Comparing the Effectiveness of Reform Pedagogy to Traditional Didactic Lecture Methods
in Teaching Remedial Mathematics at Four-Year Universities
by
Luke Alexander Smith
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 3, 2013
Keywords: remedial mathematics, reform mathematics, postsecondary education
Approved by
W. Gary Martin, Chair, Professor of Curriculum and Teaching
Marilyn E. Strutchens, Professor of Curriculum and Teaching
David Shannon, Professor of Educational Foundations, Leadership and Technology
Stephen E. Stuckwisch, Assistant Professor of Mathematics
ii
Abstract
Postsecondary remedial mathematics courses often have relatively low pass rates
compared to other courses (Bahr, 2008; Virginia College Community System, 2011) and have
contributed to the view that mathematics is a gatekeeper for college success (Epper & Baker,
2009). This study addressed this situation by exploring recommendations made by various
organizations including the National Council of Teachers of Mathematics (NCTM) (2009, 2006,
2000, 1989), the American Mathematical Association of Two-Year Colleges (AMATYC)
(2006), and the Mathematical Association of America?s Committee on the Undergraduate
Program in Mathematics (2011) to improve student learning in mathematics courses through
various pedagogical techniques; in this study, the pedagogical practices advocated by these
organizations are collectively referred to as ?reform mathematics.?
The study was conducted at a mid-sized university in the southern United States. A quasi-
experimental design was used to investigate the effectiveness of incorporating reform
mathematics practices as compared to didactic lecture techniques in improving student success in
remedial mathematics courses. Student success was measured in terms of pass rates, procedural
ability, application ability, and change in mathematical self-efficacy. Repeated Measures
ANOVAs, t-tests, and Fisher Exact tests were used to determine if the treatment had an effect on
student achievement variables. Additionally, qualitative data were also gathered from students
who were enrolled in the reform-oriented course to examine their perceptions of key aspects of
reform mathematics instruction. While the results were not statistically significant, the trends
within the data suggest that students may benefit from reform-oriented instruction.
iii
Acknowledgments
First and foremost, I would like to thank my Lord and Savior Jesus Christ for providing
me with the physical, mental, and emotional resources to complete this degree. I am awed by
how He put all the pieces together to make this accomplishment possible.
I would like to thank my wife, Michelle, for enduring all the nights that I was away
studying while she diligently cared for our children. Your love and support throughout this
program strengthened my resolve to see to its successful conclusion. To my mother and father,
Guadalupe and Edgar Smith, I thank you for raising me and instilling within me your values and
for putting me in schools where the teachers instilled a love for learning; I hope that I too may be
able to do the same for my children. To my mother and father-in law, Cheri and Gary Maxwell, I
thank you for taking care of my wife and kids while I was away from home. To my grandmother,
Ruth Smith, I thank you for advocating the benefits of higher education throughout my
childhood. To my precious children, Katelyn, Kimberly, Kara, and Karlie, I thank you for being
such great kids; although you may not understand it now, each of you gave me an immediate
reason to understand better how to teach.
I must also thank my coworkers for their support in making this study possible. To my
supervisor, Susan Barganier, I thank you for providing me unrelenting support towards
completing this program. To Dr. Lee, Dr. Schmidt, Dr. Peele, Dr. Smith, Dr. Boronski, Dr. Ray,
Ms. Tomblin, and Mrs. Warren, I thank you for your continued support; each of you contributed
towards my degree in invaluable ways.
iv
To Anna Wan, I thank you for the tremendous amount of support you gave through both
providing key resources and by making available your time, energy, and expertise. To Lisa Ross,
Beth Hickman, and Dr. Gilbert Duenas, I thank you for helping me to set up my project for my
study.
To Dr. David Shannon, I thank you for meeting with me many times throughout this
project and for helping me with the methodology and statistical processes used within this study.
To Dr. Stephen Stuckwisch, I thank you for modeling how to make mathematics fun in your
classes.
To Dr. W. Gary Martin and Dr. Marilyn Strutchens, I thank you for showing me a
completely new way of viewing teaching and for modeling that manner of teaching within your
own classes. I would like to thank Dr. Martin specifically for the many meetings we had
throughout the years in which he helped me understand how to improve my work; I am grateful
for his investing his time and energy into making me a better teacher and researcher.
v
Table of Contents
Abstract ???????????????????????????????? ??.. .ii
Acknowledgments?????????????????????????????.?.iii
List of Figures??????????????????????????????? ?..x
List of Tables????????????????????????????????..xi
1. Introduction .............................................................................................................................1
Strengths and Weaknesses of Developmental Education ..................................................3
Reform Mathematics Pedagogy .......................................................................................4
Purpose of the Study ........................................................................................................6
2. Review of Related Literature ...................................................................................................8
Characteristics of Students Who Take Remedial Courses.................................................8
Effectiveness of Remedial Mathematics Courses in Postsecondary Education................ 11
Efforts Made to Improve Student Success in Remedial Mathematics Courses ................ 15
Computer-Based Assistance for Students in Remedial Mathematics Courses ..... 15
Shortening the Length of Developmental Mathematics Programs ....................... 24
A Promising Approach: Reform Mathematics Pedagogy................................................ 26
Recommendations for K-12 Mathematics ........................................................... 27
vi
Principles................................................................................................ 27
Process Standards ................................................................................... 28
The Common Core ................................................................................. 30
The Equitable Nature of Reform Mathematics Pedagogy ........................ 32
Recommendations for Post-secondary Mathematics Students ............................. 34
Recommendations for Underprepared Post-secondary Mathematics Students ..... 36
The Effects of Reform-Oriented Classrooms on Student Achievement ............... 39
Student Achievement in Middle School Reform-Oriented Classrooms .... 40
Student Achievement in Secondary Reform-Oriented Classrooms. ......... 44
Student Achievement in College-level Reform Oriented Classrooms ...... 52
Student Achievement in Remedial Postsecondary Reform Oriented
Classrooms ............................................................................................. 58
Effects of Self-Efficacy on Student Achievement .......................................................... 61
Synthesis of Relevant Studies ........................................................................................ 67
Theoretical Framework .................................................................................................. 68
Research Questions ....................................................................................................... 70
CHAPTER 3: METHODOLOGY ............................................................................................. 72
Design ........................................................................................................................... 72
Context .......................................................................................................................... 73
My Personal Background ................................................................................... 75
vii
Description of Sample ................................................................................................... 77
Instrumentation.............................................................................................................. 79
Dependent Measures .......................................................................................... 79
Pass rates ................................................................................................ 80
Procedural skills. .................................................................................... 80
Application skills .................................................................................... 81
Mathematical self-efficacy...................................................................... 82
Perspectives of treatments ...................................................................... 82
Validity and Reliability ...................................................................................... 82
Procedural and application scores ........................................................... 83
Reformed Teaching Observation Protocol. ............................................. 83
Covariates .......................................................................................................... 84
Procedure ...................................................................................................................... 84
Control Group .................................................................................................... 85
Experimental Group Treatment .......................................................................... 86
Data Analysis ................................................................................................................ 91
Establishing Validity .......................................................................................... 91
Selecting Covariates ........................................................................................... 92
Analysis of Effects ............................................................................................. 93
Qualitative Analysis ........................................................................................... 95
viii
Summary ....................................................................................................................... 96
CHAPTER 4: RESULTS .......................................................................................................... 97
Summary of Events ....................................................................................................... 97
Integrity of Treatment .................................................................................................... 98
Inter-rater Reliability of Tests ........................................................................................ 99
Quantitative Results ..................................................................................................... 100
Selecting Covariates ......................................................................................... 100
Research Question 1: Procedural Skills ............................................................ 102
Research Question 2: Application Skills ........................................................... 107
Analysis of Procedural vs. Application Skills ....................................... 111
Research Question 3: Pass Rates ...................................................................... 116
Research Question 4: Students? Change in Mathematics Self-Efficacy ............. 116
Summary of the Quantitative Results ............................................................... 119
Qualitative Results....................................................................................................... 120
Comparison of Treatments ............................................................................... 121
Efficacy of Control Treatment .......................................................................... 122
Research Question 5: Students? Views about Reform Mathematics .................. 123
Summary of Qualitative Results ....................................................................... 125
CHAPTER 5: CONCLUSIONS AND IMPLICATIONS ......................................................... 126
Limitations .................................................................................................................. 126
ix
Conclusions ................................................................................................................. 127
Research Questions 1 and 2: Procedural and Application Skills ........................ 127
Research Question 3: Pass Rates ...................................................................... 128
Research Question 4: Change in Mathematics Self-Efficacy............................. 129
Research Question 5: Student Response to the Experimental Treatment ........... 129
Implications ................................................................................................................. 130
Teachers........................................................................................................... 131
Administrators ................................................................................................. 134
Conclusion .................................................................................................................. 138
References .............................................................................................................................. 140
Appendix A: Permission Forms ............................................................................................... 156
Appendix B: Student Surveys .................................................................................................. 160
Appendix C: Sample Application Problems ............................................................................. 165
Appendix D: Reformed Teaching Observation Protocol .......................................................... 167
Appendix E: Paired Lesson Plans ............................................................................................ 173
Appendix F: Responses to Open-ended Student Surveys ......................................................... 200
x
List of Figures
Figure 1: A sample procedural problem with corresponding grading rubric ............................... 81
Figure 2: A sample application problem with corresponding grading rubric .............................. 82
Figure 3: Mean adjusted procedural scores for control and experimental groups ...................... 106
Figure 4: Mean adjusted application scores for control and experimental groups ..................... 111
Figure 5: A solution obtained through the use of pictures ........................................................ 113
Figure 6: A solution obtained through systematic trial and error .............................................. 114
Figure 7: Mean pre- and post-mathematical self-efficacy scores .............................................. 118
xi
List of Tables
Table 1: Computer-based mathematics instruction..................................................................... 22
Table 2: Shortening the length of the developmental sequence .................................................. 26
Table 3: Effects of reform-oriented instruction in middle school mathematics courses............... 43
Table 4: Effects of reform-oriented instruction in secondary school mathematics courses .......... 51
Table 5: Effects of reform-oriented instruction in postsecondary mathematics courses .............. 56
Table 6: Effects of reform-oriented instruction in postsecondary remedial mathematics courses 60
Table 7: Effects of self-efficacy on student performance ........................................................... 65
Table 8: Demographics of sample ............................................................................................. 78
Table 9: Summary of differences betweeen traditional and reform-oriented instruction ............. 88
Table 10: Differences in RTOP scores between control and experimental sections .................... 99
Table 11: Summary of inter-rater reliability Pearson correlation values .................................. 100
Table 12: Differences in continuous variables between groups ................................................ 101
Table 13: Differences in dichotomous variables between groups ............................................. 102
Table 14: Summary of procedural scores for control and experimental groups ........................ 103
Table 15: Statistical analysis of procedural scores between groups .......................................... 104
Table 16: Summary of procedural scores adjusted for race ...................................................... 105
Table 17: Comparison of final exam scores between control and experimental groups ............ 107
Table 18: Summary of application scores for control and experimental groups ........................ 108
Table 19: Statistical analysis for the difference in application scores ....................................... 109
xii
Table 20: Summary of application scores adjusted for race...................................................... 110
Table 21: Comparison of non-algebraic strategies on application questions between groups .... 115
Table 22: Summary of pass rates ............................................................................................. 116
Table 23: Summary of students? change in mathematics self-efficacy...................................... 117
Table 24: Statistical analysis for students? change in mathematics self-efficacy ....................... 119
1
1. Introduction
Increased levels of education have been shown to have positive impacts on individuals
and society as a whole. Compared to students with lesser education, students who earn a
bachelor?s degree or higher are more likely to earn higher salaries, generate more tax revenue,
live a healthier lifestyle, obtain health insurance, acquire pensions, perform civic duties and are
less likely to receive public assistance (Baum & Payea, 2004; Perna, 2005). However, in an
effort to earn a postsecondary degree, many students have found that they were underprepared
for postsecondary mathematics and were required to take remedial mathematics courses (Fike &
Fike, 2007; Alliance for Excellent Education [AEE], 2011; Radford et al., 2012).
Remedial mathematics classes are available to help students develop mathematical skills
that should have been obtained in secondary mathematics courses. In 2008, roughly 72% of all
tertiary schools and 90% of public tertiary schools in the United States offered remedial courses
(National Center for Education Statistics [NCES], 2008). Roughly 42% of first-time
postsecondary students in 2003-2004 were required to take remedial mathematics courses
(Radford et al., 2012), and students who took remedial mathematics classes often met all other
admission standards (Duranczyk & Higbee, 2006). However, since the attrition rates of remedial
mathematics courses have often been reported around 50% (Phoenix, 1990; Ellington, 2005;
Attewell et al., 2006; Fike & Fike, 2007; Bahr, 2008; Virginia College Community System
[VCCS], 2011), and the likelihood of a student?s departure from the remedial mathematics
program increases significantly with the number of remedial courses that the student is required
to take (Hern, 2012; Bahr, 2012; Complete College America [CCA], 2012), it is not surprising
that as many as 72% of students in developmental mathematics sequences never attempted a
college-level mathematics course (Wolfle, 2012). Thus, mathematics has been viewed as a
2
gatekeeper for college success (Massachusetts Community College Executive Office, 2006; Fike
& Fike, 2007; Epper & Baker, 2009).
Since students who successfully complete remedial mathematics courses often perform as
well in their academic pursuits as students who did not need remedial courses (Attewell et al.,
2006; Bettinger & Long, 2009; Bahr, 2010), researchers have investigated various areas related
to the successful completion of mathematics courses taken by college freshmen, including the
benefits of online assessment and the effectiveness of implementing pedagogical practices that
align with the reform mathematics movement. For the purposes of this paper, ?reform
mathematics? represents the pedagogical practices that are advocated by organizations such as
the National Council of Teachers of Mathematics (NCTM) (2009, 2006, 2000, 1989), the
American Mathematical Association of Two-Year Colleges (AMATYC) (2006), and the
Mathematical Association of America?s Committee on the Undergraduate Program in
Mathematics (2011). These practices include active student learning, a diminished role of the
instructor as a source of knowledge, and student exploration and experimentation before formal
presentation of mathematical theorems. More details about reform mathematics will be presented
in the review of literature.
These methods have been found successful in some contexts. Thus in this paper, I will
present literature that advocates the need to more closely align the pedagogical practices within
remedial mathematics classrooms with pedagogical practices advocated by the reform-
mathematics movement and that diverging from traditional didactic lecture towards a more
reform-oriented style of instruction will improve the quality of instruction for students in
remedial courses.
3
Strengths and Weaknesses of Developmental Education
Before continuing, it is important to briefly clarify the meaning of remedial education and
developmental education, since both the general public and many scholars use both terms
interchangeably (Institute for Higher Education Policy, 1998; Kozeracki, 2002; Parmer & Cutler,
2007; Radford et al., 2012). Developmental education programs emphasize a holistic approach
(Boylan, Bonham, & White, 1999) to assist individuals who have failed to meet placement
requirements by providing them a variety of courses and services that focus primarily on reading,
writing, mathematics, studying strategies, and other affective variables that are important for
college success (Tomlinson, 1989; Boylan & Bonham, 2007). Remedial courses are a subset of
developmental education and refer exclusively to courses that are not at college level (Boylan,
Bonham, & White, 1999; NCES, 2004) and have served as the core of developmental education
(Brothen & Wambach, 2004). For the purposes of this paper, the term developmental will refer to
the programs enacted by colleges that provide a range of services for underprepared students,
and the term remedial will refer to the coursework that is taken at postsecondary institutions but
is below college level.
Developmental education offers significant benefits to students, institutions, and society
as a whole by providing access and equal opportunity to higher education (Tomlinson, 1989;
Mills, 1998; McCabe & Day, 1998; Goldrick-Rab, 2010; Gallard, Albritton, & Morgan, 2010;
VCCS, 2011). Since an individual?s educational attainment is a significant predictor of
occupational status and financial earnings (Kerckhoff, Raudenbush, & Glennie, 2001),
developmental education offers individuals a ?last chance? to obtain benefits associated with
higher education by preparing them for postsecondary work (Tomlinson, 1989; McCabe & Day,
1998; Gerlaugh et al., 2007; Gallard, Albritton, & Morgan, 2010). Postsecondary remediation
4
develops in students the minimum skills that are necessary to function in the economy and
democracy (Bahr, 2008).
Many of the jobs in today?s society require skills that are made available to students
through developmental mathematics programs (McCabe & Day, 1998; Goldrick-Rab, 2010).
Because many of the students who benefit from developmental education are able to improve
their skills, and thus not have to compete for the increasingly fewer low-skill jobs that are
available, developmental education plays an essential role in reducing the number of individuals
in welfare and prison populations by helping students to become independent and self-sufficient
(McCabe & Day, 1998; Gallard, Albritton, & Morgan, 2010).
Objections are sometimes raised regarding the costs associated with developmental
education programs (Bahr, 2008; Gallard, Albritton, & Morgan, 2010; AEE, 2011). For many
legislatures, postsecondary remediation has symbolized the devaluation of academic standards in
tertiary education and the failure of America?s precollegiate educational system (Mills, 1998;
Boylan & Bonham, 2007); and many legislatures are only recently recognizing the importance of
developmental education (Boylan & Bonham, 2007). Despite these objections, the benefits to
society far exceed the costs associated with implementing developmental education (McCabe &
Day, 1998; Saxon & Boylan, 2001; Gallard, Albritton, & Morgan, 2010), and developmental
education programs consistently generate sufficient revenue to cover the costs of delivering their
services (Saxon & Boylan, 2001).
Reform Mathematics Pedagogy
The high failure rates present in many developmental programs may exist because a
significant proportion of remedial students? academic backgrounds are so weak that they are
unable to succeed in even pre-collegiate courses (Adelman, 1995). The traditional lecture
5
techniques that are commonly used in college classrooms provide these students little benefit
(Adelman, 1995); if lecture techniques had worked in middle and secondary education, these
students would not need to enroll in remedial courses at the postsecondary level (Boylan &
Saxon, 1999; Trenholm, 2006).
A high percentage of students fail remedial mathematics courses (Hern, 2012). On the
other hand, students who pass them often do as well as students who do not need remedial
mathematics courses; thus, it becomes clear that remedial mathematics courses work well for
some students but not for others (Bahr, 2008). Because of the substantial benefits to students and
society that come with college success, improvements need to be made to remedial mathematics
courses so that more students can complete these courses and move closer to achieving their
college degree. The pedagogy advocated by the reform mathematics movement may be a
solution to improving the level of student understanding in postsecondary remedial mathematics
courses.
The current reform movement in school mathematics advocates that students engage in
exploring mathematical phenomena, making conjectures, and analyzing the validity of those
conjectures. Recommendations made by the above organizations include a shift from traditional
didactic lecture (the teaching method in which the teacher is the primary dispenser of knowledge
to a group of passively engaged students) towards student-oriented classrooms that encourage
active student participation in the learning process through engagement in worthwhile problem
solving, collaboration among students, multiple representations, and technology.
Many instructors still present the material to students through rote lecture?the process
whereby the instructor provides information to passive, uninvolved students (Fry, Ketteridge, &
Marshall, 2003; White-Clark, DiCarlo, & Gilchriest, 2008). In comparison, students enrolled in
6
mathematics courses that adhere to reform pedagogy generally perform at least as well as
comparison lecture-based courses. These findings have held at the middle-school (Reys et al.,
2003; Thompson, 2009), high school (Hirschhorn, 1993; Schoen, Hirsch, and Ziebarth, 1998;
Thompson & Senk, 2001; Cichon and Ellis, 2003), and postsecondary levels of education
(Lawson et al., 2002; Erickson & Shore, 2003; Ellington, 2005; Gordon, 2006).
Purpose of the Study
Current research does not adequately address the effectiveness of various teaching
strategies employed within remedial mathematics classrooms in colleges and universities. For
example, although research has been done on the effectiveness of computer-based assistance in
remedial mathematics courses in which lecture-based instruction was either supplemented or
replaced by computer-based instruction (Villarreal, 2003; Walker & Senger, 2007; Squires,
Faulkner, & Hite, 2009), the scope of these studies were limited to either the effects of stimuli
outside classroom instruction or to the effects of replacing instructors with computers; neither
approach examined the teaching practices of the instructors. Furthermore, several studies have
been performed on the effectiveness of remedial mathematics instructors? pedagogical decisions
with respect to cooperative learning, use of technology, and problem-oriented approaches to
learning (Phoenix, 1990; Erickson & Shore, 2003; Ellington, 2005); however, these studies
possessed limitations in their comparative designs. Multiple studies have shown that students
who received instruction in accordance with reform mathematics pedagogy tend to do at least as
well as traditionally taught students in procedural skills and often better in application problems
(Hirschhorn, 1993; Schoen, Hirsch, & Ziebarth, 1998; Senk & Thompson, 2006).
This study compared the effectiveness of reform pedagogy to didactic lecture methods in
teaching remedial mathematics at a four-year university. The study was guided by the following
7
broad research question: Is teaching remedial mathematics in a reform-oriented manner
beneficial to university students? Five subquestions were addressed as follow:
1. Is there a significant difference in the pass rates in the remedial mathematics courses
between university students who receive instruction consistent with reform pedagogy
versus university students who receive instruction through traditional didactic lecture
methods?
2. Is there a significant difference in mathematical procedural ability between university
students who receive instruction consistent with reform pedagogy versus university
students who receive instruction through traditional didactic lecture methods?
3. Is there a significant difference in mathematical problem solving ability between
university students who receive instruction consistent with reform pedagogy versus
university students who receive instruction through traditional didactic lecture methods?
4. Does the self-efficacy of university students in the reform classes improve as a result
of instruction received in the reform classes?
5. What views about reform instruction will university students who are enrolled in a
reform-oriented remedial mathematics course express upon completing one semester of
reform-oriented mathematics instruction?
8
2. Review of Related Literature
Having discussed the importance of improving student success in remedial mathematics
courses, it is important to examine the efforts made by others to enable students to learn
mathematics. In this chapter, I review the literature that is relevant to the study. First, I present a
description of the characteristics of students who take remedial courses. Next, I present studies
that address the effectiveness of remedial mathematics courses. Third, I describe efforts made to
improve student success in remedial courses through computer-based assistance as well as by
shortening the length of developmental mathematics programs. Fourth, I present the main tenets
of reform mathematics pedagogy, a promising alternative approach to improving student success
in remedial mathematics courses. I will present an overview of the reform mathematics pedagogy
as advocated by the National Council of Teachers of Mathematics (NCTM) for K ? 12
mathematics, followed by recommendations made by the American Mathematical Association of
Two-Year Colleges (AMATYC) and the Mathematical Association of America (MAA) for
undergraduate mathematics courses that service underprepared students. I will also present
studies that address the effects on student achievement that can occur when values that are
aligned with reform pedagogy are adopted within mathematics classrooms. Fifth, I review
literature that address the impact that mathematics self-efficacy can have on student success in
mathematics. Sixth, I present my case for developing a study that would examine the effects of
reform pedagogy on student achievement in post-secondary remedial mathematics courses.
Lastly, I explain the theoretical framework that will serve as the underpinning of my study.
Characteristics of Students Who Take Remedial Courses
This section presents socio-demographic information regarding race, gender, age, and
income levels of students in remedial classes. Descriptions of common prior academic
9
experiences and obstacles faced by students in remedial courses are also presented. According to
NCES (Sparks & Malkus, 2013), the percentage of students who took remedial courses dropped
sharply from 1999 to 2003, but increased slightly from 2003 - 2007; thus, the net difference
between 1999 data and 2007 data showed that a lower percentage of students were taking
remedial courses. This trend occurred in characteristics such as race, gender, and age. The follow
data describe the trends for first-year undergraduate students who attended public institutions.
According to the data collected by NCES (Sparks & Malkus, 2013), in 2007 - 2008,
23.3% of all first-year students reported enrolling in a remedial course, as compared to 22.1% in
2003 - 2004 and 28.8% in 1999 - 2000. During the 2007 - 2008 academic school year, the
percentages of African American, Hispanic, Asian/Pacific Islander, and White students who
reported taking a remedial course were 30.2%, 29.0%, 22.5%, and 19.9%, respectively.
Although slightly higher than 2003 - 2004 data in which the percentages of African American,
Hispanic, Asian/Pacific Islander, and White students who reported taking a remedial course were
27.4%, 26.8%, 20.1%, and 19.7%, respectively, the 2007 - 2008 data are still lower than the
1999 - 2000 data in which 37.7%, 37.8%, 34.9%, and 24.7%, respectively reported taking a
remedial course. Thus, two points should be emphasized from these sets of data. First, remedial
courses continue to be needed by students entering postsecondary education. Second, minority
students continue to be significantly overrepresented in remedial courses, a phenomenon
documented by other research (Bailey, Jenkins, & Leinbach, 2005; Bailey & Morest, 2006; AEE,
2011).
