attendance
problem
“Defiant” in
6th
And stolen
money in 11th
Proficient and advanced
scores in all grades on
standard asseessments,
yet teachers in middle
school want to retain,
low grades by teachers
in upper years because
no work was turned in
1. Heated
divorce in 4th
2. Weight
problem
3. continually
failed PE
Just below average,
grades are C’s then in
middle sch, drop lower
1. HS 5 years,
then dropped
out
Parents alcoholics
and violent
1. lived with
mom and
stepdad, until
10th
2. failed PE
3. focus became
boys
Teachers
complain of
too talkative
and drama
magnet
Tabatha Attendance Problems with
problems
talking and
until late
focusing
middle
school,
then good
Basic in ELA and
below basic in math,
low grades in math, C’s
in other
Summary
Students benefited in sharing thoughts and insight regarding the difficulty of
learning algebra. Discussing the issue appeared to help students see the situation more
clearly and led to higher cooperation during the algebra class. The students knew they
60
were part of a research project and the teacher/researcher believes this made them
more inclined to be self reflective. Discussing and identifying students’ math
difficulties also led to a feeling of hope for current success in the present algebra class.
Seniors’ (12th graders’) maturity level increased as they went through the algebra
class. This maturity could be due to normal adolescent development but the
opportunity to reflect and make a statement, and contribute to research may have been
a factor as well.
Based on attendance and social skills, the cumulative folders identified six of
the eight students as at risk in kindergarten. The two other students were identified in
the third grade, coinciding with their parents’ divorce. The students never recovered or
improved their standing.
The teacher found the inquiry process invaluable for relating to high risk
students. Understanding the background and perceptions of young adults for
motivating, encouraging, and facilitating academic growth can not be understated as a
tool.
61
Chapter 5
DISCUSSION, LIMITATIONS, RECOMMENDATIONS, AND CONCLUSIONS
Discussion
Interpreting “No Child Left Behind” means that every child will learn and
graduate from high school. Teachers, schools, and families must work together with
each individual student to ensure no child is left behind in mathematics. Identifying at
risk students as early as possible, formulating and following a plan for each student,
educating parents and student regarding the student’s deficits and strengths, are steps
in improving the educational process.
Social Problems
The eight students researched in this algebra study had difficulty relating with
their peers throughout their 12 years of school. Six of the students participating in the
algebra class were “drama queens,” very needy of personal attention. To address that
need, Kyle was given the role of student teacher and Amy was allowed to bring her lap
dog to class each day. Emily was frequently at the white board showing off her work.
Elaine was regularly peer tutored by classmates. Tabatha worked many hours
individually with the teacher with a few peers hanging around. Jocelyn spent time in
the class talking with peers, yet did not progress mathematically for lack of desire to
do so. All but this one student learned and progressed in algebra when allowed to gain
attention, and praise for their work (Davis, 2007).
Two of the eight students were extremely quiet, “under the radar.” During their
early years in school, the cumulative histories pictured these two students as out going,
62
friendly and leaders. Yet at middle school both became resigned to failure. For one
student, Sharon, this shift in personality appeared to be caused from her parents’
heated divorce. The other student, John, shifted his attitude, it appears, from constant
negative school authority issues (Linnenbrink & Pintrich, 2002). Interestingly, both
students scored Advanced, the highest category, in math on standardized tests, yet
teachers labeled them failures in math on report cards, wanting both students to repeat
grades in middle school. Consequently, during middle school years, both students quit
trying in school. Sharon explained that she attended high school just enough for her
mother to continue collecting child support from her father, and dropped out of high
school on her 18th birthday.
The other student avoiding attention, John, was kicked out of his mother’s
home before acquiring the last credit of algebra. He reappeared during the final two
weeks of school, and earned the credits with individual teacher tutoring. His attitude
was so negative regarding school he would not sign up for an adult school night class
that would have assured him of a high school diploma (Zimmerman, 1996).
Interestingly, his negative attitude was not towards education, remember he was
classified as highly skilled, Advanced, on the standardized testing, but was negative
towards school. His mother had badgered John throughout his school career to
perform. She described him as, “The laziest person I have ever known.” He seemed
very alone in the world. His mother was mostly absent and his sister had moved out.
The mother and student had moved in his senior year to an adjacent town, away from
where they had lived his entire life. His career goal was to join the military. The
63
researcher believes his need was to become part of a community, yet school had failed
to fulfill this need (Glasser, 1997).