According to data gathered by NCES (Sparks & Malkus, 2013), female students were
more likely than male students to take a remedial course in 2007 - 2008 (24.7% and 21.6%,
respectively), in 2003 - 2004 (23.1% and 20.7%, respectively), and in 1999 - 2000 (29.1% and
10
28.5%, respectively). When comparing the data across the three collection points, it becomes
clear that the overrepresentation of females in remedial courses continued to be an issue, a
phenomenon voiced by research a decade earlier (Hagedorn et al., 1999).
Approximately 23.8% of traditional college age students (ages 15 to 23 years old)
reported having taken a remedial course during their first year, whereas 22.0% of older students
ages 24 to 29 and 20.3% of students between 30 and 39 years of age reported taking a remedial
course during their first year. Supplementing the data not provided by Sparks and Malkus (2013),
Goldrick-Rab (2010) found that many of the students in community colleges who are enrolled in
noncredit instruction are older adults from disadvantaged backgrounds. The consideration of
adult learners is important because adult learners can face more difficulties in obtaining higher
level mathematics skills than recent graduates do; adult learners often face more logistical and
financial challenges. For example, adult learners are often the sole household earner and must
coordinate daycare and time off from work (Woodard & Burkett, 2005; Golfin et al., 2005;
Duranczyk & Higbee, 2006; AMATYC, 2006). Additionally, adult learners have often
functioned at low levels of quantitative literacy and have a history of education failure (Golfin et
al., 2005).
Students in remedial mathematics courses often meet all other admission standards but
are limited in educational opportunities due to poor mathematical skills (Duranczyk & Higbee,
2006), a fact that reinforces the view of mathematics as a gatekeeper for college success (Epper
& Baker, 2009). Many of the students in developmental courses face difficulties that are not
experienced by traditional students; Duranczyk and Higbee (2006) aptly summarized this
situation: ?Nontraditional students?whether in terms of age, heritage, socioeconomic status, or
educational history?often do not have the luxury of approaching higher education as full-time
11
residential students, employed for fewer than 20 hours per week, supported primarily by their
parents, and without the responsibility of caring for dependent family members? (p. 23).
Effectiveness of Remedial Mathematics Courses in Postsecondary Education
Proponents for remediation have stated that remedial courses help students develop skills
to improve their chances of collegiate success (Bettinger & Long, 2009); however, not all
researchers agree that remediation is effective (Perin, 2006; Attewell et al., 2006). The following
studies describe various effects that remedial courses have had on underprepared students. The
terms remediation and remedial in the following studies refer to courses that are below college
level.
Bahr (2010) investigated the effectiveness of post secondary remediation for students
who were deficient in mathematics, English, or both mathematics and English. His sample
consisted of 68,884 first-time, non-English Second Language college freshmen enrolled in one of
California?s community colleges during 1995. He continued to monitor these students for six
years and found that students who completed remediation in either mathematics or English as
well as students who completed remediation in both mathematics and English ?experienced rates
of credential completion and upward transfer that are comparable, or slightly superior, to those of
students who attain college-level competency in math and English without remediation? (p. 195).
In other words, students who successfully completed their remedial courses tended to do as well
as students who were not required to take remedial courses.
Although it is encouraging to find that remedial courses adequately prepare students for
future academic coursework, student persistence is problematic for remedial education. From a
similar set of data, Bahr (2008) noted that only 1 in 4 students successfully completed the
remedial courses, and of these students who did not successfully complete the remedial course,
12
roughly 80% of them did not complete a program of study or transfer to a 4-year institution. Bahr
stated that future research should examine why remediation does not work for some students.
Johnson and Kuennen (2004) studied the impact that delaying remedial mathematics had
on students? scores in freshman microeconomics, a quantitative-intensive course. From a sample
of 1,462 freshman microeconomics students, the researchers found that students who did not
need to take remedial mathematics scored higher than remedial students who had already passed
their remedial mathematics courses, and these remedial students scored higher than remedial
students who had not yet taken their remedial mathematics courses. The researchers found that
the differences between all three groups were statistically significant. Because Johnson and
Kuennen (2004) only examined microeconomics students, the researchers stated that further
study could be done related to physics, chemistry, accounting, and other quantitative courses.
The results of Johnson and Kuennen?s (2004) study differed slightly from Bahr?s (2010) study.
In Bahr?s (2010) study, developmental students who completed the remedial classes did as well
as students who did not need remedial classes; whereas in Johnson and Kuennen?s (2004) study,
developmental students who had completed their remedial classes scored slightly lower in
microeconomics than students who did not need remedial classes.
Parmer and Cutler (2007) studied the performances in Math 101 (a college-level
elementary algebra course) of students who had completed remedial mathematics (n = 591) as
compared to students who did not need to take remedial mathematics (n = 437) at Sinclair
Community College, Ohio. The researchers conducted a three-part project. First, the researchers
issued a 15 question pre-course assessment on topics including writing percents as decimals,
simplifying expression containing fractions and exponents, squaring negative numbers, and
solving linear equations. The researchers found that former remedial students answered on
13
average 9.86 questions correctly as compared with 10.22 correctly answered questions for
students who did not take remedial mathematics; the researchers did not state whether the
differences were statistically significant but did state that both groups of students were similarly
equipped for Math 101. Second, when the researchers analyzed the academic performance of the
students throughout the course, they found that former remedial students scored the same as non-
remedial students only on the first test; on all subsequent tests, remedial students scored lower
than non-remedial students. Further, significantly higher percentages of non-remedial students
passed Math 101 than remedial students (53% and 46%, respectively). Lastly, the researchers
issued an anonymous survey that asked students to report perceived difficulty on various topics
throughout the course; these topics included factoring trinomials, solving linear equations,
solving linear inequalities, and operations with polynomials. Former remedial students gave a
higher difficulty rating to learning all 10 topics on the survey than did non-remedial students.
Attewell et al. (2006) compared students who successfully completed all their remedial
mathematics courses on their first attempt to students who never enrolled in remedial
coursework. The researchers analyzed data from students whose information was gathered from
the 1988 - 2000 National Educational Longitudinal Study. After controlling for high school
experiences and socio-demographic background, the researchers' logistic regression model found
that students in two-year colleges who completed remedial mathematics courses were more
likely to earn a degree than were comparably equipped students who did not enroll in remedial
mathematics courses (n = 2,009, p < 0.001). However, for students (n = 3,833) enrolled in four-
year universities, no significant difference in graduation rates was found between successful
remedial students and non-remedial students.
14
Bettinger and Long (2009) found that when they controlled for students? ACT scores,
high school GPA, family income, gender, and several other factors, remediation for
underprepared mathematics students had a positive effect on helping these students to succeed at
the college level. Their results came from tracking 28,000 full-time, traditional freshmen
students in 42 Ohio universities over a period of 6 years. The researchers noted that the
placement of similarly prepared students (as indicated by their ACT scores and high school
GPA, for example) into remedial classes was often determined by the university that they
attended. By analyzing where these students were placed, the researchers found that students
who successfully completed remedial courses were more likely to persist in college than were
similar ability students who had not enrolled in remedial courses. Further, remediated students
were more likely to complete their degree programs and less likely to transfer to a less selective
college. Specifically, underprepared mathematics students who took remediation courses were
13.9% less likely to drop out of the program and 1.5% more likely to complete their degree
within 6 years. One of the strengths of this study is that the results were based on data from
multiple universities.
Several patterns emerge regarding the effectiveness of remedial mathematics courses.
First, students who successfully completed their remedial mathematics courses tended to have
similar graduation rates as students who did not need to enroll in remedial mathematics courses
(Bahr, 2010; Attewell et al, 2006). Second, students who took remedial mathematics courses
often did not perform as well in their quantitative classes as students who did not need to take
remedial mathematics (Johnson & Kuennen, 2004; Parmer & Cutler, 2007). However, students
who enrolled in remedial mathematics courses were more likely to graduate (Bettinger & Long,
2009) and perform better in quantitative courses (Johnson & Kuennen, 2004) than were students
15
of equal ability who did not enroll in remedial mathematics courses. In other words, even though
students who took remedial mathematics courses may not have performed as well in their
quantitative courses as their counterparts who did not need remedial coursework, students who
completed their remedial courses have graduation rates similar to those of students who do not
need remedial coursework.
Efforts Made to Improve Student Success in Remedial Mathematics Courses
Two common approaches to improving student success in remedial mathematics courses
or improving students? understanding of remedial mathematics topics have been documented in
the literature: the use of computer-based instruction and the decrease in the length of
developmental mathematics sequences. The following two sections describe effects that these
two strategies have had on improving student success related to understanding remedial
mathematics material.
Computer-Based Assistance for Students in Remedial Mathematics Courses
Implementing computer-based instruction to improve student learning in remedial
mathematics courses is a common form of intervention initiated by universities (Villarreal, 2003;
Walker & Senger, 2007; Squires, Faulkner, & Hite, 2009). Computer-based instruction has also
been used to improve and remediate students? algebraic skills in credit-bearing mathematics
courses (McSweeney & Weiss, 2003; Brouwer et al., 2009). The majority of the following
researchers stated that the technologies used in their studies were either as effective as or more
effective than traditional measures in remediating students? algebra skills; however, several of
the following authors stated that the computer-based instruction implemented in their studies did
not significantly improve student achievement, or the authors did not provide sufficient data to
support the claim that computer-based instruction benefited their students.
16
Villarreal (2003) described efforts made by the University of Texas at Brownsville,
where 49% of its students required help to begin college credit courses. Three developmental
mathematics courses were offered at the college: Basic Mathematics, Introductory Algebra, and
Intermediate Algebra. The mathematics department first experimented with Computer Directed
Instruction (CDI) in which students were enrolled in a computer-based, self-paced course that
allowed students to attend the computer laboratory at their convenience; the department soon
found that the students were not disciplined enough to complete the coursework in a timely
manner. The mathematics department eventually constructed their Intermediate Algebra classes
with both lecture and laboratory components. Students met for three hours per week in classroom
instruction and three hours per week in the computer laboratory. The researcher noted that the
passing rate for Intermediate Algebra increased an average of 12% over the following two years;
however, the researcher did not provide statistical data to support this claim. The researcher also
noted that the mathematics department offered students several paper/pencil laboratory sections
in which students were encouraged to work together with active peer tutoring instead of working
in computer laboratories; the instructor was able to work with small groups of students or
individuals as necessary. Unfortunately, Villareal (2003) did not provide any data regarding the
success of this alternate approach. Further, the type of instruction that was offered during
paper/pencil laboratories was not described in detail.
Walker and Senger (2007) studied the effect of a computer software program called The
Learning Equation (TLE) on the student achievement of 120 minority developmental students
enrolled in an intermediate algebra at Alabama State University. Roughly half of the students
were randomly placed into traditional courses whereas the other half were placed into courses
that used TLE. All of the students were given a pretest and posttest to determine the
17
effectiveness of the software program, and the researchers found no significant difference in
student achievement between the computer and the non-computer groups. Unfortunately, details
about classroom instruction were unclear. For example, Walker and Senger (2007) reported that
students in both the control and experimental classes received instruction through direct lectures
that included ample use of PowerPoint and had access to tutors in a computer laboratory;
however, the researchers did not describe the pedagogical practices (such as group work and
classroom discussion) employed by the instructor. Additionally, the researchers did not provide
a general list of topics covered in the course.
Squires, Faulkner, and Hite (2009) studied the effects of a ?one-room schoolhouse? at
Cleveland State Community College. The project involved a total of three developmental
mathematics courses (basic mathematics, elementary algebra, and intermediate algebra) and
three college level mathematics courses (college algebra, introductory statistics, and finite
mathematics). In 2008, students met for class in a computer lab one hour each week during
which time an instructor was available to help students and monitor their progress. Students were
also required to attend a computer lab an additional two hours each week where they continued
to learn the material in the course. All of the courses were delivered using an online learning
system. Each course consisted of 10 to 12 modules where students watched a brief instructional
video, completed homework, and then passed a quiz by scoring 70% or better. Once students
completed all of the modules for the course, they could start working on the material in the next
course, thus allowing students to complete multiple courses during the semester. The researchers
stated that the pass rate in remedial mathematics classes at Cleveland State had increased from a
54% to 72%. The pass rates in college algebra (a college-level course) increased from 65% to
74%; however, the pass rates in remaining college-level courses have remained at 72%. The
18
researchers also noted that the costs in the mathematics department decreased 10% because of
the restructuring of these courses.
McSweeney and Weiss (2003) performed a comparative study to examine the
effectiveness of Math Online in improving students? algebra skills so that they may succeed in
college-level Calculus 1 and Calculus 2 courses. Math Online is a self-paced, computer-based
online course that is designed to give students extra practice and reinforcement in their algebra
and precalculus skills outside the classroom. Students enrolled in the calculus section that
utilized Math Online were required to complete a set number of proctored multiple-choice
quizzes outside the classroom during the semester in a local computer facility. Each instructor
who participated in the experiment taught one traditional Calculus section and one section of
Calculus in which students used Math Online to practice mathematical skills in addition to the
lecture. Further, students did not know which type of course they would be taking until their first
day of class.
The researchers assessed student performance by 1) giving each instructor?s students pre-
tests and post-tests consisting of multiple-choice questions and 2) including common exam
questions in each instructor?s midterm and final exams (McSweeney & Weiss, 2003). When the
researchers compared the pre-test and post-test scores for each of the instructor?s two classes for
Fall 2000 and Fall 2001 (a total of 24 classes containing roughly 25 students per class), they
found that the experimental group scored an average of 1 question higher than the control group
on the 15-question tests (p < 0.05). When the researchers examined the results of the common
test questions, they found that the experimental groups did significantly better (p < 0.05) than the
control group on roughly 25% of the questions and that there was no significant difference
between the two groups on the other questions. Lastly, the researchers found that instructors
19
teaching the experimental courses could teach the same amount of material in less time (7.5%
less time) than when they taught the material without using Math Online.
Zavarella and Ignash (2009) examined the effectiveness of distance learning courses,
traditional courses, and hybrid courses for students enrolled in Beginning Algebra at a large
urban Florida community college. In their study, the researchers described distance learning
courses as online courses in which students used packaged software that was delivered at a
distance. In hybrid courses, students met on campus, and computers were used as the primary
delivering agents of the course material; however, instructors acted as facilitators and delivered
personalized instruction as needed. In the traditional courses, content was delivered in a face-to-
face classroom setting through a lecture style format. Zavarella and Ignash (2009) found that the
students who enrolled in traditional lecture courses were significantly less likely to withdraw
from the course than were students who enrolled in the distance learning sections and the hybrid
sections (20%, 39%, and 40%, respectively). One limitation of the study was the lack of
description regarding the type of software that the students in the computer-based instruction
used.
Brouwer et al. (2009) wanted to know if frequently completing online tutorials with
corresponding online assessments enhanced the students? experiences in Calculus 1 and Business
Statistics at the University of Amsterdam. The researchers studied a total of 650 freshmen
students who were required to take a concurrent remedial algebra course focused on algebraic
skills during the first part of the semester. For Calculus students, the remedial course took place
during the first five weeks of the semester; students were given a practice test on the third and
fourth week, and a final test on the fifth week. For the Statistics students, the course took place
during the first ten weeks, and students were given two tests each week. Based on results from
20
student surveys, the researchers found that the majority of both the Calculus and Statistics
students found the remedial course to be designed appropriately (79% and 67%, respectively).
Unfortunately, the researchers did not state if the remedial course improved student performance
in the Calculus and Statistics courses, nor did they describe the mathematical content that was
assessed in the remedial algebra course. In other words, even though the students felt that the
remedial course was designed appropriately, it was unclear if the experimental students? scores
significantly different from the scores of students who did not take the remedial course.
Similar to the previous study by Squires, Faulkner, and Hite (2009), Bassett and Frost
(2010) described the efforts made by Jackson State Community College to reduce the time that
students spent in remedial mathematics courses. The college transformed its three remedial
lecture-based mathematics courses into 12 computer modules ran by the mathematics software
program MyMathLabsPlus. In the new design, students could progress through the modules at
their own pace and could complete a module by demonstrating 80% mastery of its content;
therefore, students could complete their developmental course work in just one term if they were
motivated to do so. Faculty helped students by leading small group discussions on topics that
students found difficult. The pass rate for the traditional remedial courses historically averaged
41% through Spring 2008; however, when the school transferred to remedial instruction being
delivered primarily through MyMathLabsPlus, the pass rate rose to 60% (n = 1,324) by Fall
2009. Additionally, student retention rates increased from 74% during the use of traditional
lecture instruction to 83% in Spring 2009 during use of the computer-based instruction.
Consistent with the study by Squires, Faulkner, and Hite (2009), the present study also stated that
the mathematics department reduced costs as a result of having to hire fewer instructors to teach
21
the material. The study could have been strengthened by describing the type of instruction that
was demonstrated by the MyMathLabsPlus software.
Vassiliou (2011) reported the efforts of the Florida community college Kendall Campus
to use computer assisted instruction to reduce the number of remedial mathematics, reading, and
writing courses that students needed to take. Kendal Campus used a computer based tutorial
system called Advance College Readiness Online which prescribes individualized lessons based
on perceived student deficits in specific content areas. In the study, students first took a
placement test to earn a baseline mathematics score. Second, the Advancer software program
prescribed a series of individualized lessons to address deficits in arithmetic and elementary
algebra. After students worked with the software on their own time (typically between 6 - 13
hours), students took the placement test again to establish a post test score. Of the 180 students
who participated in the study from 2006 - 2008, students increased their post test scores in
algebra and arithmetic 45% and 57%, respectively. The author also stated that 136 of the 216
students (63%) in the study placed into a higher remedial course, and 62 of those 136 students
were able to avoid a remedial course altogether; additionally, the persistence and success rates of
the students who used the Advancer tutorial system was greater than the persistence of the
students who received traditional remedial classroom instruction. However, with respect to rates
of testing out of remedial courses and rates of persistence, the author did not distinguish between
remedial mathematics courses and remedial reading and writing courses. Additionally, the study
did not describe the type of instruction used by the Advancer software.
The studies in this section addressed the effectiveness of computer-based instruction for
post-secondary mathematics courses and found mixed results. The combination of lecture and
computer-based laboratory instruction improved student achievement in some studies (Villarreal,
22
2003; McSweeney & Weiss, 2003), as did the complete abandonment of traditional lectures in
favor of self-paced, computer-based instruction (Squires, Faulkner, & Hite, 2009; Bassett &
Frost, 2009). However, the benefits of computer-based instruction were limited. Villarreal (2003)
reported that the self-paced, purely computer-based instructional design had to be modified to a
lecture and laboratory design due to students? lack of discipline, and Zavarella and Ignash (2009)
found that students were more likely to withdraw from computer-based mathematics courses.
Additionally, Walker and Senger (2007) found no significant benefit in student achievement
from using computer-based instruction, and Brouwer et al. (2009) provided no information
regarding the effect of computer-based instruction on student achievement. See Table 1 for a
summary of studies that examined the effects of computer-based instruction on student success
in remedial courses.
Table 1
Computer-based mathematics instruction
Approach Used Researcher Results
Computer Directed Instruction
(Lab only) in remedial algebra
courses
Villarreal (2003) Pure lab program was
abandoned in favor of
Lecture/Lab combination
because of lack of student
discipline
Computer Directed Instruction
(Lab only) in remedial algebra
and college algebra courses
Squires, Faulkner, & Hite
(2009)
Increase in pass rate from 54%
to 72% for remedial courses;
65% to 74% for college
algebra courses
23
Lecture/Lab Combination in
remedial algebra courses
Villarreal (2003) 12% increase in pass rate over
2 years
Lecture/Lab Combination in
Calculus courses
McSweeney & Weiss (2003) Lab groups did significantly
better than non-lab groups
Lecture/Lab Combination in
remedial algebra courses
Walker & Senger (2007) No significant difference
Lecture/Lab Combination in
Calculus and college-level
statistics courses
Brouwer et al. (2009) No results given about student
achievement
Computer Directed Instruction
(Distance Learning) in
remedial algebra courses
Zavarella & Ignash (2009) Distance learning groups had
significantly higher
withdrawal rates than face-to-
face groups
Module-based Curriculum
(Computer Directed
Instruction)
Bassett & Frost (2009) Pass rate in remedial courses
rose from 41% to 60%
Enhanced Placement Scores
through Computer Tutorials
Vassiliou (2011) Arithmetic and algebra
placement scores increased
45% and 57%, respectively.
Many students placed out of
remedial courses.
24
Shortening the Length of Developmental Mathematics Programs
Some schools are attempting to improve the success rate of students in developmental
programs by decreasing the number of remedial courses that students must take before being
permitted to take college-level courses (Merseth, 2011; Hern, 2012). The lower a student places
in a developmental mathematics sequence, the more opportunities that student will have to exit
the sequence (Bahr, 2012); thus, students who pass one remedial course may decide not to enroll
in the subsequent course (CCA, 2012). This section describes efforts to reduce the number of
required remedial courses in developmental mathematics sequences.
Merseth (2011) reported the efforts made by the Carnegie Foundation for the
Advancement of Teaching to create Statway and Quantway, programs designed to improve
student persistence and student engagement in developmental mathematics courses. These
courses promoted two aspects that can benefit non-STEM (science, technology, engineering, and
mathematics) students: a path in which students could obtain college credit in only two semesters
and a curriculum that concentrated on quantitative literacy. Primarily focused on students
enrolled in community colleges, Quantway and Statway promoted student success by engaging
students in sense making about real-world issues and by compelling students to make decisions
through numerical reasoning and argumentation. In Statway, instruction focused on statistical
concepts and quantitative reasoning; mathematics served as a subplot that reinforced learning
these topics. In Quantway, instruction focused on the understanding and application of
mathematical concepts instead of the memorization of disconnected processes and procedures.
Because Statway and Quantway were recently launched in Fall 2011 and Winter 2012,
respectively, credit completion data is available only for Statway courses.
25
Byrk (2012) reported the results from the first cohort of Statway students. Roughly 50%
of Statway students earned college credit in one year. Byrk (2012) compared these results to
California community college students who enrolled in traditional developmental mathematics
sequences during Fall 2009 ? Spring 2012: 17.4% of students who needed only 1 remedial
mathematics course completed a college-level mathematics course in one year, 39.9% of students
who needed only 1 remedial courses earned college credit in three years, and 16.5% of students
who needed 2 remedial courses earned college credit in three years. In order to demonstrate that
the sequence of courses in Statway was comparably rigorous to other credit statistics courses, a
statistics test was distributed to a national reference sample of students who had successfully
completed a statistics course. The average score on the common exam was 64%, and the average
score on the exam for the Statway cohort was 62.8% (Byrk, 2012). One limitation of the study is
its focus on non-STEM students.
Hern (2012) reported the effect of implementing Path2Stats, a one-semester
developmental course that prepared students for college statistics. The study was done in seven
California community colleges during the 2011 ? 2012 school year. There were no prerequisites
for the course, and students began learning statistics on the first day of class. Any remedial
arithmetic and algebraic concepts were reviewed when the current statistical topics deemed them
necessary. In the study, 71 of the 119 (60%) of the Path2Stats students completed a college-level
statistics course at the end of one year, as opposed to 362 of the 1756 (21%) of the students who
elected to enroll in the traditional remedial courses offered by the community colleges. One
limitation of the study is that the researcher did not provide a description of the topics and
classroom activities within the Path2Stats course. Another limitation of the study was the lack of
focus on STEM students.
26
The studies in this section presented efforts made by institutions to increase student
success by reducing the time that it took students to complete a college-level mathematics
course. Some institutions used computer-based instruction, whereas other institutions redesigned
the developmental mathematics curricula to complete remedial coursework in one semester. The
studies presented generally positive results regarding the effectiveness of decreasing the required
length of mathematics developmental programs. See Table 2 for a summary of studies that
examined the effects of a shortened developmental sequence on student success in remedial
courses. Although the efforts in these studies showed promise, it may be difficult for departments
to implement these changes since they would have to significantly redesign their developmental
programs.