Interestingly, level of intelligence was not the cause of the students’ situation.
Social skills, attendance, attention span, personal attention needs, and family divorce
prohibited eight students from succeeding to their intellectual best. These students had
unique, complicated issues. It would take a special teacher with exceptional
counseling skills, prolific background information, and abundant time for researching,
to tackle and help solve these students’ issues with school.
Personal Characteristics
A defining personal characteristic of the student group was “stubbornness.”
For example four of the students held their pencils in a non standard way. The students
conveyed they had been told all their lives to hold the pencil in another manner, but
expressed they wanted to hold it their way. All eight students had attitude problems
with their peers. Previous teachers described them as bossy, disrespectful, having poor
attitudes, having difficulty getting along with others. The students in this study had
participated in four team building activities during the first nine weeks of school. The
teacher/researcher believes this helped students realize each others’ strengths and
helped them find value in their differences. Peer interaction was an issue in the algebra
class but manageable due to the team building activities and the expectation of the
teacher/researcher.
64
Learning Strategies
Students again and again said they wished they could have earlier learned
algebra in a small class setting, working together, at a pace everyone found acceptable
(Kortering et al., 2005). Students in this study expressed they liked that tests were
community tests, and that they were allowed to teach each other and talk. Students felt
confident in the small algebra class that someone would help them understand how to
do the algebra problems.
During this small group algebra instruction in their final year of high school,
focusing on the algebra problems was difficult for the students. The learning strategy
most needed by them was to learn to wrestle with algebra problems, to stick with the
problems and feel confident in the answer, to verbalize methods used in computing
answers; to not quit (Hopkins, 2005). Students felt much more confident if they could
have the answer confirmed by the teacher’s edition. Students were happy just marking
the problems wrong; they didn’t want to take the time to figure out how to correct
them. Students calculated their correct score and if it was above 69%, they were
finished! Passing with 70% was all they needed.
During the algebra tests, students were able to communicate strategies,
answers, and share results, yet during the beginning of the year students did not talk
during tests to each other, would not share, or check their work, probably due to
previous habits during testing (Tanner & Jones, 2003). They also wanted to spend the
least amount of time on the test. Talking to someone about the answers just meant
spending more time talking math. To improve and motivate student discussion, the
65
teacher demanded that the tests have all the same answers and that the class receive
100% correct answers (Winstead, 2004). Students began to talk and share methods.
Computation of the numbers rather than conceptualization of the problem solving
method frustrated and demoralized students frequently. Slowly confidence was built,
and students began to realize they could learn algebra (Epstein et al, 2008; Hopkins,
2005; Shipro, 1993; Turner et al., 1998).
Teacher Behavior
Not in journals, interviews, or observations, did any student express a feeling
of connection to a previous math teacher. Most of the students did not like their math
teachers and told wild stories of classroom life during math classes (Fullan &
Hargreaves, 1996; Noddings, 2005). Many of the students did have teachers that they
“liked,” though not math teachers. One student expressed exasperation with a teacher
who gave a learning styles questionnaire then did not change the teaching style or
assignments. This explains the importance of direct communication regarding learning
strategies and assignments relating to different styles (Hopkins, 2005; Tanner & Jones,
2003). The connection would allow students to recognize their own strengths.
The holes in mathematical skills were debilitating. When working in the small
class, students and teacher/researcher were able to address gaps in learning (Turner et
al., 1998), yet the time to truly master the incremental learning was not available to
seniors. The teacher felt a sense of buoyancy when all students proved proficient at
manipulation of negative numbers, yet this was only one fundamental layer of learning
algebra.
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Parental Support
The cumulative school histories and the experience in algebra class revealed
seven of the eight students had excessive attendance problems. It appeared the families
learned how to stay under the reporting radar early in their child’s school career. One
student in first grade had 26 absences, and then after truancy letters adjusted in second
grade to 13 absences and 47 tardies, and in third grade to 13 absences and 23 tardies.
This pattern continued throughout the student’s school career, and was a pattern
similar to other students. Several female students had wads of notes in the cumulative
folders to get out of Physical Education classes. For three students, the divorce of their
parents coincided with absence problems that continued throughout the school history.
A fourth student had severe absences up to middle school, then became interested in
boys and no longer missed school.
The researcher believes students did not have early patterns for school success.
With five students, the teacher observed parent and student interactions where the
student used verbal attacks to avoid being held accountable for school success. The
parents did not fend off the attacks, but retreated. The eight students appeared to have
no structured environment; no one was overseeing them, either because parents were
working or because the students did not allow parental interaction. Parents were
unable to make sure the student got to school. Parents were unable to ensure students
did homework. All but two students were from divorced families. Often the parents
responded to teacher phone calls with comments such as, “She said she is studying at
home today.” Students’ family life had events that focused around non student issues,
67
such as acquiring roommates to help with the mortgage, a mother’s boyfriend, career
and vacation plans for father, and parental health issues.
Limitations
The findings in this study cannot be generalized to the regular school
population. Only eight students were studied. This number represented all the 12th
grade students at an independent study high school in need of algebra credits for
graduation. Therefore no comparable group was available. The students as a group
lacked a multicultural perspective and were skewed toward the female gender with
two boys and six girls, corresponding to this independent study high school’s total
population of 15% Hispanic and 62% female students.
The students were asked to look back on their school career and analyze and
discuss what they remembered. Brain research shows memories to be inaccurate much
of the time. “…episodic memory details are often fuzzy or even completely
inaccurate, and…eye-witnesses of events are generally unreliable” explains brain
specialist Patricia Wolfe (2001, p. 31). Also students may try to suppress negative
memories. The timing also could account for a lack of awareness regarding events
three years prior to questioning.
The teacher was using a new text book to teach algebra. The state of California
approved in November of 2007 certain books be used with students at risk of failing,
therefore the independent study high school changed texts at the beginning of the
2008/2009 school year. In addition, the teacher/researcher had not taught algebra in a
68
small class setting before. Teacher/researcher prior experience had been with one on
one tutoring.
Another limitation was the cumulative school histories. The teacher/researcher
assumed they would be comprehensive and convey a more complete picture of
students’ needs and interventions. In fact, the cumulative folders varied greatly; some
folders did not show a complete history of absences, few had more than one or two
comments per year from the teacher. Occasionally the dates were not filled in on
behavior reports. Only once did a student history include the type of math intervention
tried with the child.
Recommendations
Teacher Behavior
One student folder contained information from a Student Study Team meeting.
The meeting was held in late spring, after a year of student versus teacher battles. The
responsibility of the succeeding intervention was then put on the following year’s
teacher. The next grade teacher may not have been aware of the Student Study Team
meeting, as no follow up papers or responses were found in the folder. The student
showed no improvement in behavior or grades.
Another complaint of the teacher/researcher was the discrepancies found
between subject grade given by the teacher and standardized testing grades. How
could a student be recommended to repeat a year of school when they had scored
Advanced in English/ Language Arts and Math? Given that teachers and
administrators understand the controversy surrounding retention, how could teachers
69
and administrators have recommended retention in the face of such high scores on
standardized tests? If standardized testing results are a school report card, should it not
also be reflected on the student report card? The student knew the curriculum but did
not follow the instructions regarding homework and work assignments; therefore the
teacher gave a failing grade to the student. This happened to three of the eight
students.
Cumulative Folders
Another situation compared a student’s performance in the classroom and on
the standardized tests as congruent, but only at the Basic level of understanding the
curriculum. What specific learning strategies were used to help the students in this
situation? Nothing was noted in the cumulative folder. Recording, following and
tracking the yearly interventions could drastically alleviate frustrations for the students
and teachers.
The researcher believes the cumulative folder is a tool that is underutilized.
School districts have varied requirements for the folders with some districts using
organized forms. One of the eight students’ folders listed small group learning as a
strategy tried for Writing. No other cumulative school history mentioned learning
strategies. The high school information consisted of only the standardized testing and
report cards with computerized teacher comments. Additionally, rarely has the
registrar at this independent high school seen teachers seek information from the
cumulative folders except in Special Education circumstances. So, year after school
year, the teachers gather and use information regarding learning strategies but do not
70
track or document that information for at risk individuals for use in the next school
year. Consequently, the next year’s teacher starts all over. How frustrating for the
student.