Table 2
Shortening the length of the developmental sequence
Approach Used Researcher Results
Two Semester College-Credit
Track for non-STEM Students
Byrk (2012) 50% of Statway cohort earned college
credit in 1 year vs. 17.4% of traditional
remedial students
Two Semester College-Credit
Track for non-STEM Students
Hern (2012) 60% of Path2Stats cohort earned college
credit vs. 21% of traditional remedial
students
A Promising Approach: Reform Mathematics Pedagogy
The preceding studies described efforts made by colleges to improve student achievement
through computer-based instruction or by decreasing the length of the developmental
27
mathematics sequence. Computer-based instruction was often used to reinforce mathematical
concepts outside the classroom, and decreasing the length of the developmental mathematics
sequence was applied primarily to non-STEM students. The following set of studies will describe
efforts to improve student achievement by modifying pedagogical practices inside the classroom;
additionally, these practices can be used to improve instruction for both STEM and non-STEM
students. Since the following studies are based on the ideas advocated by reform documents that
are published by the National Council of Teachers of Mathematics (NCTM), the American
Mathematical Association of Two-Year Colleges (AMATYC), and the Mathematical
Associations of America (MAA), I will first describe their main tenets before describing the
effects that adopting such practices have had on student achievement in middle, secondary, and
postsecondary mathematics classrooms.
Recommendations for K-12 Mathematics
The current standards-based reform movement began toward the end of the eighties with
the publication of NCTM?s Curriculum and Evaluation Standards (1989) followed by NCTM?s
Professional Standards (1991) and Assessment Standards (1995); in 2000, NCTM published
Principles and Standards for School Mathematics which synthesized into a single volume much
of the information presented in the previous three publications (Piburn & Sawada, 2000).
Principles and Standards for School Mathematics presented six ?principles? and five ?process
standards? that articulated and guided the reform mathematics movement by presenting a
strongly coherent picture of mathematics reform (Piburn & Sawada, 2000). A brief description of
these principles and standards are provided below.
Principles. The principles described in NCTM?s Principles and Standards for School
Mathematics (2000) were designed to provide teachers and administrators guidance. The
28
following six principles describe components of high-quality mathematics education. The Equity
Principle states that all students, regardless of their personal characteristics, physical challenges,
or backgrounds, should have the opportunity to study mathematics, have the support they need to
learn mathematics, and have access to a challenging, coherent curriculum that is taught by
capable mathematics teachers who hold high standards for their students. The Curriculum
Principle states that coherent curricula demonstrate to students how different strands of
mathematics relate to, and build on, one another; additionally, mathematics teachers should
organize their lessons around fundamental mathematical concepts that can be extended and
developed. The Teaching Principle states that teachers need to understand the big ideas in
mathematics and carefully create experiences that help students develop an understanding of
those ideas.
The Learning Principle states that students learn by actively building upon prior
knowledge, and students who learn with understanding are more likely to know when and how to
use what they know. The Assessment Principle states that assessment should focus on both
students? conceptual understanding and procedural skills, and mathematics teachers who include
formative assessment throughout their lessons can furnish useful information to both teachers
and students. The Technology Principles states that technology (such as computers and graphing
calculators) can help students to explore mathematical conjectures more easily than if they were
to create representations by hand; also, students can use technology to perform routine
procedures more quickly and accurately and thus explore a wider range of problems.
Process Standards. The standards described in NCTM?s Principles and Standards for
School Mathematics (2000) describe the math content and processes that students in high-quality
mathematics programs should learn. The Problem Solving Standard states that teachers who
29
select worthwhile problems and create environments that encourage exploration can solidify and
extend what students know, stimulate students? interest in learning mathematics, and enable them
to persist in challenging problems. The Reasoning and Proof Standard states that students need to
develop reasoning skills to be able to understand mathematics; and students at all grade levels
should see that mathematics makes sense through exploring phenomena, making mathematical
conjectures, and justifying results. The Communication Standard states that students who
communicate their ideas to their teachers and peers build meaning and permanence to the ideas,
and students who listen to others? explanations can deepen their own understanding, particularly
when they disagree.
The Connections Standard states that when teachers emphasize the interrelatedness
between mathematical concepts and other disciplines, students can better learn those concepts as
well as learn about the usefulness of mathematics; further, teachers should take advantage of the
ample opportunities in science, medicine, commerce, and social science to provide their students
mathematical experiences in a context. The Representations Standard states that multiple
representations?such as diagrams, graphs, tables, and symbolic expressions?should be
emphasized throughout a student?s mathematical education. As students develop their
mathematical abilities, they develop a repertoire of mathematical representations and an ability
to determine which representation is more advantageous based on the problem at hand.
Additionally, multiple representations allow for students to move toward abstraction so that
students can better understand the role that mathematics plays in revealing patterns.
A subsequent extension to high school mathematics standards is the idea that students
should reason through and make sense of mathematics. Reasoning and sense making are the
foundations for NCTM's Process Standards (NCTM, 2009). Students who are able to reason
30
through and make sense of newly presented mathematical concepts can organize their knowledge
in ways that can improve their mathematical abilities. These students will be more likely to
understand and retain new information because they will be able to link the new topics to skills
and concepts they have already acquired. Teachers can help their students achieve mathematical
competence by consistently encouraging students to develop increasingly sophisticated levels of
reasoning (NCTM, 2009).
The Common Core. The Common Core State Standards for Mathematics is a set of K-12
mathematics standards adopted by most of the United States that defines what students should
understand and be able to do throughout their study of mathematics (National Governors
Association & Council of Chief State School Officers [NGA & CCSSO], 2010). Although the
Common Core does not describe methods of teaching mathematical concepts, the Common Core
provides a set of grade specific standards that students should meet as they become prepared for
their colleges and careers. Grounded in evidence regarding what knowledge and skills are
necessary for postsecondary success, the Common Core is important to postsecondary education
because it will provide the basis of knowledge and skills that students across America should
have upon entering postsecondary institutions (Jones & King, 2012).
Building upon years of work by NCTM and the National Research Council to define the
mathematics that students need to understand, the Common Core articulates mathematical
standards that can be implemented at the state level (NCTM, 2011). The Common Core and
NCTM share a vision of a focused curriculum and identify critical areas in mathematics through
12th grade; further, both institutions generally agree upon the types of mathematical practices that
students should be able to demonstrate (NCTM, 2011). Similar to NCTM?s (2000) Process
Standards described above, the Common Core proposed Standards for Mathematical Practices
31
which required students to be proficient with tables and graphs, to reason abstractly and
quantitatively, to construct viable arguments and evaluate the arguments of others, to solve
everyday problems, to use technology appropriately, to develop precision in communicating
mathematics, to look for patterns within problems, and to evaluate the reasonableness of their
solutions (NGA & CCSSO, 2010). Students will be prepared to enter a wide range of
postsecondary-level courses if they are proficient with the Standards for Mathematical Practices
that are listed in the Common Core (Conley et al., 2011).
A common theme in many documents advocating reform of mathematical instruction is
the need for students to develop problems solving skills in addition to computational fluency and
conceptual understanding (NCTM, 2000; NGA & CCSSO, 2010). These abilities are mutually
supportive and facilitate the learning of one other (NCTM, 2000). Teachers who use context-
based problems to introduce mathematical principles can improve students? conceptual
understanding of the mathematics by helping students 1) provide rich representations of a
problem, 2) know when to apply mathematical principles, 3) know if their solutions are
reasonable, and 4) judge the reasonableness of their solutions (Schroeder & Lester, 1989).
Students who learn the reasons behind the mathematical principles that they are taught are more
likely to remember them correctly and apply them appropriately when confronted with new
situations (Skemp, 2006). In contrast, because each of these components supports one another,
students who are unable to determine when and how to use their knowledge will find their
mathematical abilities to be fragile (NCTM, 2000). Teachers should therefore emphasize the
interrelations between conceptual understanding and computational fluency in order to help
students become more effective at problem solving (National Mathematics Advisory Panel
[NMAP], 2008).
32
Jones and King (2012) described several implications of the Common Core for
postsecondary education. First, postsecondary instructors will be able to increase the rigor of
their courses, and institutions should be able to redirect funding to credit-level mathematics
courses due to a decreased need for remedial mathematics courses. Second, because the
expectations within the Common Core are clearly articulated and upheld by postsecondary
education, students will know that meeting these expectations will produce real benefits at the
college level. Thus, students will be much more likely to meet those expectations because of the
impending real-world consequences. Third, the Common Core adopted standards that correspond
to the highest-performing states in the United States and countries around the world. Since the
academic rigor within a curriculum is the most important factor towards achieving postsecondary
success (Adelman, 1999), the coordination between K-12 and postsecondary education regarding
the effective implementation of the Common Core should lead to less remediation and higher
success rates at the college level (AEE, 2011; Jones & King, 2012).
The Equitable Nature of Reform Mathematics Pedagogy. The classroom practices
that are advocated by the National Council of Teachers of Mathematics have been identified as
equitable with respect to increasing the achievement level of developmental students. The
following paragraphs provide a brief description of what equity means, followed by a description
of common equitable practices in mathematics classrooms.
Gutierrez (2007) stated that equity means fairness instead of sameness; and at a basic
level, equity can mean the inability to predict an individual?s mathematical achievement based
solely on student characteristics such as race, gender, and ethnicity. Stenmark (1989) stated that
equity means having the same opportunities as others but also includes a support structure by
which to take advantage of those opportunities. Banks and Banks (1995) stated that equity may
33
not always mean treating differing groups the same; rather, sometimes it is necessary to treat
groups differently in order to create equal-status situations for marginalized students.
Many paths exist to develop equitable instruction (Boaler & Staples, 2008), and equitable
instruction does not necessarily need to include curricula that is designed to be culturally
sensitive by using examples of students? cultures or students? practices outside of school (Banks
& Banks, 1995; Boaler & Staples, 2008). Conceptually oriented mathematics materials that are
consistently well taught produce more equitable results for students than do procedure-oriented
curricula that are taught through a demonstration and practice approach (Banks & Banks, 1995).
Maintaining high cognitive demand, emphasizing the importance of effort over innate ability to
learn mathematics, providing clear expectations for learning practices, showing students how to
explain and justify their answers followed by requiring students to explain and justify their
answers, and encouraging students to help other students as well as ask for help themselves have
all contributed towards making instruction more equitable (Boaler & Staples, 2008).
In equity-oriented classrooms, students are encouraged to actively construct knowledge
and to learn from their peers through social interactions; further, students benefit from
cooperative learning strategies when instructors take into account status differences among
students (Banks & Banks, 1995). Instructors who incorporate cooperative learning strategies
require students to clarify their thinking through talking and writing, test their ideas against other
students, appreciate the perspectives of other students, and develop group communications skills;
thus students are encouraged to assume responsibility for their learning by expressing their
opinions and asking questions (Boylan, Bonham, & Tafari, 2005). Because many developmental
students will not be accustomed to cooperative learning activities, instructors should take care to
34
help students to become accustomed to such activities; failing to do so may produce additional
inequities (Boylan, Bonham, & Tafari, 2005; Boaler & Staples, 2008).
Recommendations for Post-secondary Mathematics Students
The Mathematical Association of America and the American Mathematical Association
of Two-Year Colleges (AMATYC) made recommendations that are specifically intended for
college-level introductory mathematics courses. The following recommendations address the
type of mathematical content that undergraduate students should learn, the development of
intellectual abilities within these courses, and the pedagogical approaches that teachers should
use when teaching introductory college-level mathematics courses.
Undergraduate mathematics curricula should develop the mathematical knowledge and
skills of students so that they may pursue and achieve their career goals (AMATYC, 2006).
By reducing the number of topics within undergraduate mathematics courses and covering the
remaining topics in greater depth, students can learn the material with greater understanding and
flexibility (AMATYC, 2006). Mathematical content that contain practical applications are
especially important for adult learner (Goldrick-Rab, 2010), and although real world problems do
not help students with procedural skills, they do help students do well on other real world
problems (NMAP, 2008).
The Mathematical Association of America?s Committee on the Undergraduate Program
in Mathematics (CUPM) made a number of recommendations for college algebra courses. They
advocated that students should become proficient with using systems of equations to model real
world situations, and they should understand the concepts of rate of change and be familiar with
linear, polynomial, exponential, and logarithmic functions (CUPM, 2011). It is also important for
students to learn how to collect data and analyze it through statistical techniques such as fitting a
35
curve to a scatter plot and using that curve to make predictions based on the trends within the
data (CUPM, 2011).
Mathematics courses should also develop students? intellectual abilities. Students should
develop their logical reasoning skills and their ability to communicate mathematical ideas in both
oral and written form (CUPM, 2011). Instructors should help students analyze and synthesize
information, and instructors should help students to work collaboratively to explore
mathematical phenomena and report their findings (CUPM, 2011). Students should be able to
engage competently and confidently in problem-solving activities. Problem solving includes the
ability to create and interpret mathematical models based on real world situations (CUPM,
2011). When faced with a problem, students should develop a personal method of attacking a
problem. For example, such a method of attack may include rereading the problem, defining
relevant variables, drawing a diagram, using appropriate methods of solution (analytic,
numerical, graphical), interpreting the appropriateness of the solution, and revising the model if
necessary (CUPM, 2011).
Instructors should emphasize conceptual understanding of mathematics when teaching
students and should provide opportunities for students to explore mathematical material (CUPM,
2011). Such emphasis on conceptual understanding is important since students enter the
classroom with preconceived notions, thereby necessitating that instructors engage students?
initial understanding and help them to make analogies between new concepts and what they
already know (Donovan, Bransford, & Pellegrino, 1999; Golfin et al., 2005). To improve
conceptual understanding, algebraic techniques should be developed in the context of solving
problems (CUPM, 2011). Additionally, technology (such as computers, calculators,
spreadsheets) can assist students in their mathematical explorations (CUPM, 2011). Instructors
36
should also incorporate student-centered instruction through small group activities and projects
(CUPM, 2011). Instructional techniques that involve personal interaction seem to benefit
students who are struggling with the material (AMATYC, 2006).
Instructors should use a variety of assessments (in addition to individual quizzes and
tests) to assess a student?s level of understanding. Listening to students, asking them appropriate
questions, and giving them opportunities to demonstrate their knowledge in a variety of ways is
an effective strategy to increase student learning (AMATYC, 2006). Group homework, projects,
presentations, activities, and quizzes can help instructors assess students? levels of understanding
(CUPM, 2011).
In summary, instruction within a postsecondary introductory mathematics course should
improve students? attitudes towards mathematics and prepare them for the mathematics they will
encounter in future courses (CUPM, 2011). Instructors can improve students? conceptual
understanding by promoting mathematical exploration through technology and group activities
(AMATYC, 2006; CUPM, 2011). These courses should also prepare students to engage in
mathematics that they might encounter in their own personal lives (CUPM, 2011).
Recommendations for Underprepared Post-secondary Mathematics Students
Recommendations have also been made by the Mathematical Association of America,
AMATYC, and the U. S. Department of Education that are specifically intended for
postsecondary remedial mathematics courses. The following recommendations address the type
of mathematical content that underprepared students should learn, the development of
intellectual abilities within these courses, and the pedagogical approaches that teachers should
use when teaching remedial mathematics courses.
37
In order to pursue successfully college-level mathematics, students need to have a solid
foundation in arithmetic, geometry, trigonometry, algebra 1 and 2, and statistics (Golfin et al.,
2005), and students should come to view the mathematics within these areas as interrelated
concepts instead of unrelated facts to be memorized (AMATYC, 2006). Solving proportions and
knowing its applications to their daily lives is a key concept that students in remedial courses
need to understand (AMATYC, 2006). Additionally, remedial mathematics courses should
minimize some algebraic topics such as factoring, radicals, and operations with rational
expressions while instead emphasizing modeling, communication, and quantitative reasoning
(AMATYC, 2006).
In addition to specific types of mathematical knowledge, students must also be able to
think critically, present sound solutions to problems using multiple representations, and apply
knowledge in new contexts (Golfin et al., 2005). Students also need to gain confidence in solving
real-world problems and build a reservoir of problem-solving strategies (AMATYC, 2006). As a
result of the course, students should develop appropriate time-management skills and study
habits, be comfortable working collaboratively, and have successful experiences using
technology to organize and analyze data, and become comfortable executing multistep problems
(AMATYC, 2006).
Mathematics instructors should create classrooms that are authentically welcoming and
supportive (Boylan, Bonham, & Tafari, 2005). Instructors can build trust in their classrooms by
taking the time to learn about students as individuals and by creating spaces where students can
learn more about themselves and their classmates (Boylan, Bonham, & Tafari, 2005). Instructors
also need to provide positive experiences for underprepared students (AMATYC, 2006).
Instructors can improve students? experiences in a course by modeling multiple problem-solving
38
approaches, engaging students actively in the learning process, and providing students with
adequate time to explore problems and to reflect upon and understand multiple approaches to
solving problems (AMATYC, 2006).
A community-centered environment in which the instructor encourages small group
discussions can increase student learning for several reasons. First, students in small group
settings are more inclined to express disbelief and challenge ideas, thus providing a need for
explicit mathematical argumentation (Golfin et al., 2005). Second, the members of the group
bring with them insights and experience that can assist in the problem solving process (Golfin et
al., 2005). Third, by encouraging students to ask questions and express their opinions,
collaborative learning encourages students to assume responsibility for their learning (Boylan,
Bonham, & Tafari, 2005).
Technology can also be an effective strategy in increasing student learning. Although
NMAP (2008) stated that no clear consensus could be reached regarding the effectiveness of
technology-based delivery methods, instructors who used calculators to emphasize problem
solving, real-world problems, or the development of critical thinking skills are finding greater
success than instructors who use calculators to emphasize basic skills (Golfin et al., 2005).
Similarly, AMATYC (2006) emphasized the importance of integrating technology into
mathematics instruction in order to help students recognize numerical and graphical patterns.
Instructors should also provide tasks during which students have successful experiences with
technology, including calculators, spreadsheets, and other computer software (Golfin et al., 2005;
AMATYC, 2006).
In summary, students in postsecondary remedial mathematics courses need to view
mathematics as a balance of analyzing problems and using appropriate techniques to arrive at
39
meaningful answers (AMATYC, 2006). When their first attempts are unsuccessful, students in
remedial mathematics courses need to be comfortable switching to alternative strategies to attack
the problem (AMATYC, 2006). By using technology and fostering a supportive community-
centered classroom environment, instructors can employ classroom activities that improve
students? confidence and problem-solving abilities (AMATYC, 2006).
The Effects of Reform-Oriented Classrooms on Student Achievement
The previous sections provide recommendations consistent with reform pedagogy which
emphasizes a balance of procedural fluency and conceptual understanding. Students actively
participate in the learning process by exploring mathematical concepts in groups with the aid of
technology and discussing with their classmates what they discovered. Students also develop
conceptual understanding by understanding the reasons behind the mathematical principles they
are taught. Further, teachers help their students develop their problem-solving abilities by
presenting mathematics in real-world contexts.
The studies in the following paragraphs provide data regarding the effectiveness of
reform-based curricula. Although different curricula are used throughout the studies, the reform-
based curricula in the following studies mostly adhered to several pedagogical practices. First,
mathematics should be presented in context and should have applications to real world situations
(Robinson & Robinson, 1998; Schoen, Hirsch, & Ziebarth, 1998; Thompson & Senk, 2001;
Thompson, 2009). Second, exploration (often small group) and experimentation are important in
helping students to understand formal theory (Robinson & Robinson, 1998; Schoen, Hirsch, &
Ziebarth, 1998; Webb, 2003; Thompson, 2009). Third, graphing calculators and other technology
are valuable tools for helping students to understand concepts (Schoen, Hirsch, & Ziebarth,
1998; Webb, 2003; Thompson, 2009). Fourth, representations (pictures, graphs, or other objects
40
that illustrate concepts) can help students to make connections in mathematics (Thompson &
Senk, 2001; Robinson & Robinson, 1998). Lastly, both routine and non-routine problems are
presented during instruction (Robinson & Robinson, 1998; Thompson & Senk, 2001; Webb,
2003).
The content in college remedial mathematics courses includes many of the same concepts
that are covered in middle school and high school mathematics courses (Bahr, 2008). Such topics
include order of operations, signed numbers, solving first and second degree equations, factoring
polynomials, and introduction to graphing (Parmer & Cutler, 2007). Because of the similarity in
mathematical content between middle and secondary courses to content in tertiary courses, I
include in the literature review studies that address the effectiveness of reform-oriented
pedagogy in middle and secondary classrooms.
Student Achievement in Middle School Reform-Oriented Classrooms. The following
studies describe the effect of implementing reform curricula or reform-oriented pedagogical
practices in middle school mathematics classrooms. Each study was conducted for at least two
years. Reys et al. (2003) compared the mathematics achievement between students who had used
reform-based curriculum for at least two years (Grades 6 and 7) and students who used
traditional curricula during that time. Three districts who had implemented reform-based
curricula?either Connected Mathematics Project (Lappan et al., 1997) or MATH Thematics
(Billstein & Williamson, 1998)?beginning in fall 1996 were compared with three individually
matched comparison districts based on prior student achievement and socioeconomic levels. The
Missouri Assessment Program (MAP) mathematics exam was used to establish a baseline by
which to identify comparison districts and was also used as the posttest to measure mathematical
41
achievement. Beginning in 1997 through 1999, the researchers compared the mathematical
achievement of eighth grade students on the MAP.
Reys et al. (2003) found the students using reform-based curricula for at least two years
during middle school performed as well as or better than students from the matched comparison
districts. Additionally, all significant differences (at least p < 0.05) on the MAP were in favor of
the students who used reform-based materials. The authors made two important comments
regarding the strength of their study. First, the authors noted that their study would have been
improved if all students used the same textbook series throughout the middle grades; but such a
scenario is rarely found in the real world. Second, because the authors had no direct information
regarding the quality of teaching within the classrooms, they assumed that considerable
variability in teaching existed across all of the schools in the study.
Mac Iver and Mac Iver (2009) examined the relationship between mathematical
achievement growth and the number of years that urban schools implemented a whole school
reform (WSR) model that included reform-based mathematics curricula. In 1999 ? 2000, 12 of
the 86 schools in the study used reform-based mathematics curricula; the remaining 74 schools
either lacked a coherent mathematics program of instruction for their WSR model, or they lacked
a WSR model altogether. The researchers used the Pennsylvania System of School Assessment
(PSSA) to measure achievement growth of 9,320 eighth grade students across 86 Philadelphia
schools. Since the PSSA is offered in both fifth and eighth grade, and since the test is vertically
equated so that it is possible to measure scale score growth over time, the researchers compared
the scores of the students from fifth grade to the scores these same students achieved in 8th
grade. Using multi-level change models, the researchers analyzed scale scores and found that
students enrolled for three years in schools that implemented a mathematics component to its
42
whole school reform gained significantly higher mathematical achievement growth over students
who attended schools that did not have a mathematics component to their whole school reform.
Mac Iver and Mac Iver?s (2009) study could have been strengthened in two areas. First, it
is important to note that their study only compared schools that had a reform-based mathematics
component to schools that did not have any mathematics component. If the researchers had also
analyzed schools that adopted a WSR model with a computation-focused curriculum, the
researchers could have determined the following relationships: 1) how does consistently
implemented reform-based mathematics curricula compare to a consistently implemented
computation-focused mathematics curricula? (For example, do students who experience three
straight years of a reform-based curriculum demonstrate better mathematical understanding than
students who experience three straight years of mathematics curriculum that is not reform-
based?), and 2) how does a consistently implemented computation-focused mathematics
curricula compare to schools that do not have a consistently-implemented mathematics curricula
at all? (For example, do students who take three straight years of a non-reform mathematics
curriculum demonstrate better mathematical understanding than students who experience
different mathematical curricula from year to year?) Second, the researchers did not take into
account the quality of instruction within the classrooms.
Thompson (2009) compared the effects of reform-based instruction and non-reform-
based instruction on students' mathematical achievement as measured by the Iowa Test of Basic
Skills (ITBS). Observers, who were trained to use an observation instrument adapted from math
and science education standards and the TIMSS survey, documented mathematical reform-based
activities and behavior and non-reform-based activities and behaviors within classrooms.
Examples of reform-based mathematical activities included: 1) students using manipulatives, 2)
43
students engaged in self-assessments, and 3) students working in pairs or small groups.
Examples of non-reform-based activities included 1) students listening to a teacher lecture and 2)
students working on pencil/paper worksheets. From 2000 to 2002, 408 observations were made
of randomly selected Oklahoma City mathematics and science classrooms (204 mathematics and
204 science) in grades 6 to 9 containing roughly 10,000 students. Using specific reform-based
and non-reform-based practices as independent variables and using student achievement as the
dependent variable, the researchers analyzed the data using stepwise multiple regression
procedures to identify variables for elimination. Thompson found that the multiple effect
contributions of manipulatives, self-assessment, and group-based projects significantly
contributed to students' mathematics achievement (3% of the variance in ITBS math, p < 0.05).
Thompson also found that none of the non-reform-based practices significantly contributed to
mathematics achievement.