If we are to truly leave no child behind, then an individual education plan for at
risk students must be made, followed, and adjusted regularly. Currently this is the
norm of students in need of Special Education services. The researcher believes
students at risk of failure are also entitled to it under the No Child Left Behind Act of
2001. Research has identified at-risk students as early as the 6th grade level using
attendance, failure of an academic class, and citizenship citations as criteria for
accuracy to 86% of dropping out of high school (Balfanz, Herzog and MacIver, 2007)
All of the eight students in this research project were identified as at risk by their
social skills and attendance rates before the third grade.
Parent Support
Regular attendance is not just an educational issue but one of economics. The
business community shows attendance as an essential attribute of a good employee.
The absences and tardies for the students in this study appeared to be a pattern set in
the early years of school, continuing throughout each year. Improved outreach to
parents to instill the importance of attendance is vital.
Additionally, two of the students began having problems in the classroom
when their parents divorced. Acknowledgment and education of the risks facing
children of divorcing parents conceivably falls with the schools.
Learning Strategies
71
Diagnosing and implementing learning strategies that work for students is the
first step in improving the attainment of mathematical knowledge for them. When
students show hints of being at risk- when parents are divorcing, when student
attendance rates are below average, when standardized tests are not congruent to
teacher grades, when student social skills are in need of improvement- the educational
system, under the No Child Left Behind Act of 2001, is liable to provide strategies that
work for each individual student. Specifically, and of outmost importance is our atrisk youth.
Conclusions
We have all heard the statement ‘Do unto others as we would want them to do
unto us’. And we have all heard it changed to ‘Do unto others as they would want us
to do unto them’. Imagine a tofu turkey for the meat eater at Thanksgiving, and in
contrast a steak dinner for the vegan. In education, the focus is on student learning.
Every child can learn, and teachers can facilitate the learning, but it will take vast
sums of tax money to enable schools the resources to ensure every child learns.
Effective legislators and administrators, efficient teachers, supportive and
knowledgeable families, and enlightened students are pieces in this complex puzzle.
Until society is willing to fund a shift to individual education plans for at risk
students, educators must continue to refine their craft. Rather than starting anew with
students at the beginning of each school year, teachers can utilize an existing tool
already in schools, the cumulative folders. Knowledge of learning strategies that have
been tried and work for the student, teacher behavior that fits the learning needs of
72
each student and a record of continuing social and family issues for the new teacher to
be aware of is essential for efficiency.
It is the researcher’s experience that all children want to succeed in
mathematics; and adults with students as partners must make sure they do.
73
APPENDIX A
Consent to Participate in Research
74
Appendix A
Consent to Participate in Research
Maidu High School is participating in a research project exploring the algebra
requirement for high school students. To participate in this research, your student will
be participating and answering questions pertaining to their algebra experience
through middle school and high school. The questions are formulated to discern why
your student had difficulty in their first algebra classes.
Your student will not be treated or taught differently if they do or do not participate in
this research, but your permission to participate will be greatly appreciated. A copy of
the questions asked will be provided upon request.
Please sign below if your student may be included. At no time will their names be
used or will they be identified in the research or findings without your explicit
permission. A copy of the report will be given to you and your student upon request.
I give, student name _____________________________________ permission to
participate in the research study at Maidu High School regarding algebra requirement
at middle school.
Parent name ________________________________________________
Date ______________________________________________________
75
APPENDIX B
Student Journal Questions
76
Appendix B
Student Journal Questions
Student Name _____________________ Today’s date____________
Algebra 1
Close your eyes, relax, we are going to go back in time to your first
algebra class. There are noises, feelings and stress with this first algebra
class experience. Reflect for a minute of two. As you open your eyes,
please jot down a few of your first memories. Then answer the questions
regarding elements of learning in this research packet as fully as possible.
Quick write of memories:
What grade were you in?______
What school?_______________
Room Setting
Think about the way this classroom: grouped or rows of desks, approximate number of
students, posters on the wall, noisy, early in the morning or later in the afternoon.
Describe what your classroom was like:
Did the teacher have classroom control?
Teaching Strategies
Think about your daily routine in this class. What did you like and not like about the
way this teacher taught math.
I liked…….
I didn’t like……
Student Engagement
Was the class easy to be involved in? Think about your commitment to learning, your
homework level and how you felt.
Compare how you did in this class to how you did in other classes at that time.
Fractions are a big part of algebra, did you do ok with fractions?
77
Student Maturity Level
How would you describe your maturity level at this point in your life? Give an
example or two.
Example:
Family Support
How did your family help and/or hinder you during this time in school?
78
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