The preceding studies illustrated that students who were taught using reform-based
pedagogy in the middle grades tended to do at least as well as students who received traditional
instruction; however, the effect of reform-based instruction was more pronounced for students
who had received such instruction for at least two years. Table 3 summarizes the studies that
examine the impact of reform-oriented teaching on student success in middle school mathematics
courses.
Table 3
Effects of reform-oriented instruction in middle school mathematics courses
Approach Used Researcher Results
Use of reform curricula
(Connected Mathematics and
Reys et al. (2003) Students using reform-based curricula for
two years scored as well as or better than
44
MATH Thematics) matched traditional students on the
Missouri Assessment Program
Incorporation of reform
mathematics curricula in the
mathematics component of
Whole School Reform
programs
Mac Iver & Mac
Iver (2009)
Students who were enrolled for three
years in schools that implemented a
reform mathematics component to its
whole school reform gained higher
mathematical achievement growth over
students who attended schools that did
not have a mathematics component to
their whole school reform.
Reform-oriented instruction Thompson (2009) The combination of manipulatives, self-
assessment, and group-based projects
affected 3% of variance of students?
mathematical achievement on the Iowa
Test of Basic Skills; non-reform practices
had no significant affect.
Student Achievement in Secondary Reform-Oriented Classrooms. The following
several studies describe the effects of implementing reform curricula or reform-oriented
pedagogical practices in secondary mathematics classrooms. All of the studies are comparative
and base their results on students from multiple schools.
Hirschhorn (1993) reported the effects that a reform-based curriculum, the University of
Chicago School Math Project (UCSMP), had on student achievement and attitudes towards
45
mathematics. UCSMP began in 1983 with funding from the Amoco Foundation. The goal of the
foundation was to improve school mathematics education by designing effective teaching
materials. In an ex post facto study, the researchers compared students who completed four years
of the reform curriculum to a carefully matched set of comparison students who received
traditional curricula. A total of 141 students across three sites participated in the study. In spring
1990, students took three instruments as posttests: a) the Mathematics Level 1 Achievement Test
which covered geometry and second-year algebra, b) an "Application Test" which covered
applications of arithmetic, geometry, algebra, and advanced algebra, and c) a student opinion
survey. The results showed that the students using the reform curriculum outperformed
comparison students on the Applications Test at all three sites. The reform students at sites A and
B significantly outperformed the comparison students on the Level 1 Achievement Test and the
Application Test. For the eleventh grade cohort at site C, the comparison students outperformed
the reform students on the Level 1 Achievement Test. For the 10th grade cohort at site C, the
reform students outperformed the comparison students on the Application Test. The researchers
noted that the comparison students tended to perform better on factoring topics, whereas the
reform curriculum deemphasized such topics. Additionally, The student opinion survey showed
1) very little difference in attitudes towards mathematics between UCSMP and comparison
students, 2) reform students who used a scientific calculator for at least 4 years were more likely
to agree that calculators helped them to learn mathematics, and 3) reform students were more
likely to agree that using a calculator too much makes you forget how to do arithmetic. Lastly,
the researchers stated that a conservative measure was necessary to assess the validity of the
results since they did not formally examine the quality of teaching within the classrooms.
46
Schoen, Hirsch, and Ziebarth (1998) examined the effects of the reform-oriented
curriculum, the Core-Plus Mathematics Project (Coxford et al., 1998), on student achievement
including the Iowa Tests of Educational Development and a test based on the National
Assessment of Educational Progress. Beginning in 1997, the researchers performed a
longitudinal study of high school freshman located in several states through their first year of
post-high school education. To establish a baseline and properly match students in the Core-Plus
group to students in the comparison group, the researchers administered the Ability to Do
Quantitative Thinking (ATDQT) standardized test as a pretest to all students; the ATDQT is a
subtest of the Iowa Tests of Educational Development. At the end of each year, the researchers
administered open-ended posttests that were developed by the Core-Plus Mathematics Project
evaluation team. At the beginning of the study, 2,944 Core-Plus students from 33 schools and
527 comparison students from 11 schools participated. After one year, 2,270 Core-Plus and 201
comparison students remained in the study, and after two years, 1,457 Core-Plus students and 0
comparison students remained in the study. The researchers found that Core-Plus students
demonstrated better reasoning in quantitative situations in the ATDQT than did comparison
students; Core-Plus students were better able to apply algebra and geometry concepts on
posttests; and while comparison students outperformed Core-Plus students at the end of the first
year in algebraic procedures, a significant difference no longer existed at the end of the second
year.
Thompson and Senk (2001) examined the difference in student achievement between 150
students who used UCSMP high school curricula and 156 students who used traditional
curricula. A total of 16 second-year algebra classes located in 4 schools across four states
participated in the study; the schools represented a variety of educational and socioeconomic
47
conditions. Each UCSMP class at a school had a paired non-UCSMP class at that same school
where both sets of students were of comparable mathematical abilities. The researchers
administered a pretest to measure entering algebra and geometry knowledge to determine if the
students were comparably matched. At the end of the school year the researchers administered a
posttest, the Advanced Algebra Multiple-Choice Posttest, which measured students' content
knowledge. On the entire multiple choice posttest, the researchers found that the differences in
the mean percentages between the paired classes were statistically significant for five of the eight
classes, all favoring UCSMP classes. No significant difference existed between the remaining
three pairs of classes. The authors cautioned that although the posttest was designed to be fair to
both types of classes, teacher feedback indicated that major differences in content coverage
existed among classes. However, for the Fair Test (which included items that both sets of
teachers reported that their students had opportunities to learn the needed content), UCSMP
classes again outperformed the comparison classes seven out of eight times, with four of these
differences being statistically significant in favor of the UCSMP classes; the remaining
differences were not statistically significant. On the Conservative Test which emphasized
mathematical skills, the difference in achievement between the two groups was not statistically
significant. On the Problem-Solving and Understanding Test, all but one set of differences was
statistically significantly in favor of UCSMP classes. Thus, the UCSMP curricula tended to help
students understand mathematics and did not adversely affect procedural skills.
Continuing the string of studies on UCSMP, Senk and Thompson (2006) reported a
secondary analysis of the solutions written by the second-year algebra students from Thompson
and Senk's (2001) study. The students in the analysis used either UCSMP Advanced Algebra or a
traditional second-year algebra curriculum. The researchers found that UCSMP students scored
48
higher than non-UCSMP students overall on the Problem Solving and Understanding Test and
on a multiple choice achievement test. The researchers also found that UCSMP students used
graphical and numerical strategies more frequently than students who used comparison
textbooks. Moreover, since UCSMP students left fewer questions blank, the researchers
hypothesized that the emphasis of UCSMP on multiple dimensions of understanding and
multiple solution approaches better helped students begin a problem than other curricula studied.
Researchers investigated whether enrollment in the reform-oriented curriculum, the
Interactive Mathematics Program (IMP) (Fendel et al., 1999), 1) increased the percentage of
students who took college-qualifying high school mathematics courses and 2) impacted student
achievement as measured by the Comprehensive Test of Basic Skills (CTBS) and the Scholastic
Aptitude Test (SAT) (Webb, 2003). The Interactive Mathematics Program is a four-year college-
preparatory curriculum for grades 9 - 12 that integrates a wide range of mathematics and
frequently uses technology throughout the program. IMP encourages students to use graphing
calculators and work cooperatively to solve both routine and non-routine problems. Students are
expected to experiment with examples, search for and articulate patterns, and provide conjectures
to be tested. Students are encouraged to verbalize their thinking as evidenced by classroom
activities including presentations, small-group activities, and written explanations. By using their
teachers, classmates, textbook, and other resources, students are encouraged to become
independent learners.
A total of 1,121 student transcripts from the class of 1993 across three diversely
populated California high schools were analyzed. The researchers found that a significantly
higher percentage of IMP students decided to pursue a fourth year of high school mathematics
than did students who were enrolled in more traditional mathematics courses (64% and 38%,
49
respectively). When analyzing student performance on the Comprehensive Test of Basic Skills
(CTBS), the researchers found no significant difference between IMP students and non-IMP
students. With respect to the Scholastic Aptitude Test (SAT), the researchers found that the IMP
students in one of the high schools scored significantly higher than the non-IMP from that same
school; for the other two schools, no significant difference was found. Thus, students enrolled in
the IMP from the 9th grade performed at least as well as non-IMP students when considering
SAT and CTBS scores. A weakness of the study was that students volunteered for the IMP
curricula; thus, the study?s lack of random assignment made it difficult for the researchers to
ensure that mathematical abilities of each cohort of students were similar at the beginning of the
study.
Cichon and Ellis (2003) reported that researchers gathered data from the graduating
classes of 1997, 1998, 1999 who used MATH Connections (Robinson & Robinson, 1998), a
reform-oriented curriculum. MATH Connections is a curriculum designed according to the goals
of NCTM. This curriculum blends different areas of mathematics, technology, cooperative
learning, and real-world situations to help students understand the mathematical concepts
presented (Cichon & Ellis, 2003). From the eight schools selected, the MATH students were
matched with comparison students based on their performance on the math portion of the
standardized Connecticut Mastery Test (CMT) administered in the eighth grade. Observations
were made of both MATH and non-MATH classes, and items were assessed on the scale in
which 1 = none and 5 = extensive. Both classes had a similar amount of group interaction, but
MATH classes had significantly more on-task student behavior than did comparison classes (4.4
and 3.8, respectively) and significantly more complex cognitive levels of discourse than
comparison classes (2.3 and 1.9, respectively). The researchers noted that several factors in
50
MATH classes promoted an environment of successful conceptual understanding and problem
solving experiences. These features included frequent use of multiple representations of
mathematics such as visual and symbolic representations, the routine incorporation of graphing
calculators, the focus on classroom arguments and open-ended questions that could be answered
in multiple ways, and the use of problem-solving activities that involve real-world situations.
When the researchers analyzed the performance of both groups on the Connecticut Academic
Performance Test (CAPT) (taken in the tenth grade), they found that 60% of the 558 MATH
students met or exceeded the state goal compared to 55% of the 745 comparison students; the
result was statistically significant. The researchers found no significant difference between the
CAPT mean scores of students from both groups who had matching CMT scores. However,
when the researchers used the CMT score of students as a covariate to control for incoming high
school mathematics ability, they found that MATH students significantly outperformed the
comparison students on the CAPT. Thus the students in using MATH Connections were at least
as successful in learning mathematics as the comparison students.
The previous studies demonstrate that students in reform-oriented secondary mathematics
courses often outperformed equally matched comparison students in reasoning skills and
application problems. Additionally, any differences in procedural and algebraic ability between
the two groups diminished after a couple of years. Table 4 summarizes the studies that examine
the impact of reform-based mathematics curricula on student success in secondary mathematics
courses.
51
Table 4
Effects of reform-oriented instruction in secondary school mathematics courses
Approach Used Researcher Results
University of Chicago School
Math Project (UCSMP)
Hirschhorn (1993) Student using UCSMP consistently
outperformed comparison matched
students on the Applications Test
Core-Plus Mathematics
Project (CPMP)
Schoen, Hirsch, &
Ziebarth (1998)
Compared to matched students, students
using CPMP demonstrated better
reasoning in quantitative situations, were
better able to apply algebra and geometry
concepts, and eliminated deficits in
procedural ability by the end of the
second.
University of Chicago School
Math Project (UCSMP)
Thompson & Senk
(2001)
Students using UCSMP performed as
well as or better than matched students
on problem-solving tests; no significant
difference existed between the groups in
procedural ability.
Interactive Mathematics
Program (IMP)
Webb (2003) Compared to non-IMP students, students
using IMP scored were more likely to
enroll in a fourth year of high school
mathematics
MATH Connections Cichon & Ellis Students in MATH classes had more on-
52
(2003) task student behavior, more complex
cognitive levels of discourse, and were
more likely to meet state math standards
than students in comparison classes
University of Chicago School
Math Project (UCSMP)
Senk & Thompson
(2006)
Students using UCSMP used graphical
and numerical strategies more frequently
than matched comparison students
Student Achievement in College-level Reform Oriented Classrooms. The following
studies address efforts made by post-secondary institutions to improve student performance in
college-level mathematics. Although several of the authors did not explicitly refer to reform
documents for their motivation to implement the described classroom changes, the changes
implemented in the studies often aligned with pedagogical practices advocated by reform
documents. All of the following studies are comparative studies.
Hurley, Koehn, and Ganter (1999) reported that the University of Connecticut conducted
a 5-year longitudinal study (Fall 1989 ? Spring 1994) of 579 students who took either a
traditional calculus course or an experimental computer-integrated calculus course. Although
both sections used the same text, the experimental course included the following: 1) students
participated in a computer-laboratory period for one class hour per week, 2) students engaged in
a group problem-solving session that addressed both conceptual and computational questions,
and 3) students were encouraged by instructors in both sessions to explore and analyze the
problems provided. When analyzing the students? performance on a common final exam which
included both conceptual and procedural questions, the researchers found that the experimental
53
sections outperformed the traditional sections each semester; however, the authors did not
indicate that the differences were statistically significant. The data also showed that taking the
computer-integrated calculus course was the only statistically significant factor that correlated
with persistence in technical majors among females. For males, persistence in technical majors
correlated significantly with the calculus course taken as well as Mathematics SAT score. Lastly,
students who took the computer-integrated calculus course completed significantly more (p <
0.02) post-calculus major courses than did students who took the traditional calculus course.
Lawson et al. (2002) observed six sections of Math Theory for Elementary Teachers.
Three sections were taught by instructors that were influenced by the Arizona Collaborative for
Excellence in the Preparation of Teachers (ACEPT), and the other three sections were taught by
instructors who were not influenced by ACEPT. ACEPT is a National Science Foundation-
sponsored program that attempts to improve mathematics instruction at Arizona State University
by incorporating reformed teaching methods into undergraduate mathematics and science
courses. At the beginning and end of the semester, students were administered a test that
measured computational skills, number sense, and conceptual understanding. Each instructor was
evaluated at least twice using the Reformed Teaching Observation Protocol (RTOP), a 25-
question observation instrument developed by ACEPT to measure the degree to which a
classroom?s activities align with reform pedagogy (Piburn & Sawada, 2000). The researchers
found correlations between the following pairs of items: 1) student post-test scores and the mean
instructor RTOP scores (r = 0.94, p < 0.001), 2) normalized student achievement gains and mean
instructor RTOP scores (r = 0.86, p < 0.001), and 3) student post-test number sense scores and
mean instructor RTOP scores (r = 0.92, p < 0.001). However, the researchers did not find a
relationship between student post-test performance on the computational skills test and the
54
instructors' mean RTOP scores. Unfortunately, the researchers did not state the number of
students that were involved in the study.
Ellington (2005) reported that Virginia Commonwealth University (VCU), an urban
institution with over 28,000 undergraduate and graduate students, developed a college algebra
course that attempted to focus on mathematics topics that were important to other disciplines,
develop students' abilities to work as a team and communicate quantitative ideas orally and in
writing, and emphasize the development of mathematical models and the use of technology. In
Fall 2004, the researchers compared 284 students across 8 sections of the modeling-based classes
to 989 students enrolled in 28 sections of traditionally taught skills-based classes. The
experimental students took tests that consisted of 70% modeling questions and 30% skills
questions, and they spent the majority of the class period working in groups of 2-4 students on
modeling problems with intermittent pauses for whole or partial-class discussion on issues or
skills that needed to be addressed. Additionally, graphing calculators were emphasized on a daily
basis, often to find and evaluate mathematical models.
The author found that roughly 72% of the students in the experimental group earned a
grade of A, B, or C as compared to 50% of the traditional students (p < 0.01). The DFW rates
(the percentage of students who earned a final grade of ?D?, ?F?, or ?Withdrawal? for the
course) for the experimental and traditional classes were 28% and 51%, respectively; however,
the researchers did not state if these values were statistically significant. Students in both
sections were administered ten common questions on their final exams that covered algebraic
computations and modeling applications. The experimental students scored significantly higher
than the traditional students (p < 0.001) on the 13 common final exam questions. When
comparing the students in both courses who earned a C or higher, the experimental students
55
outperformed the traditional students on the skills questions, modeling questions, and the
combined set of questions. In the subsequent mathematics courses (Spring 2005), significantly
more traditional students than experimental students earned an A, B, or C in precalculus (70%
and 56% respectively; p < 0.01). However no significant difference in ABC rates for the business
mathematics existed between the two groups (Ellington, 2005).
Several limitations regarding the results of the study should be noted. First, each
experimental instructor was assigned two teaching assistants to attend all class meetings to help
students who were having difficulty and to facilitate group activities; outside of class, the
assistants tutored students and ran help sessions before each test. The authors reported no such
advantage for the traditional students. Second, the experimental final grades included group
projects (20%) and class activities (10%) whereas the traditional final grades only included
homework and tests. Lastly, many of the experimental sections had an attendance policy that
significantly penalized students' grades for unexcused absences; the traditional sections did not
have such a policy.
Gordon (2006) reported that in Fall 1999, researchers at New York Institute of
Technology (NYIT) compared student achievement and student attitudes of 37 students enrolled
across two reform-modeling precaluclus classes to 27 students enrolled across two traditionally
taught precalculus classes. The traditional classes were lecture-based and focused on routine
algebraic manipulations whereas in the reform-modeling classes, algebraic manipulations arose
in the context of problem solving. The reform-modeling course focused on conceptual
understanding of mathematical ideas, problem-solving, and realistic applications. Additionally,
all classes used graphing calculators. When the researchers compared the students' answers on
ten common questions on the final exam (primarily procedural in nature), they found that the
56
reform students significantly (p < 0.05) outperformed the traditional students. The researchers
also conducted a student attitudinal survey through a pre-post survey design. The results showed
that the reform students expressed higher positive attitudes towards mathematics, were more
likely to view mathematics to be connected to situations beyond math courses, and were more
likely to view technology as important to learning mathematics.
The studies in the previous section demonstrate that students in college-level
mathematics courses who receive problem-oriented instruction, combined with appropriate use
of technology and cooperative learning, can perform at least as well as comparison students in
terms of pass rates and performance on examinations. Additionally, students who received
reform-oriented instruction tended to have higher positive attitudes towards mathematics.
Ellington?s (2005) study was the only study in which students who received reform-oriented
instruction performed worse in subsequent precalculus courses, even though the other students
who received reform-oriented instruction performed as well as the comparison students in the
subsequent business mathematics course. Table 5 summarizes the studies that examine the
impact of reform-oriented instruction on student success in postsecondary mathematics courses.
Table 5
Effects of reform-oriented instruction in postsecondary mathematics courses
Approach Used Researcher Results
Integration of computer
laboratory meetings involving
group work and exploration
into a Calculus course
Hurley, Koehn, &
Ganter (1999)
Compared to traditional courses,
students in the computer-integrated
course scored higher on the final
exam, and females in the course were
more likely to pursue technical
57
majors
Reform-oriented instruction in
a college-level mathematics
course for elementary teachers
Lawson et al. (2002) Significantly high correlations existed
between reform-oriented instruction
and students? post test scores,
achievement gains, and number sense
scores
Integration of problem-
oriented approach and cross-
disciplinary content into a
remedial algebra course
Erickson & Shore
(2003)
Students in the experimental course
earned higher test scores and reported
more positive attitudes than students
in the traditional remedial course
Integration of cross-
disciplinary topics, group
work, and technology in a
college-level mathematics
course
Ellington (2005) Students in the experimental group
had higher pass rates and higher
scores in the algebra course, lower
pass rates in the subsequent
Precalculus course, and no significant
difference in subsequent the business
mathematics course
Reform-oriented instruction in
a college-level Precalculus
course
Gordon (2006) Students in the reform course
demonstrated higher procedural skills
and more positive attitudes towards
mathematics
58
Student Achievement in Remedial Postsecondary Reform Oriented Classrooms
The previous studies addressed efforts to improve student performance in college-level
mathematics courses. The following studies address efforts made by post-secondary institutions
to improve student performance in remedial mathematics courses. Although several of the
authors did not explicitly refer to reform documents for their motivation to implement the
described classroom changes, the changes implemented in the studies often aligned with
pedagogical practices advocated by reform documents. With the exception of Phoenix (1990), all
of the following are comparative studies.
Phoenix (1990) examined the effect that the following classroom practices had on her
students' achievement in a remedial mathematics course: 1) student verbalization and immediate
feedback, 2) cooperative learning, 3) concept/discovery-based approach, and 4) creative
classroom activities. Students? placement into the course was based on their performance on the
college's mathematics placement test. The average score on the placement test was 15.1 with a
5.1 standard deviation (a score of 25 out of 40 is passing). At the end of the semester, 25 of the
original 30 students scored an average of 28.1 on the placement test with a standard deviation of
6.6. The researcher also reported that 16 students passed the course, 9 students failed the course,
and 5 students withdrew from the course; thus, the class had a 53% pass rate. Compared to other
sections of the course, the instructor's class outperformed 10 of the 12 other sections. Although
the pedagogical techniques reported seemed promising, the study could have been strengthened
in several areas. First, the researcher, who was also the instructor, did not address teacher effect.
The instructor may have been a significantly better teacher than the other instructors teaching the
course. Second, the researcher could have buttressed her claim that her class was taught
significantly differently from the other 12 classes by providing a scoring device that evaluated
59
the activities within her classroom. Lastly, to her credit, the researcher noted that the results of
the study were inconclusive. Future studies of this nature could be strengthened by presenting
qualitative data that demonstrated students' appreciation for the classroom activities or
quantitative data that allowed for comparisons between types of instructions between classrooms
to be made.
Erickson and Shore (2003) studied the effects of integrating a problem-oriented approach
to learning as well as cross-disciplinary content from health disciplines (such as nursing and
physical therapy) into a remedial intermediate algebra course provided at the Physical Therapist
Assistant program at Allegany College of Maryland. The faculty of the health departments
designed problems that were intended to demonstrate to students how mathematics was used in
the health disciplines. The intermediate mathematics course covered polynomials, linear and
quadratic equations, radicals, systems of equations, and graphing of functions. Students received
instruction through lecture, classroom discussion, and in-class problem solving. Although the
authors did not provide the size of the sample in the study, the authors found that the students
who were enrolled in the cross-disciplinary, problem-oriented courses yielded significantly
higher test scores and reported more positive attitudes than students in the traditional
mathematics courses.
Hooker (2011) studied the effect that collaborative learning had on students enrolled in a
remedial algebra course at a small Tribal community college. The researcher used an
experimental design in which the control group (n = 31) and the experimental group (n = 30)
were taught concurrently during Fall 2008. In the experimental group, students sat in groups of 4
? 8; each group was assigned a group leader who was a former student that the instructor had
chosen and trained. During the first three days of each week, students worked on problems and
60
assignments; on the fourth day of class, the students engaged in a special in-class workshop in
which an activity was given to each group of students. The activity was chosen to engage
students in challenging real-life applications of the content that was taught during the week. The
activity was also designed to cause students to 1) talk about the problem, 2) practice using new
vocabulary and concepts, 3) encourage students to think about different ways to apply the new
concepts, and 4) learn how to work together. The researcher found that the experimental group
had a higher pass rate than the control group (43 % and 35%, respectively), and a higher
percentage of the students in the experimental group persisted to the end of the course compared
to the students in the control group (47% and 32%, respectively). However, the author did not
state if these results were statistically significant.
The studies in the previous section demonstrate that students in postsecondary remedial
mathematics courses who received problem-oriented instruction or cooperative learning
instruction performed at least as well as comparison students in terms of pass rates and
performance on examinations. Additionally, students who received problem-oriented instruction
tended to have higher positive attitudes towards mathematics. Table 6 summarizes the studies
that examined the impact of reform-oriented instruction on student success in postsecondary
remedial mathematics courses.
Table 6
Effects of reform-oriented instruction in postsecondary remedial mathematics courses
Approach Used Researcher Results
Incorporation of student
verbalization, cooperative
learning, and concept/
Phoenix (1990) Inconclusive since the instructor, who
was also the researcher, taught only
one section and compared the results
61
discovered-based approach
into a remedial algebra course
from her section to sections taught by
other instructors
Integration of problem-
oriented approach and cross-
disciplinary content into a
remedial algebra course
Erickson & Shore
(2003)
Students in the experimental course
earned higher test scores and reported
more positive attitudes than students
in the traditional remedial course
Collaborative learning in a
remedial algebra course
Hooker (2011) Students in the collaborative learning
course had higher pass rates and
higher persistence rates to the end of
the course
Effects of Self-Efficacy on Student Achievement
Another important component in improving students? success is self-efficacy. Bandura
(1997a) described self-efficacy as the ?beliefs in one?s capabilities to organize and execute the
courses of action required to produce given attainments? (p. 3). In other words, self-efficacy
refers to a person?s confidence in their own abilities to accomplish the goals at hand. Bandura
(1997b) stated that since people try to exercise control over the events in their lives, they have a
much stronger incentive to act if they believe that their actions will be effective. Thus,
individuals with a low self-efficacy will have low aspirations, weak commitment to their goals,
and avoid difficult tasks; in contrast, individuals with high self-efficacy set high goals, sustain
strong commitment, and view difficult tasks as challenges to be mastered (Bandura, 1997b). The
following studies describe the effects of self-efficacy on student learning and present either the
direct effects or the mediating effects of self-efficacy on performance. In all of the following
62
studies, students with higher self-efficacy either demonstrated higher mathematical performance,
or they demonstrated higher performance on variables affecting mathematical performance.
Pintrich and De Groot (1990) examined the effects of self-efficacy on cognitive strategies
employed by students and on students? academic performance. ?Cognitive strategies? was
defined by the researchers as strategies that students used to learn, remember, and understand the
material; such strategies included rehearsal, elaboration, and organizational strategies. The
sample consisted of 173 seventh graders from science and English classes from a middle-class,
predominantly White Michigan city school district, and data consisted of students? responses on
the Likert-style Motivated Strategies for Learning Questionnaire and also students? performance
on class work, quizzes, tests, essays, and reports. The researchers found that higher levels of self-
efficacy were correlated with high levels of cognitive strategy use (r = 0.33, p < 0.001) and that
students with high self-efficacy were significantly more likely to use cognitive strategies than
were students with low self-efficacy (p < 0.02). However, when controlling for cognitive
engagement variables in regression analysis, self-efficacy was not significantly related to
students? academic performance. The researchers suggested that although cognitive engagement
was more directly related to academic performance, self-efficacy played a facilitative role in
relation to cognitive engagement.
Pajares and Kranzler (1995) studied 329 high school students from two Southern public
schools, roughly 79% non-Hispanic White. Students completed a mathematics self-efficacy
instrument called the Mathematics Confidence Scale (Dowling, 1978). The researchers found a
significant correlation between math self-efficacy and mathematics performance (r = 0.64, p <
0.0001). Using path analysis, the researchers found that the direct effect of self-efficacy on
mathematical performance (? = 0.349) was as strong as the direct effect of ability on
63
performance (? = 0.324). However, the high majority of students in their study demonstrated a
high level of mathematical confidence that was often not matched by their mathematical
competence, and the high school students in their sample demonstrated higher levels of
mathematical overconfidence than did college undergraduates in their previous investigations.
Pajares and Graham (1999) tracked 273 students from grade 6 through grade 8. The
sample contained roughly the same number of boys as girls and consisted of 70% non-Hispanic
Whites. The Southern suburban public middle school followed a mathematics curriculum that
was consistent with the standards of the National Council of Teachers of Mathematics. After
controlling for several variables including previous mathematics achievement, perceived value of
mathematics, self-regulation, and anxiety, the researchers used multiple regression to determine
the contribution that self-efficacy made to mathematical performance. They found that self-
efficacy accounted for a modest but statistically significant portion of the variance in
mathematical performance (r2 diff = 0.03, p < 0.05). Additionally, they found that students
tended to be biased towards overconfidence.
Pietsch, Walker, and Chapman (2003) studied the relationship between mathematics self-
efficacy and mathematical performance of 416 high school students ages 13 to 16 in Sydney,
Australia. The students came from low socioeconomic backgrounds, and 80% of the students
were from non-English-speaking backgrounds. The researchers designed a self-efficacy survey
to assess students' mathematics self-efficacy, and they assessed students' mathematical
performance by using end-of-term examinations. Using confirmatory factor analysis and
structural equation modeling techniques, the researchers found that mathematics self-efficacy
significantly impacted mathematical performance and cited the results of one model (Goodness
64
of fit index = 0.92) that demonstrated the path from self-efficacy to mathematics performance to
be 0.53 (p < 0.05).
Mousoulides and Philippou (2005) analyzed 194 sophomore pre-service teachers who
enrolled in a mathematics course during Fall 2004. Of the participants, 18% were male and 82%
were female. Using the MSLQ 26-item questionnaire from Pintrich et al. (1993), the researchers
devised a model that contained seven variables including self-efficacy, task value, and
metacognitive strategies. They found that the goodness-of-fit index of their model was good in
relation to typical standards (Comparative Fit Index = 0.923, chi2 = 451, df = 303, RMSEA =
0.056). The researchers found that self-efficacy had a causal effect of 0.33 on achievement.
Thus, the researchers concluded that self-efficacy was a strong predictor of academic
performance in mathematics and that their study corroborated the study of Pintrich and De Groot
(1990).
Kitsantas, Cheema, and Ware (2011) analyzed data from the 2003 Program for
International Student Assessment (PISA) and school questionnaires from NCES (2003). Based
on the mathematics literacy of 3,776 15-year-olds enrolled in grades 9, 10, and 11 across 221
schools, the researchers found a significant correlation between mathematics self-efficacy and
mathematics achievement (r = 0.54, p < 0.001). Using multiple regression analysis, the
researchers found that mathematics self-efficacy accounted for 20% of the total variation in
mathematics achievement (p < 0.001) after controlling for gender, race, relative time spent on
mathematics homework, and homework support. The researchers concluded that educators
should help their students feel efficacious in using the mathematics to which they have been
exposed. Although the researchers controlled for several variables, they did not control for
students? prior mathematics achievement or mathematical abilities.
65
The previous studies demonstrate that self-efficacy is positively related to student
engagement and student performance. A major finding in many self-efficacy studies is that after
controlling for students? previous performance, students? beliefs in their abilities strongly predict
their mathematical performance (Wigfield & Eccles, 2002). Further, the confidence that students
possess in their own mathematical abilities helps to determine how they use the knowledge and
skills that they possess (Pajares & Kranzler, 1995). Because high mathematics self-efficacy helps
students to possess greater interest in and perseverance towards solving mathematical problems
(Pajares & Kranzler, 1995; Olani et al., 2011), educators should help all students to feel
efficacious in handling the mathematics to which they have been exposed (Kitsantas, Cheema, &
Ware, 2011). Educators can help to develop mathematical self-efficacy in their students by
developing a learning environment in which students? ideas are valued and respected and in
which students develop ownership of their ideas. Such an environment can develop positive
dispositions in students towards mathematics which in turn encourages students to engage in
mathematical reasoning and thus acquire conceptual understanding (Mueller, Yankelewitz, &
Maher, 2011). Table 7 summarizes the studies that examined the effect that efficacy had on
student performance.
Table 7
Effects of self-efficacy on student performance
Approach Used Researcher Results
Examined effects of self-
efficacy on cognitive
strategies and academic
performance on middle school
Pintrich & De Groot
(1990)
Higher levels of self-efficacy were
correlated with increased use of as
well as higher levels of cognitive
strategy use
66
students in science and
English courses
Examined relationship
between mathematics self-
efficacy and mathematical
performance on secondary
students
Pajares & Kranzler
(1995)
A significant correlation existed
between mathematics self-efficacy
and mathematical performance
Examined relationship
between mathematics self-
efficacy and mathematical
performance on middle school
students
Pajares & Graham
(1999)
Mathematics self-efficacy accounted
for 3% of the variance in
mathematical performance
Examined relationship
between mathematics self-
efficacy and mathematical
performance on secondary
students
Pietsch, Walker, &
Chapman (2003)
Mathematics self-efficacy
significantly affected mathematical
performance
Examined impact of
mathematics self-efficacy on
preservice teachers?
mathematical performance
Mousoulides &
Philippou (2005)
Mathematics self-efficacy was a
strong predictor of mathematical
performance
Compared secondary students?
responses on questionnaires to
Kitsantas, Cheema, &
Ware (2011)
Mathematics self-efficacy accounted
for 20% of variation in mathematics
67
their mathematical literacy achievement
Synthesis of Relevant Studies
The first part of the literature review describes characteristics of students in remedial
courses, the effectiveness of remedial mathematics courses in improving the chances of
academic success for underprepared students, and the effectiveness of computer-based
instruction and decreased length in developmental sequences have had in improving student
achievement in remedial mathematics courses. The second part of the literature review describes
the major tenets of the reform mathematics movement for K ? 12 mathematics classrooms as
voiced by NCTM, recommendations that specifically address post-secondary mathematics
courses for prepared and underprepared students, studies that describe the effects that reform-
oriented curricula and pedagogical practice have had on student achievement, and studies that
describe the effect that mathematics self-efficacy has had on improving student achievement.
Based on the literature, attempts at improving remedial mathematics courses through
computer-based instruction were mixed, and the implementation of such instruction required
considerable financial and logistical support. Decreasing the number of required remedial
mathematics courses showed promise, but the studies involved drastic redesigns in the
mathematics department?s developmental program (Squires, Faulkner, & Hite, 2009; Bassett &
Frost, 2010) and focused primarily on non-STEM students (Byrk, 2012). However, students in
secondary and postsecondary mathematics reform-oriented courses tended to do at least as well
as students in traditional lecture courses in terms of pass rates (Hooker, 2011), overall test scores
(Erickson & Shore, 2003), procedural ability (Reys et al., 2003; Thompson, 2009), and problem-
solving ability (Hirschhorn, 1993; Schoen, Hirsch, & Ziebarth, 1998; Thompson & Senk, 2001).
68
In conclusion, based on 1) the fact that the topics covered in tertiary remedial
mathematics courses are equivalent to those covered in middle and secondary school
mathematics courses and 2) the successes described by the studies that incorporated reform-
oriented curricular and pedagogical changes within middle and secondary classrooms, I
conducted a quasi-experimental design in which I taught one set of students using primarily
traditional didactic lecture techniques and another set of students using pedagogical practices that
more closely align with pedagogical practices advocated by reform documents. My study
examined the effectiveness of reform-based pedagogy in terms of pass rates, procedural ability,
and problem solving ability. Additionally, since studies have shown that students? beliefs in their
own mathematical abilities can influence their mathematical performance (Wigfield & Eccles,
2002), my study also examined the effect that reform-based pedagogy has on students?
mathematical self-efficacy. I also provided documentation regarding the daily classroom
practices within the study (such as a series of RTOP scores), and I gathered qualitative data from
the students in the reform-oriented class to learn how they felt about key aspects of reform-
oriented instruction.
Theoretical Framework
I hold to the theoretical perspective of constructivism, a perspective that has significantly
influenced mathematics education in the past several decades (Ernest, 1997). Constructivism is
the belief that individuals create their own knowledge by modifying their existing concepts when
presented with new evidence and experiences (Annetta & Dotger, 2006). Constructivism views
learning as an active process since individuals wrestle to reconcile their perceptions of the world
with their existing knowledge framework (Anderson et al., 1994).
69
Ernest (1997) cautioned that researchers who adhere to the constructivist epistemology
should do so only with caution and humility. Since knowledge is attained through individual and
social experiences, knowledge constructed by individuals may not align with that of an objective
reality (Cooner, 2005). Additionally, since constructivism highlights the subjective
interrelationship between the researcher and the participants, as well as the coconstruction of
meaning between the two groups, the researcher is not considered an objective observer and
must therefore acknowledge his values to his readers (Mills, Bonner, & Francis, 2006).
Lerman (1989) stated that constructivism has been described as consisting of two
hypotheses: ?1) knowledge is actively constructed by the cognizing subject, not passively
received from the environment, and 2) coming to know is an adaptive process that organizes
one?s experiential world; it does not discover an independent, pre-existing world outside the
mind of the knower? (p. 1). Lerman (1989) further stated that researchers who accept only the
first hypothesis are considered ?weak? constructivists, and researchers who accept both
hypotheses are considered ?radical? constructivists. With respect to a research perspective, I
consider myself a weak constructivist since I assume that individuals construct their own
knowledge and that an objective reality does exist.
Ernest (1997) described several important implications of a constructivist framework for
mathematics education research. Researchers need to 1) attend to their own beliefs about
knowledge, 2) attend to the constructs that participants bring with them into the study 3) attend
to the social contexts of learning, 4) carefully use methodological techniques since truth can be
acquired in more than one manner, 5) attend to the negotiation and shared meaning of the
knowledge constructed by the participant, and 6) question the learner?s subjective knowledge.
70
Thus, a constructivist researcher needs to consider participants as whole persons in light of the
complex social context that exist among the participant, teacher, and researcher.
Research Questions
In an effort to improve my understanding regarding the effectiveness of reform-based
pedagogical practices in the context of postsecondary remedial mathematics courses at a four-
year university, I conducted a mixed methods study that examined both quantitative and
qualitative data. The quantitative data helped me verify empirically the success of each
treatment, and the qualitative data helped me understand the strengths and weaknesses of reform-
based pedagogy from students? perspectives. My study was guided by the following broad
research question: Is teaching remedial mathematics in a reform-oriented manner beneficial to
university students? Five subquestions were addressed in my study as follow:
1. Is there a significant difference in the pass rates in the remedial mathematics courses
between university students who receive instruction consistent with reform pedagogy
versus university students who receive instruction through traditional didactic lecture
methods?
2. Is there a significant difference in mathematical procedural ability between university
students who receive instruction consistent with reform pedagogy versus university
students who receive instruction through traditional didactic lecture methods?
3. Is there a significant difference in mathematical problem solving ability between
university students who receive instruction consistent with reform pedagogy versus
university students who receive instruction through traditional didactic lecture methods?
4. Does the self-efficacy of university students in the reform classes improve as a result
of instruction received in the reform classes?
71
5. What views about reform instruction will university students who are enrolled in a
reform-oriented remedial mathematics course express upon completing one semester of
reform-oriented mathematics instruction?
As outlined in the following chapter, the first four subquestions were answered quantitatively
and addressed pass rates, procedural ability, application ability, and change in mathematical self
efficacy. The fifth subquestion was answered qualitatively and addressed students? perceptions
of the reform-oriented course.
72
CHAPTER 3: METHODOLOGY
This mixed-methods study was designed to gather knowledge about the effectiveness of
reform-oriented instruction in postsecondary remedial mathematics courses. This chapter will
first present the design of the study followed by the context of the study. Next, the
instrumentation and data analysis plan will be discussed. Last, the procedure describing the
treatments that were implemented in the study will be presented.
Design
Creswell (2007) stated that the goals of research influence the approaches that are used in
research. The broad goal of this study was to determine whether or not reform-oriented
instruction was an effective means to teach mathematics. Due to my constructivist theoretical
framework, students? perceptions regarding the effectiveness of the treatments were important in
ascertaining the effectiveness of reform-oriented instruction (Ernest, 1997). This theoretical
perspective therefore necessitated that I use a mixed methods design for this study. The quasi-
experimental portion of the study attempted to determine the effect that reform-oriented
instruction had on the following student achievement outcomes: procedural abilities, application
abilities, pass rates, and change in mathematics self-efficacy. The success of the reform-oriented
instruction was viewed in contrast to the success of the didactic lecture instruction. The study
was quasi-experimental because the students were not randomly selected; instead, students
selected their course according to what best accommodated their schedules. However, no policies
existed that systematically placed students into one particular class or the other, and students did
not know until the first day of class which treatment they would receive.
A quasi-experimental design was chosen in order to establish a cause-and-effect
relationship between the treatment and students? success in the course, and any results were
73
controlled for variables that could influence the results (Gravetter & Wallnau, 2004). This quasi-
experimental design was an appropriate choice for this study since 1) it is a commonly used
method to discover generalizations about phenomena, 2) it attempts to validate empirically the
relationships between teaching practices and student learning, and 3) a successful experiment can
provide replicable and objective generalizations (Ernest, 1997; Carnine & Gersten, 2000). In the
experiment, the didactic lecture course was defined as the control group, and the reform-oriented
course was defined as the experimental group. Covariates were used in the study to temper
results based on significant differences in demographic between the two groups.
However, focusing only on the quantitative aspects of a study can neglect important
qualities that are worth examining (Miles & Huberman, 1994). I also wanted to understand
students? views of reform-oriented instruction. Specifically, to what extent and in what areas did
students perceive that the reform-oriented instruction benefited them? Therefore, I incorporated
a qualitative component into the study which would allow me to understand the perspectives of
the students who received reform-oriented instruction (Merriam, 2001); students? responses on
anonymous end-of-course surveys served as the basis for this analysis. The qualitative portion of
the study examined the views that students in the reform-oriented course had on reform-oriented
instruction; these views were solicited from students through anonymous end-of-course surveys.
Context
The South East University (SEU) (a pseudonym) at which this study took place offers a
range of undergraduate degrees and graduate degrees in the schools of Business, Liberal Arts,
Education, Nursing, and Sciences. In Fall 2011, 5,305 students were enrolled at SEU. During
the 2011 ? 2012 school year, a total of 874 students graduated from SEU, with 63% and 37% of
those students earning a bachelor?s degree and master?s degree, respectively. With respect to
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ethnicity, approximately 55% of the student population was White, 31% Black, 2% Asian, 2%
Hispanic, and 10% other. Additionally, 37% of the students were male, and 63% were female.
Since SEU also encourages part-time studies, 38% of its students were part-time. Lastly, the
average ACT score for entering freshmen at SEU was 22.0.
The remedial course sequence for mathematics at SEU consisted of Math 0700
(Elementary Algebra) and Math 0800 (Intermediate Algebra), followed by credit-level courses
such as Finite Mathematics and Precalculus. In order to take Math 0800, a student must either
pass Math 0700 (the preceding remedial mathematics course) or place directly into Math 0800 by
taking a computer placement test generated by computer software purchased by SEU?s
mathematics department. Traditionally, Math 0800 (Intermediate Algebra) at SEU was taught in
the following sequence: 1) Techniques for factoring, 2) Rational expressions and equations, 3)
Graphing quadratic, square root, absolute value, and linear functions and performing operations
with functions, 4) Simplifying radical expressions and solving radical equations, and 5) Solving
and graphing quadratic equations. During the 2010 ? 2011 school year, 606 students and 487
students enrolled in Math 0700 and Math 0800, respectively. The pass rates for Math 0700 and
Math 0800 were 51.8% and 46.6%, respectively. Collectively, 49.5% of the 1,093 students who
enrolled in remedial mathematics courses passed their remedial courses.
Dr. Jones (a pseudonym), head of the SEU mathematics department, agreed that an
experimental section using alternative instructional methods could be offered on the condition
that the experimental students would receive instruction that was on essentially the same level?
in terms of difficulty level and type of problems?as the students in the traditional course. The
remedial mathematics tests were designed by the SEU mathematics department and stressed
primarily algebraic manipulations with one to two application problems on each test. However, I
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had been given permission to modify the tests slightly by adding or removing questions as I
deemed necessary. I also received permission from Dr. Jones to modify the grading scales of the
experimental and control classes to allow for homework assignments and class participation
grades, as well as to remove from the course the concepts of completing the square and graphing
of circles.
Students in remedial classes at SEU were required to attend a Math Lab once per week
in addition to the classroom lectures, during which time students used a computer-based format
to work on homework and quizzes. Each Math Lab session lasted between one to two hours in
duration. Students in remedial courses (including those in the Control and Experimental groups)
were required to pass the lab in order to pass their course, which was based on their attendance,
homework, and quiz averages within the Math Lab.
Before proceeding with the study, I sought permission from the Institutional Research
Board (IRB) to conduct my study. (Consent forms are located in Appendix A.) I obtained
permission from the Institutional Research Boards of both Auburn University and SEU. Students
were given the option of ?opting into? the study by permitting me to use their data in the study.
My Personal Background
Since I was the instructor for both courses in this study, it is important to consider my
teaching background. I earned a Bachelor?s degree in Education, double majoring in both
Secondary Mathematics and General Science, after which time I immediately pursued a Master?s
degree in Mathematics Education. Upon completing my Master?s degree, I taught the spectrum
of secondary mathematics and science courses for three years in a high school before obtaining a
position managing a mathematics and sciences tutoring facility at the university in which the
present research study was conducted. I also served as an adjunct for the mathematics
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department at this university and taught many freshman level mathematics courses, including a
substantial number of remedial mathematics courses.
I experienced traditional lecture instruction in my primary, secondary, and most of my
postsecondary mathematics courses. Upon earning my undergraduate degree, I entered the
teaching arena and taught in the same manner that I was taught: traditional lecture instruction.
When I entered graduate school, I was exposed to the pedagogy advocated by reform
mathematics organizations; however, I was hesitant to modify my perspectives until I could
experience an impetus to justify such a modification. I perceived the lack of student performance
to be largely due to a lack of student effort. Then one day during class a professor made the
comment, ?In the end, teachers do not have any control over what students do outside the
classroom. All the teacher can control is what happens inside the classroom.? This statement
helped to change my focus from the deficiencies of my students to the deficiencies of the course
and my teaching style.
Ironically, it was during my second semester of graduate school that I became
disheartened by the abysmal performance of students in my remedial mathematics courses. I felt
that I had maximized the benefits of traditional instruction, and yet the students in my remedial
mathematics classes were still failing at unacceptable rates. I was no longer satisfied with my
current way of teaching, and I was prepared to consider different methods of teaching
mathematics. Through the pedagogical courses in my graduate program, a reform-oriented
graduate mathematics course that taught me mathematics using reform pedagogy, and extended
feedback from colleagues who were trained to use reform pedagogy, I gradually became
comfortable teaching in a reform-oriented manner and even developed a preference for that style
of teaching.
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I began modifying the Math 0800 course in order to determine if it were feasible to teach
such an intensely procedurally-oriented course in a reform-based manner. Including my first
pilot course in Summer 2010, I conducted three pilot courses over several semesters and found
each course to be fairly successful in terms of students? pass rates and general student feedback.
By the time I taught the reform-based course for this study in Spring 2012, I had become
comfortable teaching the Math 0800 course in a reform-based manner. I had also earlier taught
this course numerous times using a more tradition, lecture-based manner.
Description of Sample
Students in the study chose to be in their respective courses based on what best fit their
schedule. No policy was in place that systematically placed a disproportionate number of
students into one class or the other. The reform-oriented course was taught in Spring 2012, and
the traditional lecture-based course was taught in Fall 2012.
Students in this study were recruited from their respective classes within the first few
class meetings of each course. On the third class meeting, a representative from the SEU?s IRB
board described the nature of the study to the students and gave them the opportunity to
participate in the study. I (the instructor) was not present during this interaction. Students elected
to participate in the study by filling out an ?informed consent form? and a ?grade release form?
(see Appendix A). The IRB representative collected the forms and kept them confidential from
me until the final grades had been distributed at the end of the course.
Surveys were used to collect demographic information from students in the sample
regarding their age, race, gender, prior mathematical knowledge, number of hours employed
each week, and number of credit hours attempted for the current semester (see Appendix B).
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Table 8 summarizes the demographics data for the sample. The differences between the two
groups will be analyzed later when I present the results of the study.
Table 8
Demographics of sample
Overall
(n = 29)
Treatment
(n = 18)
Control
(n = 11)
Mean (SD) Mean (SD) Mean (SD)
Age 21.8 (5.6) 22.4 (6.5) 20.7 (3.7)
Prior Mathematical Knowledge 24.7 (6.0) 26.1 (6.2) 22.6 (5.2)
Hours of Employment per week 10.8 (14.1) 11.7 (15.5) 9.5 (12.0)
Attempted Number of Credit Hours 14.0 (2.6) 14.8 ( 2.2) 12.8 (2.8)
Race n (%) n (%) n (%)
Black 11 (37.9) 4 (22.2) 7 (63.6)
White 18 (62.1) 14 (77.8) 4 (36.4)
Hispanic 0 (0.0) 0 (0.0) 0 (0.0)
Native American 0 (0.0) 0 (0.0) 0 (0.0)
Asian/Pacific Islander 0 (0.0) 0 (0.0) 0 (0.0)
Other 0 (0.0) 0 (0.0) 0 (0.0)
Gender
Male 9 (31.0) 6 (33.3) 3 (27.3)
Female 20 (69.0) 12 (66.7) 8 (72.7)
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Two students who agreed to participate were removed from the study. One student was
removed from the Control group because she only attended the first two days of class, and one
student was removed from the Experimental group because the student missed the first two
weeks as well as the last four weeks of the course. Removing the students from the study was
appropriate because of the lack of exposure to the treatments in their respective groups.
Instrumentation
Various levels of measures were employed in this study. First, measures were used to
address the primary research questions about pass rates, procedural skills, application skills, and
change in mathematical self-efficacy. Second, the Reformed Teaching Observation Protocol and
the use of an additional test grader were used as measures to establish that the study maintained
an acceptable level of validity and reliability. Third, covariates acted as measures to temper
differences in class data based on demographic differences between groups.
Dependent Measures
A lack of consensus appears to exist regarding the appropriate metrics that should be used
in evaluating the effectiveness of tertiary developmental mathematics programs, perhaps in part
because there exists a lack of consensus regarding the ultimate role of these programs. The issue
that emerges is whether developmental courses should aim to ensure that students who complete
the program attain a high level of mathematics competency, or whether such programs should
aim to ensure that larger numbers of students complete the course at a slightly lower, yet
acceptable, level of competency (Golfin et al., 2005). Some researchers examined the pass rates
(typically a C or higher), average exam scores, or final class GPA to assess the level of
mathematical competency gained by the students (Phoenix, 1990; Squires, Faulkner, & Hite,
2009). However, researchers may report these results at the expense of higher withdrawal rates
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(Golfin et al., 2005). On the other hand, some researchers are more concerned with pass rates
than with content mastery because their concern is whether an instructional approach can help
larger numbers of underprepared students succeed in basic skills instruction (Golfin et al., 2005).
My study addressed both the pass rates of students and their level of content mastery in
Math 0800. Content mastery was divided into two parts: procedural skills and problem-solving
abilities. Procedural skills included students? abilities to simplify algebraic expressions or solve
equations without any type of real-world or situation-based context; for example, one problem
might be ?Solve the following equation for x: x2 + 2x = 1.? Problem solving skills included
students? abilities to solve situation-based problems by using a given equation or by devising
their own method to solve the problem if no equation is given (see Appendix C). Additionally, I
examined the mathematics self-efficacy of students in both classes since it is an important
predictor in mathematics problem solving (Pajares & Kranzler, 1995). Lastly, the study collected
data regarding students? views of the pedagogical practices used during their respective courses.
Pass rates. This study examined the difference in pass/fail rates between the two groups
of students. For each class, the number of students that initially ?enrolled? in the class was
defined as those who completed at least one test, and students who withdrew from the course or
failed due to excessive absences were grouped with other students who failed but regularly
attended class.
Procedural skills. This study examined the effect that the treatments had on students?
procedural skills. Math 0800 has five regular tests consisting entirely of short-answer questions,
and each test consists primarily of procedural questions. Students? performance on the
procedural questions was used to determine if a significant difference in procedural skills existed
between the two classes. A rubric was developed for each test to aid in the consistency of the
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grading. Emphasis in the rubric was placed on students? demonstrating understanding of key
concepts, and arithmetical mistakes were not severely punished. An example of a procedural
problem and the corresponding grading rubric is provided in Figure 1.
Figure 1: A sample procedural problem with corresponding grading rubric
Application skills. This study examined the effect that the treatments had on students?
application skills. Each of the five regular tests in Math 0800 contained one to two short-answer
application problems. Students? performances on the application questions were used to
determine if a significant difference in application skills existed between the two classes. A
rubric was also developed to aid in the grading of the application problems. Emphasis was placed
on students? demonstrating understanding of key concepts, and arithmetic mistakes were not
severely punished. See Figure 2 for an example of an application problem and its corresponding
rubric.
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Mathematical self-efficacy. Students? mathematical self-efficacy was defined by their
responses on a survey. I used Midgley et al.?s (2000) five-question, Likert-scale Mathematics
Self-Efficacy Survey. The alpha for the survey was 0.78. See Appendix B for copy of the survey.
Perspectives on instruction. The students in the control and the experimental groups
were given anonymous surveys at the end of the course to solicit their likes and dislikes
regarding the teaching styles employed throughout their respective courses. The experimental
group had three additional questions on their survey than were on the control group?s survey.
These extra questions solicited students? perspectives regarding the incorporation of three key
reform practices into daily instruction: group work, student presentations, and graphing
calculators. See Appendix B for a copy of the surveys.
Validity and Reliability
The results of my study were based on two key items: 1) how well I graded my students?
tests and 2) how well I maintained fidelity to the intended treatments (traditional lecture
instruction vs. reform-oriented instruction). Since objectivity and replicability are significant
components of quantitative research (Cohen, Manion, & Morrison, 2007), I arranged for an
Figure 2: A sample application problem with corresponding grading rubric
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outside grader to confirm the accuracy of my test grading. Additionally, two colleagues in the
field of education performed multiple classroom observations to document the extent to which I
maintained fidelity to the intended treatments. The following sections explain these procedures
in more detail.
Procedural and application scores. I wanted to establish inter-rater reliability in order
to support the validity of the procedural and application scores earned by the students. For the
five free-response tests that were given each semester, I met with a colleague in mathematics
education to grade a portion of the tests after each set of tests was administered. Specifically, at
each meeting I graded six tests, and my colleague graded the same six tests. My colleague and I
used the same grading rubric, and any significant differences in test scores were analyzed and
resolved. The relationship between the researcher?s and colleague? procedural scores as well as
the application scores were intended to maintain a Pearson correlation of at least 0.8.
Reformed Teaching Observation Protocol. The Reformed Teaching Observation
Protocol (RTOP) was used to corroborate the claim that the two types of instruction were
significantly different from one another. The RTOP is a 25-item observation protocol that was
devised by Piburn and associates (2000) through the Arizona Collaborative for Excellence in the
Preparation of Teachers (ACEPT) to assess the level of reformed teaching that is present
mathematics and science lessons (see Appendix D). I arranged two paired lessons (lessons in
which I taught the same set of concepts to both classes) that were observed by colleagues in the
field of education. The selected lessons were representative of the instruction that each group of
students received. The RTOP was chosen because it aligned with reform pedagogy. Additionally,
its creators designed the instrument to be easy to administer and appropriate for K-20
mathematics and science classrooms (Piburn & Sawada, 2000).
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Covariates
In order to determine if the Control and Experimental groups were comparable, students
in both courses were given surveys at the beginning of the semester in which they provided
demographic data regarding Age, Race, Gender, Prior Mathematical Knowledge, Number of
Attempted Credit Hours, and Number of Hours of Employment (see Appendix B).
Establishing prior mathematical knowledge is important in comparative studies (Senk &
Thompson, 2003); therefore, prior mathematical knowledge was assessed by giving the final
exam from Math 0700 (the previous math course) as a pretest on the first day of class. Similarly,
data regarding students? gender, race, age, employment intensity, and course load were collected
due to their potential impact on students? mathematical achievement (Hagedorn et al., 1999;
Bahr, 2008).
Procedure
This study was quasi-experimental in that students were not randomly assigned to the
control and experimental treatments. Students did not know before the first day of class which
section would receive the experimental treatment. Soon after students were notified of the study,
one student in the experimental course transferred to a different section of the course. Both the
experimental and control sections were offered at 8 a.m. on Mondays and Wednesdays.
Both sets of students used the same textbook and attended Math Lab once per week in
addition to the classroom lectures, during which time students used a computer-based format to
work on homework and quizzes. Students were required to pass the lab in order to pass the
course. Students passed the Math Lab based on their attendance, homework, and quiz averages
within the Math Lab. Because a few topics were removed from the experimental course (and thus
the traditional course), accommodations were made for students in my classes.
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The amount of material that was covered as well as the manner in which the students
were graded was the same between the two groups. In other words, both groups covered the
same material, took the same tests, and were graded the same way. Additionally, both classes
had an attendance policy (unlike the study presented in Ellington [2005]). Due to an
administrative policy at SEU that limited SEU staff members to teach one course per semester, I
taught the experimental course in Spring 2012 and the control course in Fall 2012. An advantage
to my teaching both classes was that it prevented ?teacher effect? from becoming a factor in the
study.
Control Group
The Control group received traditional didactic instruction. In other words, I spent
roughly 95% of class time explaining to the students the concepts, with the remaining time filled
with students asking questions. Instruction proceeded in the following manner: 1) introduce the
concept, 2) explain the concept in abstract terms without any realistic context, and 3) explain
how the methods developed from the abstract presentation can be used to solve application
problems containing a real-world context and explain how to solve these problems using the
techniques currently in discussion. According to Schroeder and Lester (1989), this approach to
teaching could be called teaching for problem solving since students would first be shown how
to perform the procedural skills and then shown how to use those skills to solve both routine and
non-routine problems.
Technology was not used to reinforce or explain mathematical concepts. (Currently, the
use of graphing calculators is discouraged, usually prohibited, in Math 0800 courses at SEU.)
Students sat in individual desks and did not work in groups during class. When students asked if
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something were correct or incorrect, I answered their questions to their satisfaction, but I did not
probe their understanding to help them figure out the answer to their own question.
Experimental Group Treatment
The Experimental group received instruction slanted toward reform pedagogy as
illustrated by the NCTM (2000, 2009), AMATYC (1995, 2006), and CUPM (1998, 2011)
documents. Thus, I provided students opportunities to understand mathematical concepts on
their own or with the help of their classmates using group work. I provided these opportunities
by carefully developing the mathematical concepts to be understood either through real-life
applications or through mathematical scenarios that encouraged mathematical exploration that in
the end helped them to understand the mathematical principles in question before their
classmates formally presented their findings to the class. Schroeder and Lester (1989) referred to
this paradigm as teaching via problem solving, the process of introducing reasonable problem
situations that embody mathematical concepts and then developing mathematical techniques in
response to those problems. Additionally, I gave students opportunities to explain mathematical
principles to the rest of the class before I formally explained the concepts to the class.
Students sat at tables that fostered interaction and discussed their findings to the
questions posed to them. After students had time to explore the problems and discuss their
findings with their classmates, students were asked to present their work to the rest of the class.
As the students presented to the class, the other students were expected to critique the presented
information to determine its accuracy; thus, students were encouraged to engage in respectful
constructive criticism in an intellectually safe environment. During group work, when a student
asked me if a particular approach or answer were correct, my default response was ?What do you
think??, ?How could you check your answer??, or ?What do your classmates think?? I avoided
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directly answering the question and instead guided the student in a direction to figure out for
himself if the answer or approach were correct. In other words, if the approach or answer were
an incorrect one, I guided the student in a direction that illustrated to him or her that something
was amiss. As described by Pines and West (1986), once students reflected upon the
compatibility of their conceptions and experiences, they would be much more likely to accept
formal theories as their own. If the approach were correct, then the students reinforced their
understanding as they discussed their findings with their classmates, or they devised a way to
check the reasonableness of their solutions.
Student presentations to the class were standard practice. Having students present to the
class reinforced what the students had learned, helped other students understand the
mathematical concept, and helped students to better understand the material by justifying to their
classmates the reasoning behind their solution. Students used the document camera to present
their solution to the class since the document camera can save a significant amount of class time
by removing the need for students to recreate their solutions as they presented to the class.
Graded homework differed between the experimental and control sections in terms of the
type of homework assigned; however, the amount of time necessary to complete the homework
assignments was roughly the same for both classes. Both sections were given suggested
problems within the text that would reinforce procedural skills. However, the control group was
assigned graded homework based on exercises within the book that primarily emphasized
procedural skills. The experimental group was assigned graded homework that addressed
conceptual understanding. These conceptually oriented problems required students to relate the
mathematics to realistic applications, produce several forms of justification including tables and
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graphs, explore concepts that commonly act as stumbling blocks, and articulate clearly solutions
and the meaning of solutions.
In the experimental section, students were encouraged to use tables, graphs, and algebraic
approaches to understand mathematical concepts. These students also used graphing calculators
to understand solutions to linear, quadratic, and radical equations. Thus, students would not be
completely reliant on algebraic techniques to solve these types of problems. Rather, they would
be able to quickly construct a picture to test the reasonableness of their answers, or they could
use the graph to prompt them in the right direction. Graphing calculators were supplied to
students during class, but the students were responsible for obtaining graphing calculators for use
outside the classroom. To help students obtain access to graphing calculators, students were
encouraged to use the graphing calculators available in a nearby tutoring facility, and they were
shown how to access online graphing calculators. Table 9 summarizes the key differences
between the traditional course and the reform-oriented course. Additionally, see Appendix E to
view two sets of paired lesson plans that demonstrate the difference between traditional and
reform-oriented instruction.
Table 9
Summary of differences between traditional and reform-oriented instruction
Traditional Reform-oriented
The teacher is the sole dispenser of
knowledge and serves as a ?sage on the
stage?
The students regularly present their knowledge
and findings to the class; the teacher serves as a
?guide on the side? (NCTM, 2000; White-Clark,
DiCarlo, & Gilchriest, 2008)
Direct lecture is extensively used Direct lecture is kept to a minimum (Boylan,
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Bonham, & Tafari, 2005)
Students are passive learners Students are active learners (MCCEO, 2006;
AMATYC, 2006)
Classroom discourse consists primarily of
teacher-to-student discourse
Classroom discourse consists significantly of
student-to-student and student-to-teacher
discourse (NCTM, 1991)
Socratic questioning is not employed Socratic questioning is significantly employed
(Vosniadou & Brewer, 1987)
Student exploration and experimentation
are not encouraged before formal theorems
are presented
Student exploration and experimentation are
encouraged before formal theorems are presented
(AMATYC, 2006; Thompson, 2009)
The teacher values the most efficient means
of solving a problem
The teacher values multiple problem-solving
approaches; efficiency is a secondary concern
(AMATYC, 2006)
Algebraic techniques are presented as the
primary means of solving problems
Pictures, tables, and graphs are emphasized in
addition to algebraic techniques in order to help
students improve conceptual understanding and
solve problems (NCTM, 2000; AMATYC, 2006).
Students master algebraic techniques
before learning how to apply such
techniques to story problems
Story problems act as vehicles in which to
introduce mathematical concepts (Schroeder &
Lester, 1989)
Primarily procedural/routine problems are
emphasized during instruction
Both conceptual/non-routine problems and
procedural/routine problems are emphasized
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during instruction (Robinson & Robinson, 1998;
Thompson, 2001; Webb, 2003)
Students work on problems in isolation Students work on problems collaboratively in
small groups, engage in small-group and whole-
class discussions, and present solutions to the
class (NCTM, 2000; Boylan, Bonham, & Tafari,
2005; Golfin et al., 2005; AMATYC, 2006;
Thompson, 2009)
Limited use of technology Extensive use of technology through the graphing
and table functions of graphing calculators
(Golfin et al., 2005; AMATYC, 2006; Webb,
2003; Thompson, 2009)
Homework emphasizes primarily
procedural skills
Homework emphasizes both conceptual
understanding and procedural skills (NCTM,
2000)
Assessment is mostly summative through
homework and tests
Assessment is strongly formative through class
and group discussions, in addition to summative
assessments of homework and tests (NCTM,
2000)
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Data Analysis
The following paragraphs describe the sequence in which the collected data in this study
were analyzed. In short, the researcher verified that the RTOP scores were significantly different,
determined which covariates were necessary to include in the statistical analyses, performed the
statistical analyses, and analyzed the results from the student surveys.
Establishing Validity
The RTOP was incorporated into the study to establish that two distinct treatments had
indeed taken place in the study. With respect to the required difference in RTOP scores that
would be required in order to state that the types of instruction were significantly different,
MacIsaac and Falconer (2002) distinguished between high school and university RTOP scores.
Using physics lessons as a backdrop, the researchers stated that a traditional university lecture
that is passive in nature would produce an RTOP score less than 20, whereas a traditional high
school lecture with student questions would produce an RTOP score less than 45. When
describing observations made of high school mathematics and biology teachers, Judson and
Lawson (2007) similarly categorized RTOP scores of less than 30 to be low and an RTOP score
of 43 to be moderate. MacIsaac and Falconer (2002) emphasized that the preceding scores
approximate the amount of reform instruction implemented in a classroom but that any RTOP
score greater than 50 indicated considerable presence of reformed teaching in a lesson.
Lawson et al. (2002) based the success of their program by comparing the RTOP scores
of ACEPT-influenced teachers to non-ACEPT teachers. When examining the mean RTOP scores
for third year teachers, the ACEPT teachers scored significantly higher RTOP scores than non-
ACEPT teachers (62 and 45, respectively, p < 0.05).
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Thus an average RTOP score for the control class that is close to 20 and an average
RTOP score for the experimental class that is at least 40 would align with the classifications
presented by MacIsaac and Falconer (2002) and Judson and Lawson (2007); additionally, the
difference in RTOP scores between the two classes would meet the 17 point difference presented
by Lawson et al. (2002).
Selecting Covariates
For each of the statistical analyses that were performed, the researcher determined which
variables (Age, Race, Gender, Prior Mathematical Knowledge, Number of Attempted Credit
Hours, and Number of Hours of Employment) needed to be included as covariates. Age, Prior
Mathematical Knowledge, Number of Attempted Credit Hours, and the Number of Hours of
Employment were treated as continuous variables. Therefore, the difference in averages between
the Control and Experimental groups with respect to each of these variables were analyzed using
a t-test. Race and Gender were considered categorical variables; therefore, the differences in
percentages with respect to these variables were analyzed using a Fisher?s Exact Test. Data
regarding these variables were obtained from students through surveys that were administered at
the beginning of the course.
In this study, there was no reason to expect that the two classes would differ significantly
with respect to any of the aforementioned variables. A common rule of thumb when controlling
for variables in statistical analysis is to allow one variable into the study per 10 ? 15 students
(Osborne & Costello, 2004). Since the sample consisted of only 29 students, and since the study
had already introduced the variable treatment, I only introduced variables that were significantly
different between the two groups and which also significantly impacted the research question?s
dependent variable.
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Determining if a variable needed to be included as a covariate involved two steps. First,
the appropriate statistical test was used to determine if the Control and Experimental groups
significantly differed with respect to a variable. The differences between the two groups were
analyzed using a t-test for continuous variables (Age, Credit Hours, Work Hours, and Prior
Mathematical Knowledge) and a Fisher?s exact test for categorical variables (Race and Gender).
If the two groups did not differ significantly with respect to the variable, then the variable was
not included as a covariate. However, if the two groups did differ with respect to that variable,
then step two was invoked: treat the variable as an independent variable and determine if it alone
has an effect on the dependent variable. If the variable had a significant effect on the dependent
variable, then it would be included as a covariate in the final statistical analysis.
Analysis of Effects
The differences in students? Procedural scores and Application scores were each analyzed
using a 2 (experimental group vs. control group) x 5 (5 regular tests) repeated measures
ANOVA. The statistical analyses contained any covariates in which the two groups significantly
differed and which also significantly impacted the dependent variable. Similar to a t-test, a
Repeated Measures ANOVA is a statistical procedure used to test the null hypothesis that the
means of variables do not differ. However in a Repeated Measures ANOVA, each participant in
the study is tested multiple times and therefore contributes multiple values under the same
variable. A Repeated Measure ANOVA was appropriate for analyzing differences in procedural
and application scores because each student in the study was given multiple tests (and therefore
contributed multiple values) that evaluated the student?s skill set with respect to a specific
dependent variable (Huck, 2004).
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A Repeated Measures ANOVA produces two types of results that are relevant to this
study: 1) between-groups results and 2) within-groups interaction results. For example, in
addressing the first research question, students in the Control group and the Experimental group
were given a series of five tests over the course of the semester that evaluated their ability to
solve procedural problems. A statistically significant ?between-groups? result would imply that
the average procedural ability of the Control group was significantly different from the average
procedural ability of the Experimental group. In contrast, a statistically significant ?within-
groups interaction? result (denoted by the phrase ?Procedural * Treatment?) would imply that
each group?s procedural scores changed differently over time; that is, the change in the Control
students? average procedural scores throughout the study was significantly different from the
way that the Experimental students? average procedural scores changed throughout the study. A
graph with intersecting lines can be an indicator that a significant interaction effect exists in the
data (Huck, 2004).
The difference in Pass Rates between the two groups was analyzed using a Fisher?s Exact
Test because of the test?s usefulness in analyzing the difference in percentages between two
groups; additionally, a Fisher?s Exact Test was chosen due to its ability to accommodate small
sample sizes (Huck, 2004). The results of the analysis were controlled for any necessary
covariates.
The difference in students? change in mathematical efficacy was analyzed using a 2
(Experimental group vs. Control group) x 2 (Pre/Post Test) repeated measures ANOVA. In
contrast to the 2 x 5 repeated measures ANOVA used to analyze the difference in procedural
scores, a repeated measures ANOVA was employed simply to determine if a within-subjects
interaction took place. In other words, the test was used to determine if one group changed
95
significantly more in mathematical self-efficacy than did the other group across the period of
instruction.
The size of the treatment effects were determined by calculating Cohen?s d and partial eta
squared. Cohen?s d provides the difference in means between two groups divided by the standard
deviation of the sample. For example, a Cohen?s d of 0.3 implies that the mean difference
between the two groups was 0.3 standard deviations. A Cohen?s d less than 0.2 implies a small
treatment effect, between 0.2 and 0.8 is a medium treatment effect, and greater than 0.8 is a large
effect. Similarly, a partial eta squared value directly indicates the percentage of the variability in
the data that is due to the differences between treatments. For example, a partial eta squared of
0.425 implies that 42.5% of the variability in the data is due to the differences between
treatments (Gravetter & Wallnau, 2004).
Qualitative Analysis
I coded students? comments on the end-of-course surveys according to a coding strategy
advocated by Miles and Huberman (1994). Prior to beginning the study, I created a ?start list? of
predefined codes based on the survey questions and students? possible responses to those
questions. After administering the survey, I used a representational approach to code students?
answers. According to Sapsford (1999), researchers who use this approach represent the surface
content fairly by using key words to identify core concepts. Throughout the coding process,
some of the predetermined codes increased in bulk and seemed ill-fitting. I therefore reassessed
the strength of my original codes and created subcodes in order to produce a better fit for the
collected data. Additional codes were also created from themes that emerged from the collected
data. The coding process terminated once all of the students? statements could be readily
classified according to the existing set of codes (Miles & Huberman, 1994).
96
Summary
A quasi-experimental design was used to test the effectiveness of teaching a remedial
mathematics course in a reform-oriented manner as opposed to teaching the remedial
mathematics using didactic lecture. The effectiveness of each treatment was based on students?
course pass rates, procedural skills, application skills, and mathematical self-efficacy.
Additionally, students provided their perspectives regarding their respective treatments through
an anonymous survey that was issued at the end of the course. The results of the statistical
analyses were controlled for demographic variables in which the two groups significantly
differed, and the validity of the results was supported by the use of an outside grader and through
colleague classroom observations. Additionally, the results of the student surveys provided
qualitative data and were categorized according to students? perspectives on the teaching
techniques employed during their respective courses. In the following chapter, I will present the
findings of this study.
97
CHAPTER 4: RESULTS
The first section in this chapter will describe key events that occurred during the study.
Second will be the results of the classroom observations through the lens of the Reformed
Teaching Observation Protocol (RTOP), followed by the results of the inter-rater reliability for
the grading of tests. Lastly, the results for the five research questions will be presented based on
the data analysis methods presented in the prior chapter.
Summary of Events
When the plan for this study was first developed, the policy at the Southeast University
allowed me to teach multiple courses per semester. However, the policy at the university
changed during the planning of the study. I spoke with the head of the Mathematics Department
and asked if a waiver could be submitted that would allow the researcher to teach two classes
currently for the Spring 2012 semester. The head of the Mathematics Department agreed to file a
waiver request since it would assist the researcher in completing his dissertation project.
However, the upper administration at the Southern University denied the waiver request; thus, I
was required to teach the Experimental class in Spring 2012 and the Control class in Fall 2012.
From the early planning stages of the study, the head of the Mathematics Department
fully supported the development and execution of the study. Even though some students from the
multiple pilot studies complained about the methods in the reform-oriented classes, the head of
the Mathematics Department told me not to worry about the complaints and that I had the
department head?s full support. With administrative support in place, I began the study by
teaching the Experimental class in Spring 2012.
Several weeks into the study, the head of the Mathematics Department approached me
and stated that the Spring 2012 semester would be the last time that I could teach in a reform-
98
oriented manner. When I asked about the withdrawal of support, the department head stated that
multiple students had complained about the teaching method and that even a parent (who was an
instructor and researcher at another university) contacted the assistant dean of the School of
Sciences at the Southeast University. The administration decided to accommodate the
complaining parent by moving his child to another class. The assistant dean of Sciences later
spoke with me and stated that she defended my actions. However, the department head stated
that the culmination of complaints from the current semester as well as previous semesters
caused him to withdraw his support of reform-based teaching. Later conversations with the
department head revealed that he would allow me to teach in a reform-based manner, but such
teaching could not take place in the context of a research study. After this initial series of
complaints, I finished teaching the Experimental course without any further incidents. In
contrast, the Control course was taught in the Fall 2012 semester without incident, and I was not
informed of any student complaints regarding my style of traditional teaching.
Integrity of Treatment
The first step in my analysis was to determine the degree to which appropriate teaching
methods had been delivered to the two classes included in the study. The Reformed Teaching
Observation Protocol (RTOP) was used by two colleagues in the field of education in order to
establish that the Control class had received traditional mathematics instruction and that the
Experimental class had received reform-oriented instruction. Two paired lessons were observed
in each course: ?Completing the Square? and ?Shifting of Graphs?. The lesson on ?Completing
the Square? was taught early in the course, and the lesson on ?Shifting of Graphs? was taught
midway through the course. Both colleagues were present for each of the observations. Recall
that the RTOP produces scores from 0 through 100, where a lesson receiving a score higher than
99
50 is considered to have significant incorporation of reform-oriented pedagogy. The two lessons
that were observed in the Control group received firmly traditional scores, and the two lessons in
the Experimental group received firmly reform-oriented scores. Table 10 presents the scores for
the four classroom observations from each observer. Thus, the Experimental section did appear
to receive reform-oriented instruction while the Control section did not.
Table 10
Differences in RTOP scores between control and experimental sections
Completing the Square Lesson Shifting of Graphs Lesson
Section Rater 1 Rater 2 AVG Rater 1 Rater 2 AVG
Control 22 23 22.5 22 35 28.8
Experimental* 92 80 86.0 91 91 91.0
Note. A score greater than 50 indicates significant use of reform pedagogy
Inter-rater Reliability of Tests
The next step in my analysis was to examine the inter-rater reliability of the test scores
assigned to the students in order to support the validity of the scores earned by the students. The
relationship between the researcher?s and colleague?s graded tests for Procedural Scores for the
first five tests ranged from r = .825 to r = 1.000, with a median value of r = .996. The
relationship for Application Scores for the first five tests ranged from r = .948 to r = 1.000, with
a median value of r = 1.000. See Table 11 for a summary of the correlations between my scores
and the outside grader?s scores. This table indicates that the outside grader and I maintained the
desired Pearson correlation of at least r = .8 throughout the study. Thus, the scores in the study
were valid.
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Table 11
Summary of inter-rater reliability Pearson correlation values
Control Group Experimental Group
Test 1 Test 2 Test 3 Test 4 Test 5 Test 1 Test 2 Test 3 Test 4 Test 5
Procedural
Scores
.995 .993 .985 .999 1.000 .825 .998 .997 .988 .998
Application
Scores
.979 .948 1.000 1.000 1.000 .974 1.000 1.000 1.000 1.000
Quantitative Results
The Control group and the Experimental group were analyzed in four different ways: 1)
student performance on procedural problems, 2) student performance on application problems, 3)
students? pass rates, and 4) student change in mathematics efficacy. The first section provides a
description of how I determined which variables should function as covariates. Following that
section are the results of each of the above four analyses.
Selecting Covariates
I analyzed the degree to which the Experimental section was comparable to the Control
section. Thus, data were collected to determine if a significant difference existed between the
two sections in terms of Race, Age, Gender, Prior Mathematical Knowledge, Number of Hours
Employed, and Number of Attempted Credit Hours. I used a t-test for analyzing the four
continuous variables (Prior Mathematical Knowledge, Hours of Employment, Credit Hours, and
Age). The Experimental and Control groups differed significantly only with respect to the
number of Credit Hours that students were taking (p = 0.043). See Table 12 for the results of the
analyses. Thus, Credit Hours was identified as a potential covariate.
101
Table 12
Differences in continuous variables between groups
Variable Treatment n Mean Standard
Deviation
(t, p-value)
Prior Mathematical
Knowledge
Control 11 22.64 5.2 (-1.49, .148)
Experimental 16 26.06 6.2
Total 27
Hours of Employment
Control 11 9.50 12.0 (-.396, .696)
Experimental 18 11.67 15.5
Total 29
Credit Hours
Control 11 12.82 2.8 (-2.125, .043)
Experimental 18 14.78 2.2
Total 29
Age
Control 11 20.73 3.7 (-.801, .430)
Experimental 18 22.44 6.5
Total 29
Of the two dichotomous variables, Gender and Race, I used a Fisher?s Exact Test to
determine that only Race was significantly different between the Control and Experimental
groups (p = 0.048). See Table 13 for the results of the analyses. Thus, Race was also identified as
a potential covariate.
102
Table 13
Research Question 1: Procedural Skills
My first research question examined whether the two groups demonstrated similar
procedural skills throughout the five tests within the course. Data were gathered from students
who completed all five tests, which included ten students from the control group and seventeen
students from the experimental group. Table 14 provides a summary of students? procedural
scores. According to the table, the average difference in Procedural scores between the two
groups was 2.0 points in favor of the Control group. The Control group earned higher marks on
the first, second, and fourth tests, and the Experimental group earned higher marks on the third
and fifth tests.
Differences in dichotomous variables between groups
Variable Treatment n Male Female p-value
Gender
Control 11 3 8 1.00
Experimental 18 6 12
Total 29
Race
Control 11 7 4 .048
Experimental 18 4 14
Total 29
103
Table 14
Summary of procedural scores for control and experimental groups
Test 1:
Factoring
Test 2:
Rational
Expressions
Test 3:
Functions
Test 4:
Radicals
Test 5:
Quadratic
Equations
Average
Difference
Control
Mean 80.4 77.6 74.6 72.7 62.2
Std Dev 10.1 14.0 9.6 17.2 16.2
Experimental
Mean 76.7 67.6 75.0 65.8 72.2
Std Dev 14.5 17.4 11.4 15.3 17.4
Difference in
Means (E-C)
-3.7 -10.0 +0.4 -6.9 +10.0 -2.0
Cohen?s d -0.30 -0.63 0.04 -0.42 0.60
Note: The values listed represent the percentages of points earned
A 2 (Treatment) x 5 (Procedural Test Scores) Repeated Measures ANOVA was used
because each student in the two groups took a total of five tests. The independent variable was
the Treatment, and the dependent variable was the Procedural Tests Scores. Additionally,
Mauchly?s Test of Sphericity indicated that the assumption of sphericity for the analysis was not
violated (Mauchly?s W = .906, df = 9, p = .988).
Race was included as a covariate because of the statistically significant interaction
between Race and students? Procedural Scores (Procedural Scores * Race F = 4.625, p = .002).
Although Race was not a focus in this study, Race was used as a covariate to minimize the
104
differences between the Control and Experimental groups and therefore improve the accuracy of
the statistical model. The other potential covariate, Credit Hours, was not included as a covariate
in this analysis because it had no significant effect on students? Procedural Scores (between-
subjects effect F = 1.271, p = .316; Procedural Scores * Credit Hours F = .924, p = .580).
The two groups did not differ significantly in their overall procedural scores (Test of
Between-groups ?Treatment? effect: F=.365, p = .551, Power = .089). Additionally, the
treatment did not have a significant effect on the students? procedural scores over time (Test of
Within-groups interaction ?Procedural Scores * Treatment?: F = 1.285, p = .281, Power = .388).
Refer to Table 15 for a summary of the analysis.
Table 15
Statistical analysis of procedural scores between groups
df Mean
Square
F Sig Partial
Eta
Squared
Observed
Power
Between Groups
Race 1 149.009 .206 .654 .009 .072
Treatment 1 263.406 .365 .551 .015 .089
Error 24 721.651
Within Groups
Procedural Tests 4 4.606 .002 .161 .937
Procedure * Treatment 4 1.285 .281 .051 .388
Procedure* Race 4 1.975 .104 .076 .575
105
When Race was included as a covariate in the analysis of procedural scores, the average
difference between the two groups increased to 3.4 points in favor of the Control group. As with
the unadjusted scores, the Control group scored higher on the first, second, and fourth tests; and
the Experimental group scored higher on the third and fifth tests. The Cohen?s d values similarly
indicate a large treatment effect on the first, second, and fourth tests (d > 0.8) and a medium
effect for the third and fifth tests (0.2 < d < 0.8). Table 16 provides a summary of students?
procedural scores after being adjusted for Race.
Table 16
Summary of procedural scores adjusted for race
Test 1:
Factoring
Test 2:
Rational
Expressions
Test 3:
Functions
Test 4:
Radicals
Test 5:
Quadratic
Equations
Average
Difference
Control
Mean 81.7 76.3 74.3 73.1 66.4
Std Error 4.6 5.8 3.9 5.7 5.8
Experimental
Mean 76.0 68.4 75.2 65.5 69.7
Std Error 3.4 4.3 2.8 4.2 4.2
Difference in
Means (E-C)
-5.7 -7.9 +0.9 -7.6 +3.3 -3.4
Cohen?s d -1.41 -1.55 0.27 -1.52 0.65
Note: The values listed represent the percentages of points earned
106
As illustrated in Figure 3, the two groups experienced different trends in performance
with respect to procedural scores. The Control group started with an average adjusted procedural
score of 81.7% on the first test and steadily declined to an average adjusted procedural score of
66.4% on the fifth test, a decrease of 15.3% over the semester. On the other hand, the
Experimental group experienced increases and decreases on the five tests, but did not experience
as much of a decline (76.0% to 69.7%) over the semester.
Figure 3: Mean adjusted procedural scores for control and experimental groups
Although not part of my original design, I used the department?s comprehensive final
exam to further examine if the two groups demonstrated a significant difference in procedural
ability. The departmental final exam was multiple-choice and consisted almost entirely of
procedural problems. Data were gathered from students who completed the final exam: ten
students from the control group and fifteen students from the experimental group. I performed an
ANOVA and found that Race should not be included as a covariate since it did not have a
60
65
70
75
80
85
1 2 3 4 5
Pr
oc
ed
ur
al
Sc
ore
s (
%
)
Procedural Tests
Control
Experimental
107
significant effect on final exam scores (F = 0.007, p = 0.932). I also used multiple regression to
determine that Credit Hours should not be included as a covariate because it did not have a
significant effect on final exam scores (t = -0.413, p = 0.683). Using a t-test, I found that the
difference in final exam scores between the two groups was not statistically significant (t = -
0.223, p = 0.825). The results in Table 17 showed that the procedural ability between the two
groups of students was comparable.
Table 17
Comparison of final exam scores between control and experimental groups
n Mean Final Exam Score Standard Deviation
Control 10 71.2% 12.9
Experimental 15 72.4% 13.4
Difference (E ? C) +1.2%
Cohen?s d 0.0912
Research Question 2: Application Skills
My second research question analyzed whether the two groups demonstrated similar
application skills throughout the five tests within the course. Data were gathered from students
who completed all five tests: ten students from the control group and seventeen students from the
experimental group. Table 18 provides a summary of students? application scores throughout the
semester. According to the table, the average difference in application scores between the two
groups was 13.7 points in favor of the Experimental group.
108
Table 18
Summary of application scores for control and experimental groups
Test 1:
Factoring
Test 2:
Rational
Expressions
Test 3:
Functions
Test 4:
Radicals
Test 5:
Quadratic
Equations
Average
Difference
Control
Mean 62.0 67.5 68.1 82.0 41.2
Std Dev 27.0 23.7 33.1 21.5 40.0
Experimental
Mean 72.4 74.5 90.8 87.1 64.7
Std Dev 18.6 30.7 20.0 16.5 39.6
Difference
(E-C)
+10.4 +7.0 +22.7 +5.1 +23.5 +13.7
Cohen?s d 0.44 0.25 0.83 0.27 0.59
Note. The values listed represent the percentages of points earned
As in the first research question, a 2 (Treatment) x 5 (Application Test Scores) Repeated
Measures ANOVA was used because each student in both groups took a total of five tests. The
independent variable was the Treatment, and the dependent variable was the Application Tests
Scores. Additionally, Mauchly?s Test of Sphericity indicated that the assumption of sphericity
for the analysis was not violated (Mauchly?s W = .603, df = 9, p = .253).
Race was included as a covariate because of its statistically significant between-groups
effect (F = 4.517, p = .044). In contrast, the data showed that the Number of Credit Hours
Attempted by students had no significant effect on students? application scores (between-subjects
109
effect F = 1.122, p = .390; Application * Credit Hours F = .577, p = .948); therefore, the Number
of Credit Hours Attempted by students was not included as a covariate.
The Experimental group outperformed the Control group on the application problems of
every test. However, when Race was entered as a covariate, the overall difference between the
two groups? application scores was not significant (F =1.051, p =.315, Power = .166). Further,
the treatment did not have a significant effect on the students? application scores over time
(Within-groups interaction ?Application Scores * Treatment? F = .297, p = .879, Power = .114).
See Table 19 for a summary of the analysis.
Table 19
Statistical analysis for the difference in application scores
df Mean
Square
F Sig Partial
Eta
Squared
Observed
Power
Between Groups
Race 1 2439.224 1.628 .214 .064 .232
Treatment 1 1575.674 1.051 .315 .042 .166
Error 24 1498.514
Within Groups
Application Tests 4 1397.822 2.364 .058 .090 .664
Application Tests * Treatment 4 175.679 .297 .879 .012 .114
Application Tests * Race 4 322.167 .545 .703 .022 .177
Error (application tests) 96 591.385
110
Table 20 provides a summary of students? application scores after being adjusted for
Race. When Race was taken into account, the average difference between the two groups
decreased to 8.3 points in favor of the Experimental group. The Experimental group earned
higher marks on all of the tests and scored substantially higher on the fifth test. Although not
statistically significant, the Cohen?s d values for all five tests indicate either medium or large
treatment effects for all five tests.
Table 20
Summary of application scores adjusted for race
Test 1:
Factoring
Test 2:
Rational
Expressions
Test 3:
Functions
Test 4:
Radicals
Test 5:
Quadratic
Equations
Average
Difference
Control
Mean 65.0 68.4 76.2 83.8 44.5
Std Error 7.8 10.2 8.4 6.6 14.2
Experimental
Mean 70.6 74.0 86.1 86.0 62.8
Std Error 5.7 7.5 6.2 4.8 10.4
Difference
(E-C)
+5.6 +5.6 +9.9 +2.2 +18.3 +8.3
Cohen?s d 0.82 0.63 1.34 0.38 1.47
Note. The values listed represent the percentages of points earned
As illustrated by the graph in Figure 2, the two groups experienced similar trends in
performance. The Experimental group started with an average adjusted application score that
111
was 5.6 points higher than the Control group. The two groups roughly maintained that difference
throughout the course until the fifth test in which the gap between the two groups grew to a
difference of 18.3 points. Although the Experimental group consistently scored higher than the
Control group throughout the course, the overall difference in application scores between the two
groups was not significant.
Figure 4: Mean adjusted application scores for control and experimental groups
Analysis of Procedural vs. Application Skills. Due to a trend that appeared in each group?s
performance with respect to their procedural problems and application problems, an additional
analysis was conducted. A strong correlation existed in the control group between students?
average procedural scores and their average application scores (r = .7645); of the ten students in
the control group who completed the course, seven of the ten students (70.0%) had higher
procedural scores than application scores. In contrast, a weak correlation existed in the
experimental group between students? average procedural scores and average application scores
30
40
50
60
70
80
90
1 2 3 4 5
Sc
ore
s (
%
)
Application Tests
Control
Experimental
112
(r = .2308); of the seventeen students who completed the course, only six of the seventeen
students (35.3%) had higher procedural scores than application scores. In other words, students
in the Experimental group were more likely than students in the Control group to earn average
application scores that were higher than their average procedural scores across the five tests.
However, a 2x2 Fisher?s exact test revealed that the difference in trends exhibited between the
two groups was not statistically significant (p = 0.12)
The data showed that several students in the experimental course earned low procedural
averages but were able to earn relatively high application scores. For example, Gary (a
pseudonym) earned a 41% average procedural score on the five tests during the course and a
78% average application score on the five tests during the course; thus, his average application
score across the five tests was 37% higher than his procedural scores. Five other students
similarly earned much higher average application scores (at least 16% higher) than procedural
scores. In contrast, no student in the control group earned an average application score that was
more than 11% higher than his or her average procedural score.
In order to understand better why each group of students experienced different
correlations between their average procedural and average application scores, students?
responses on the application problems were further examined to determine what type of methods
were used to answer the problems. The examination showed that in addition to solving problems
through algebraic means, students also used pictures to understand the situation within a problem
or to supplement their algebraic ability to solve a problem. Figure 5 illustrates how the use of
pictures helped a student answer the application problem 2B.
113
Figure 5: A solution obtained through the use of pictures
The analysis also showed that students used systematic trial and error approaches by
constructing a table of values or by evaluating multiple solutions until the correct solution was
attained. These approaches were often used in the place of more formal algebraic techniques to
solve application problems. Figure 6 illustrates the systematic trial and error approach that a
student used to answer question 5A.
114
Figure 6: A solution obtained through systematic trial and error
Table 21 describes the difference in usage of supplemental methods such as pictures and
systematic trial and error between the control and experimental groups. According to Table 21,
students in the experimental group used pictures and systematic trial and error methods much
more often than students in the control group.
115
Table 21
Comparison of non-algebraic strategies on application questions between groups
Test Question Control
n (%)
Experimental
n (%)
Test Question Control
n (%)
Experimental
n (%)
1A 1B
Pictures 1 (10.0) 3 (17.6) Pictures 10 (100) 17 (100)
Tables/
Trial & Error
0 (0.0) 4 (23.5) Tables/
Trial & Error
0 (0.0) 2 (11.8)
2A 2B
Pictures 0 (0.0) 5 (29.4) Pictures 0 (0.0) 8 (47.1)
Tables/
Trial & Error
0 (0.0) 0 (0.0) Tables/
Trial & Error
0 (0.0) 0 (0.0)
3A 3B
Pictures 0 (0.0) 0 (0.0) Pictures 0 (0.0) 0 (0.0)
Tables/
Trial & Error
0 (0.0) 2 (11.8) Tables/
Trial & Error
6 (60.0) 12 (70.6)
4A* 4B
Pictures 8 (80.0) 17 (100.0) Pictures 0 (0.0) 3 (17.6)
Tables/
Trial & Error
0 (0.0) 0 (0.0) Tables/
Trial & Error
0 (0.0) 1 (5.9)
5A
Pictures 0 (0.0) 2 (11.8)
Tables/ 1 (10.0) 11 (64.7)
116
Trial & Error
* Problem 4A specifically asked students to draw a picture that described the scenario in
the problem
Research Question 3: Pass Rates
My third research question attempted to determine if the two groups exhibited similar
pass rates for the course. Using logistic regression, I established that neither Race (Wald = .423,
p = .515) nor Credit Hours (Wald = .158, p = .691) should be included as covariates because they
did not have a significant effect on students? course pass rates. Data were gathered from all
students who agreed to participate in the study: eleven students from the Control group and
eighteen students from the Experimental group. A 2 (Control/Experimental) x 2 (Pass/Fail)
Fisher?s Exact Test found that the difference in pass rates between the two groups was minimal
and likely due to chance (p = 1.00). Table 22 summarizes these results.
Table 22
Summary of pass rates
n Passed Failed Pass Rate Sig
Control 11 7 4 63.6% 1.00
Experimental 18 11 7 61.1%
Research Question 4: Students? Change in Mathematics Self-Efficacy
My fourth research question attempted to determine if the two groups demonstrated
similar changes in mathematics self-efficacy throughout the course. Because Credit Hours and
Mathematics Self-Efficacy were both continuous variables, I used linear regression to establish
that Credit Hours did not need to be a covariate (t = -0.575, p = 0.571). I also used a 2 x 2
117
Repeated Measures ANOVA to establish that Race did not need to be a covariate (Race *
Efficacy F = 0.810, p = 0.378). Data were gathered from students who completed the Pre- and
Post-Mathematics Self-Efficacy Surveys (a 5-question Likert-style survey): ten students from the
Control group and thirteen students from the Experimental group. The dependent variable was
students? Mathematics Self-Efficacy, and the independent variable was the Treatment. The
relationship between these two variables was analyzed using a 2 (Pre-Efficacy/Post-Efficacy) x 2
(Control/Experimental) Repeated Measures ANOVA. Table 23 summarizes the students? change
in mathematics self-efficacy.
Table 23
Summary of students? change in mathematics self-efficacy
n Mean
Pre-Test
Std Dev
Pre-Test
Mean
Post-Test
Std Dev
Post-Test
Change in
Means
Control 10 4.34 0.55 3.98 0.84 -0.36
Experimental 13 4.11 0.64 4.18 0.96 +0.07
Difference
(E ? C)
-0.23 +0.20 +0.43
Note. Survey results are based on a 5-point scale
The graph in Figure 7 shows that at the beginning of the course, the Control group
reported a self-efficacy score that was 0.23 points higher than the Experimental group?s score.
However at the end of the course, the Experimental group reported a self-efficacy score that was
0.20 higher than the Control group?s score.
118
Figure 7: Mean pre- and post-mathematical self-efficacy scores
The Control group reported a modest drop in mathematical self-efficacy by the end of the
course, and the Experimental group reported having a slight increase in mathematical self-
efficacy by the end of the course, resulting in a net difference of 0.43 in favor of the
Experimental group. According to the Repeated Measures ANOVA, the effect that the treatment
had on each group?s change in mathematics self-efficacy was not significant (Treatment *
Efficacy F = 1.014, p = .325, Power = .161). Table 24 provides a summary of the statistical
analysis.
1
2
3
4
5
1 2
M
ath
em
atical
Se
lf-
Ef
ficac
y
Pre/Post Scores
Control
Experimental
119
Table 24
Statistical analysis for students? change in mathematics self-efficacy
df Mean
Square
F Sig Partial
Eta
Squared
Observed
Power
Between Groups
Treatmenta 1 .002 .003 .995 .000 .050
Error 21 .659
Within Groups
Efficacy 1 .226 .426 .521 .020 .096
Efficacy * Treatmenta 1 .540 1.014 .325 .046 .161
Error (Efficacy) 21 .532
Note. ?a? denotes the result that is relevant to the researcher?s question
Summary of the Quantitative Results
With respect to procedural skills, the Control group outperformed the Experimental group
on three of the five tests; and with respect to application skills, the Experimental group
outperformed the Control group on all five tests. However, the differences in both the procedural
skills and application skills of the students were not significant. Students in the Control group
had a stronger correlation between their average procedural scores and application scores than
did students in the Experimental group; this difference may due the Experimental students using
non-algebraic strategies more often on application problems than did the Control students. With
respect to pass rates, the Control group had a slightly higher pass rate over the Experimental
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group; however, the difference in pass rates was not statistically significant. With respect to
change in mathematics self-efficacy, the Experimental group maintained its starting level of
mathematics self-efficacy, while the self-efficacy of the Control group decreased by the
completion of the course. However, the difference in the changes in mathematics self-efficacy
was not statistically significant.
Qualitative Results
The following sections provide the results from the anonymous free-response student
surveys. The sample for the qualitative data consisted of forty-five respondents, as opposed to
the sample for the quantitative data which consisted of twenty-nine participants. The qualitative
data had a higher sample size than the quantitative data because the qualitative data were
gathered anonymously through end-of-course surveys from all remaining students in each
course. The twenty-three respondents from the Control group primarily addressed my ability as
their instructor to explain the information, as well as the mathematics department?s design of the
course. The twenty-two respondents of the Experimental group addressed issues similar to those
that were addressed by the Control group. However, since the students in the Experimental group
were asked additional questions regarding various components of the experimental teaching
method, the students in the Experimental group addressed issues that did not apply to the
students in the Control group. I used the representational approach described by Sapsford (1999)
to identify core concepts in students? responses, and I terminated the coding process once all of
the students? statements could be classified according to the existing set of codes (Miles &
Huberman, 1994).
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Comparison of Treatments
The first question of the student surveys for both groups was the same: ?How does this
math class compare to other math classes that you have had? Explain.? The purpose of this
question was to support the claim that the pedagogical techniques used in the Experimental
group were significantly different from those used in the Control group.
The types of comments that students in the Control group made were consistent with
those expected from a traditionally taught classroom. For example, a significant number of
comments favorably addressed the instructor?s ability to communicate mathematical concepts
and to provide a relatively enjoyable learning experience. The remaining comments addressed
the pacing and difficulty of the course; the majority of these comments were also positive. See
Appendix F for a summary of the Control group?s comments for question 1 of the student
survey.
In contrast, the types of comments made by the students in the Experimental group were
consistent with the pedagogy advocated by reform documents. The positive responses about the
course showed that 1) students were given the opportunity to learn mathematics through
extensive interactions with their peers, 2) the instructor minimized explicit mathematical
instruction, 3) classroom instruction incorporated significant use of pictures and graphs (in
addition to algebraic techniques), 4) mathematics was related to the real world, and 5) students
were developing mathematical tools and ways of thinking that could be used in future
mathematics classes. The negative responses similarly provided insight into the daily classroom
experiences by referring to students? investigating mathematical phenomena on their own,
students being required to work with the members of their group, and the instructor?s practice of
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engaging students in questioning techniques. See Appendix F for a summary of the Experimental
group?s comments for question 1 of the student survey.
Efficacy of Control Treatment
The remaining questions on the Control group?s student surveys were gathered in order to
support the claim that the Control group was taught fairly in the eyes of the students. Students?
responses in questions 2 ? 4 were grouped together but kept separate from students? responses to
question 1. The remaining questions on the Control student survey were the following:
2) What are some things you liked about the course?
3) What are some things you did not like about the course?
4) Other comments.
A high number of the students stated that the instructor adequately and enthusiastically
explained the material. Students? comments regarding the structure of the course were more
divided. Some students disagreed over the difficulty of the course, while other students
commented negatively on the mathematics department?s design of the course (such as not
allowing calculators on the final exam). Students similarly were divided over the homework
policy. Some students felt that the homework problems helped them understand the material,
while other students felt that too much homework was assigned. Several students negatively
commented on the time of day for the course (8 a.m.). Interestingly, however, when given the
opportunity to state negative characteristics of the course, ten of the twenty-three responding
students explicitly stated that the course did not contain any negative qualities.
Overall, 62 positive comments and 16 negative comments were made by the students;
and of the 16 negative comments, 10 addressed factors beyond the instructor?s control (such as
departmental policy and the class?s meeting time.) Thus, based on the proportionally high
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number of positive comments about this particular mathematics course, it is clear that the high
majority of the students responded positively to the way that the course was taught. See
Appendix F for a summary of the Control group?s comments for questions 2-4 of the student
survey.
Research Question 5: Students? Views about Reform Mathematics
Additional questions were placed on the Experimental group?s survey in order to solicit
explicit feedback regarding several key components of reform pedagogy that were employed
during the study. Students? responses in questions 2 ? 7 (see Appendix F) were grouped together
but kept separate from students? responses to question 1 which had a different purpose. The
remaining questions were the following:
2) What are some things you liked about the course?
3) What are some things you did not like about the course?
4) To what extent did you like working with your classmates during class? Explain.
5) Did you find the graphing calculator useful? If yes, please explain how/when it was
useful.
6) To what extent did you benefit from presenting your work to the class (or watching
your classmates present their work to the class)? Explain.
7) Other comments.
Students possessed a generally positive view regarding student presentations by noting
that the presentations 1) pushed them to perform good mathematical work, 2) helped them better
understand mathematical concepts by observing how their classmates? approaches compared to
their own, and 3) helped students increase their confidence in their mathematical and speaking
abilities. Some students, however, felt that they did not benefit from peer presentations and
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thought that class time could be better used by the instructor working through additional
problems.
Students also possessed a generally positive view of working together in groups. Many
students seemed to feel that they benefited from the support structure provided by their groups,
both in terms of helping one another understand a concept as well as sharing alternative ways to
view a particular concept. Some students preferred not to work in groups because they did not
like to share their work or because they would have liked the instructor himself to communicate
explicitly the mathematical material.
Students overwhelmingly liked the use of graphing calculators. Students expressed that
graphing calculators helped them solve problems, graph functions, and verify that their answers
were correct. The calculator?s ability to create graphs and tables provided students an alternative
means to solve problems other than by using purely algebraic techniques. No negative
comments were made regarding the use of calculators.
Students provided mixed reviews regarding the teaching methods used during the course.
While some students enjoyed every facet of the course and stated that the mathematics course
was ?fun?, other students equally disliked the course. Several students would have preferred to
minimize student discussions so that more examples could be done during class, and some
students felt that the classroom instruction did not connect well with the problems that were in
the book and on the tests. A few students also stated that the mathematical techniques developed
during class as well as the type of problems solved during class did not correspond to the
techniques and problems presented in the Math Lab.
Students acknowledged the inherent trade-off between covering fewer problems in
greater depth (questioning, group discussions, student presentations) versus the teacher solving
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more problems during class, with the expectation that students will understand the material once
a sufficient number of examples are presented.
Summary of Qualitative Results
The responses from the students? anonymous end-of-course surveys suggested that the
treatments for each group were what I had thought. Based on the ratio of positive to negative
comments, students appeared to find student presentations, group work, and graphing calculators
to be beneficial. Students generally possessed positive views regarding student presentations, and
they also possessed generally positive views about working together in groups. Further, every
comment regarding the use of graphing calculators was positive. However, students provided
mixed views regarding the relatively few number of examples that were worked by the
instructor; many students felt that they would have benefited from the instructor working out
more examples during class. Overall, students made significantly more positive comments than
negative comments about the reform-oriented course and its components (109 positive and 26
negative).
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CHAPTER 5: CONCLUSIONS AND IMPLICATIONS
In this chapter, the limitations and conclusions of the study will be presented.
Subsequently, the implications of the study for teachers and administrators will be discussed.
Lastly, directions for future research studies will be suggested.
Limitations
The current study contains several limitations beyond my control. First, the study was a
quasi-experimental study in that students were not randomly assigned to a treatment. Instead,
students enrolled into the course which fit their schedule. Secondly, I could not control for all
possible variables. Such variables extended to the extent and type of resources that students
enlisted outside the classroom. Third, the study had a small sample size. A larger sample size
would have been preferred in order to increase the statistical power of the study and therefore
obtain a higher level of confidence in the study?s results. Fourth, the two classes were conducted
at two different points in time. It is possible that students in each of the classes were affected by
a different set of social events outside the classroom. The passage of time may have also resulted
in my maturing in some manner between courses.
Two important limitations that this study attempted to mitigate were a lack of fidelity to
the treatments and the possibility of researcher bias. Scores on the Reformed Teaching
Observation Protocol (RTOP) (Piburn & Sawada, 2000) supported my claim that the Control
group received instruction consistent with traditional lecture methods and that the Experimental
group received instruction consistent with reform pedagogy. Additionally, the open-ended
student surveys provided additional data that showed that students experienced two different
types of instruction in their respective courses. The survey data also indicated that students from
the Control group felt that the instructor appropriately implemented traditional teaching
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techniques, thus helping to mitigate the possibility that the researcher inadvertently provided the
Control Group with lower quality instruction than the Experimental Group.
Conclusions
Although the data did not yield statistically significantly results, the trends within the data
were consistent with those of similar studies of other reform mathematics classrooms. Key
results for each of the research questions are presented in the sections below.
Research Questions 1 and 2: Procedural and Application Skills
Though the results were not statistically significant, the trends within the data suggested
that incorporating reform-oriented pedagogy into post-secondary remedial courses may improve
students? problem-solving abilities without sacrificing procedural proficiency. The trends in this
study are consistent with prior research on secondary students in which reform students scored as
well as traditional lecture students in procedural skills and better on problem-solving skills
(Hirschhorn, 1993; Schoen, Hirsch, & Ziebarth, 1998; Thompson & Senk, 2001). With respect to
students? procedural skills in this study, students in the Control group scored higher on the first
test but gradually decreased in performance on each subsequent test throughout the course. In
contrast, students in the Experimental course experienced both increases and declines throughout
the course. The average score on the comprehensive, procedural final exam for the Control
course was nearly the same as the average score for the Experimental course. With respect to
application skills, students in the Experimental course outperformed students in the Control
course on all five tests. The change in performances of the two classes generally mirrored each
other on the application portions of the five tests.
Students in the Experimental group demonstrated a much weaker correlation between
their average procedural scores and average application scores than did students in the Control
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group. Students in the Experimental group often earned higher average application scores than
procedural scores across the five tests in the course. In contrast, students in the Control group
tended to earn lower average application scores relative to their average procedural scores across
the five tests. The reason for this reversal in trends may be due to the difference in how the
Experimental and Control students were taught the material. Students in the Control group were
taught the most efficient methods to solve problems; these methods were most often algebraic
methods that were introduced in a general context first and then demonstrated later in a specific
context (such as a story problem). In contrast, the Experimental group was taught various
methods to explore problems such as systematic trial and error and utilizing the table and
graphing functions of a graphing calculator; algebraic methods were often introduced after
students had been given time to explore problems and develop reasonable solutions based on
non-algebraic techniques. Thus, students in the Experimental group had more methods available
to solve application problems in the event that one of those methods (such as the algebraic
method) failed them. The results in this study were similar to those of Senk and Thompson
(2006) in which secondary students in reform-based courses were more likely to use graphical
and numerical strategies to solve problems than did their matched comparison students.
Research Question 3: Pass Rates
This study found no significant difference in the pass rates between the students in the
Experimental and Control groups. Recall that a lack of consensus exists regarding maintaining
higher standards in remedial mathematics courses versus adopting acceptably lower mathematics
standards in exchange for higher pass rates (Golfin et al., 2005). In this light, the results of the
study were promising; the students in this study who were subjected to the Experimental
treatment were able to maintain the high standards of the course without reducing student pass
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rates. In other words, the gains made by the Experimental group did not come at the expense of
higher rates of attrition.
Research Question 4: Change in Mathematics Self-Efficacy
This study found no significant difference in the change in mathematics self-efficacy
between the two groups. Compared to traditional lecture methods, the trends in the data
suggested that reform-oriented instruction may produce more favorable changes in students?
mathematics self-efficacy. This trend towards improving mathematics self-efficacy in the
Experimental course was not surprising. First, several students commented that presenting their
work to the class improved their confidence in their mathematical abilities despite constructive
criticisms made by their classmates. One student commented: ?When presenting my work to the
class I gained more confidence in the way I was solving my problems. Others were also able to
point out flaws in my work as I was [able to point out flaws in] theirs. It basically made the
whole class a big group.? Second, it was common practice throughout the course for the students
in the Experimental group to spend at least 10 ? 15 minutes on a problem or a set of related
problems. The daily behavior expected of and exhibited by the students in the Experimental
group aligned with Bandura?s (1997b) description of high self-efficacy: sustaining strong
commitment towards a goal and viewing tasks as challenges to be mastered.
Research Question 5: Student Response to the Experimental Treatment
The data from free-response student surveys showed that the students in the reform-
oriented course expressed generally positive comments about the style of instruction. When
asked to compare their current class to other mathematics classes in the past, more than one third
of the students commented that that their understanding of the material resulted from non-
algebraic methods of communication. With respect to group work, graphing calculators, and
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student presentations (key elements of reform pedagogy), students? comments in the reform-
oriented class were mostly favorable. Student comments revealed that discussing problems
during group work and student presentations helped them to improve their understanding of the
material by considering different perspectives; these perspectives helped students to learn from
each other and resulted in their ?knowing the problems inside and out.? Other comments
revealed that students found graphing calculators to be useful because they helped improve
conceptual understanding and provided alternative means to solve problems through tables and
graphs. Interestingly, no negative comments were made by students about the ability to use
graphing calculators in the course.
However, roughly half of the students in the Experimental course expressed a desire to
see the instructor present and solve more problems during class. Such comments were not
surprising considering the fact that many of the students? mathematical backgrounds consisted
primarily of traditional lecture teaching. One may recall that the comments made by students in
the traditional lecture course expressed very high opinions of the teaching method due in large
part because the instructor ?tried his best in giving [students] easy ways to solve the problems?
and would teach ?in detail the steps of the problem.? In other words, many of the students in the
study seemed to prefer that the teacher solve multiple problems in a clear and detailed manner.
Implications
The results from this study can inform both teachers and administrators who engage post-
secondary remedial mathematics students. I discuss the implications of this study for teachers
and administrators. Lastly, although studies have consistently shown that the most important
factor in school effectiveness is the teacher (Boaler, 2008), I conclude this section with a
reminder of the vital role that administrators play in the implementation of reform curricula.
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Teachers
Teachers who may be interested in implementing more reform-oriented pedagogy into
their remedial mathematics courses may be discouraged if the objectives within their courses are
procedural in nature; however, this study demonstrated that significant implementation of the
pedagogy advocated by reform documents into remedial mathematics courses may be possible.
Although the objectives of the remedial mathematics course in this study were primarily
procedural in nature, I was able to transform the inherently procedure-oriented course into a
reform-oriented course by incorporating into daily activities pedagogical techniques that were
consistent with reform pedagogy. Because the course textbook within the study reinforced
primarily procedural skills, I supplemented homework problems and classroom activities with
those I created myself as well as those that I obtained from the literature. Finding reform-
oriented lessons was not exceedingly difficult. The objectives in a post-secondary remedial
mathematics course are similar to the objectives at the secondary level, and ample reform-
oriented curricula addressing secondary mathematics are available for teachers to tailor to the
needs of their classes.
Based on the resistance from students in this study to the new teaching approach, teachers
who would like to incorporate more reform-oriented techniques into their classrooms need to be
aware that many of their incoming students will likely have had little exposure to reform-
oriented instruction. Teachers should therefore make certain that students understand at the
beginning of the course the expectations for the reform-oriented course (such as active student
engagement through questioning, group work, justification of answers to peers) and should
submerge the students in reform-oriented activities at the onset of the course. Teachers who
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clearly explain and demonstrate the desired learning practices to their students can improve the
equitable nature of their instruction (Boaler & Staples, 2008)
For example, I designed an activity for the first day of the reform-based class in which
students were required to answer a basic algebra problem and then explain how they knew that
their answer was correct. In order to model what was expected of them, I answered the first
question ?What is 1x + 2x?? two different ways. First, I broke the problem into ?x + x + x? and
argued that my answer was 3x because I had a total of three x?s. My second approach involved
plugging a number (say, 4) in for x and showed that 1(4) + 2(4) = 3(4). After answering students?
questions, I let them attempt the following few problems such as ?What is (x2)(x3) ??. Although
many of the students stated that the answer was x5, their sole justification was given by phrases
such as ?that?s the rule in the book? or ?that?s what I was taught by my high school teacher.?
I then asked these students if their classmates also arrived at the same answer or if there
were any way to convince an opposing viewpoint of the validity of their answer. Thus, this first
activity set the tone for the course in that students quickly saw that they were expected to find the
answers to problems without simply relying on theorems within the book; in other words, they
were expected to derive a theorem or at least to understand why a theorem made sense.
Additionally, students realized that they were expected to work with each other to make sure that
they correctly understood the material, and if students disagreed on a solution, they were to
attempt to reconcile their differences. As the course progressed, the students gradually began
developing ways to make sense of problems and verify their answers to those problems through
methods other than ?the book said so.?
Having taught the experimental section, I offer an additional observation from this
study?teachers need to remember to be patient. It may take time for students to adjust their
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classroom learning habits from passive observers to active participants. Kara (a pseudonym), one
of the students in Experimental course, at first did not appreciate the activities and types of
problems assigned in the course. Early in the course, Kara emailed me, expressing some
frustration that the classroom activities and assigned homework did not correspond to the types
of problems presented in the text (which were procedural in nature). I replied that the classroom
activities and homework were intended to address conceptual understanding and should therefore
help to minimize common student mistakes. One week later, Kara emailed me, stating, ?I have
been working on the practice test, and I am remembering all of the mistakes I was making before
you tutored me. All the times I was forgetting the steps, and I had to figure it out...I have not
forgotten! It stuck with me. Thanks again for your help.? Kara?s diligence in ?figuring things
out? ultimately helped her to do very well in the course. At the end of the course, she emailed
me and stated that she thoroughly enjoyed my teaching style. Kara?s success story represents the
reason why teachers should expect, but not become overwhelmed by, students? initial frustrations
with reform-oriented teaching. Just as the students in prior studies (Schoen, Hirsch, & Ziebarth,
1998; Reys et al., 2003) took up to two years to fully adapt to reform-oriented instruction, Kara?s
experiences similarly demonstrate that it can take time for students to realize the benefits of
learning mathematics in a reform-oriented manner.
In this experimental course and previous pilot experimental courses, I consistently found
time to be a significant opponent. Whereas traditional lecture methods often ask students to copy
the teacher?s notes from the board (with or without understanding), reform mathematics
pedagogy asks students to experiment, to discuss the findings of their experiments with
classmates, and to learn from the findings of other classmates. Simply stated, these cognitive
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processes take time. Teachers therefore need to tailor their lessons in such a way as to allocate
sufficient time for students to fully engage in the lessons.
Lastly, for teachers wishing to transform a more traditional course into a more reform-
oriented course, it is critical for them to acquire administrative support. If students are not won
over by the advantages of reform-oriented instruction, a strongly flavored reform-oriented course
may cause students to complain to the administration or to produce negative feedback on course
evaluations. Teachers need to make certain that administrators are aware of the likelihood of
student complaints and that administrators are prepared to diplomatically address complaints
presented by students.
Administrators
Administrators need to understand the advantages of reform-oriented instruction. The
goal of reform-oriented mathematics is not to develop students? problem solving ability at the
expense of procedural proficiency. Rather, its goal is to develop students? conceptual knowledge
of the mathematics with the expectation that students will better develop and retain procedural
skills, as well as understand under which conditions those procedural skills should be applied.
The goals of reform-oriented instruction coincide with the calls made by various reports for
postsecondary students to develop critical thinking and problem solving skills (Conley &
Bodone, 2002; AEE, 2011). The trends in the data that were gathered in this study are consistent
with the claims of reform mathematics: teaching in a reform-oriented manner does no harm
while potentially providing multiple benefits.
Administrators who want their instructors to teach in a reform-oriented manner need to
provide their instructors with training and support. Administrators who would like their
instructors to teach in a more reform-oriented manner should consider encouraging their
135
instructors to attend conferences, workshops, and other types of training that can help instructors
better understand how to implement reform-oriented pedagogy. Many instructors know how to
teach only in a traditional manner; thus, administrations may need to invest time and resources
into helping their instructors understand an alternative way to teach mathematics. I would never
have been able to teach in a reform-oriented manner had it not been for the tools that I acquired
from my graduate program in mathematics education. Because I had only known traditional
lecture methods until entering graduate school, the daily pedagogical modeling by my professors
were instrumental in helping me to understand how to teach in a reform-oriented manner.
Lastly, this study reiterates the warning given above to teachers: administrative support is
a critical component in any reform process. The data within this study add to the body of
literature that state that reform-oriented teaching quite often does no harm, while in many cases
has the potential to help. In her 2008 book What?s Math Got to Do with It?, researcher Jo Boaler
described a secondary mathematics department that adopted an award-winning reform
mathematics curriculum that was supported by its teachers. Despite the success of the
academically rigorous and engaging curriculum, a small group of parents used misleading
information to lobby other parents and students into signing a petition that required the
mathematics department to abandon its reform mathematics curriculum. Ultimately, the parents
prevailed against the wishes of the teachers, and the mathematics department returned to ?the
traditional books and methods of teaching that they had used for many years, with very little
success? (p. 33). Likewise, in my study, student complaints (instead of statistical data linked to
student success) caused the department to retract support of my teaching in a reform-based
manner.
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Future Research
Although not statistically significant, the results of this study were promising. Additional
studies could be done that could overcome the limitations of the present study. Several directions
for future research studies as well as factors to consider in designing those studies are presented
in the paragraphs below.
The statistical power of my study was limited by its small sample size. Researchers
should consider replicating this type of study with a much larger sample size. A larger sample
size would improve the statistical power of the study and increase the chances of finding a
difference in treatments if any existed. Larger sample sizes would also make it easier to study
various covariates (such as Age and Race) in the context of reform mathematics. Thus,
researchers could better understand the effectiveness of reform mathematics on different
subpopulations of students. As a first step, researchers may need to develop a method of
improving the participation rate of students in their research, as this was the major cause of my
small sample size.
Additionally, future studies may consider employing multiple teachers who are capable
of teaching in both traditional and reform-oriented manners. Using such a design would similarly
strengthen the results of such studies by mitigating teacher-effect. Additionally, depending on the
scope and duration of the study, conducting the paired classes during the same academic terms
may help researchers to strengthen their research design by minimizing the effects of student and
instructor maturation. In designing such studies, researchers should seriously consider designing
paired courses that meet at the same time of day; multiple students in the present study
commented that the meeting time of the class (8:00 a.m.) affected their outlook on the course.
137
Extending the duration of the treatment may also be a point of further interest. For
example, if a mathematics program contained multiple levels of remedial mathematics courses
(such as Elementary Algebra followed by Intermediate Algebra), researchers could examine if
the effect of the treatment increased with additional exposure. The present study implemented
the treatment for a total of one academic semester. However, other studies analyzing the success
of new programs advocate allowing the program to continue for at least two years in order to
properly assess the success of the program (Schoen, Hirsch, & Ziebarth, 1998).
While some studies may examine the effectiveness of teaching methods for the remedial
mathematics course in which they are currently enrolled, other studies may track cohorts of
students and examine their success in subsequent credit-bearing courses. Success in future
courses could be analyzed both in terms of overall pass rates and academic achievement within
the course. Researchers may also wish to examine the number and level of mathematics courses
that students take during their postsecondary education, based on their exposure to traditional
and reform-oriented curricula. The results of these studies could further be analyzed according to
variables such as race, gender, and socioeconomic status.
Researchers could also examine the effectiveness of teachers collaborating together to
improve their remedial mathematics courses by implementing reform-based instruction. The
Carnegie Foundation?s Networked Improvement Community is a prime example of how
instructors and researchers can work together to supply the research community with
recommendations for non-STEM postsecondary mathematics courses (Merseth, 2011). The
collaboration network used in developing Path2Stats (Hern, 2012) is another example of how
collaboration networks can provide invaluable support for teachers who wish to improve the
structure of their mathematics courses. Similarly, the body of literature could benefit from
138
studies that examined how postsecondary remedial mathematics instructors worked together to
implement reform-based strategies into their classrooms. These studies could also examine the
paths that entire postsecondary mathematics departments took to restructure their remedial
mathematics courses.
Conclusion
The results of this study extended prior research on reform-based instruction for
secondary and introductory postsecondary courses to postsecondary remedial mathematics
courses. Although its results were not statistically significant, this study demonstrated that
reform-oriented practices at the postsecondary remedial mathematics level have the potential to
improve students? problem-solving ability and mathematical self-efficacy; these benefits may be
achieved without sacrificing procedural skills or student pass rates. Students who received
reform-oriented instruction were more likely than students who received didactic lecture
methods to use non-algebraic methods such as pictures and systematic trial and error to solve
application problems. The students in the reform-oriented course may have developed this
behavior because of their consistent exposure to word problems throughout the course; they grew
accustomed to take information from stories and interpret that information in a mathematical
manner through tables, graphs, and pictures. Comments made by the students in the reform-
oriented group about the type of instruction they received were generally positive; however,
many students felt that they would have better understood the material if the instructor had
directly explained the concepts and worked many examples during class.
Although this study showed that it is possible to incorporate reform-based pedagogical
practices into a procedurally-oriented course without limiting and possibly enhancing students?
mathematical achievement, administrator support is essential if instructors are to teach in a
139
reform-oriented manner. Administrators and teachers who attempt to implement reform-oriented
instruction should be prepared to address student complaints regarding the structure of the
course. Additionally, administrators who would like to see reform-based practices in their
mathematics classrooms should provide their instructors with sufficient training and support.
Future research should further examine the effectiveness of reform-oriented pedagogy in
postsecondary remedial mathematics courses from both students? and teachers? perspectives.
140
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156
Appendix A
Permission Forms
157
Grade Release Form
158
Informed Consent Form
159
160
Appendix B
Student Surveys
161
Efficacy Survey
Below are some questions about you as a student in this math class. Please circle the
number that best describes what you think. Your responses will be kept anonymous.
1. I'm certain I can master the skills taught in math class this semester.
1 2 3 4 5
NOT AT ALL TRUE SOMEWHAT TRUE VERY TRUE
2. I'm certain I can figure out how to do the most difficult class work in math class.
1 2 3 4 5
NOT AT ALL TRUE SOMEWHAT TRUE VERY TRUE
3. I can do almost all the work in math class if I don't give up.
1 2 3 4 5
NOT AT ALL TRUE SOMEWHAT TRUE VERY TRUE
4. Even if the work is hard in math class, I can learn it.
1 2 3 4 5
NOT AT ALL TRUE SOMEWHAT TRUE VERY TRUE
5. I can do even the hardest work in this math class if I try.
1 2 3 4 5
NOT AT ALL TRUE SOMEWHAT TRUE VERY TRUE
162
Demographic Information
For the purposes of this study, please provide the following information. The information will be
kept confidential.
Name: ______________________
On average, for how many hours are you employed each week? ___________
For how many credit-hours did you enroll this semester? __________
What is your age? __________
What is your sex:
Male Female
Please specify your race:
Black/African American White Hispanic Native American Asian/Pacific Islander Other
163
Anonymous End-of-Term Student Survey (Traditional Course)
1. How does this math class compare to other math classes that you have had? Explain.
2. What are some things you liked about the course?
3. What are some things you did not like about the course?
4. Other comments (room on back)
164
Anonymous End-of-Term Student Survey (Reform-oriented Course)
1. How does this math class compare to other math classes that you have had? Explain.
2. What are some things you liked about the course?
3. What are some things you did not like about the course?
4. To what extent did you like working with your classmates during class? Explain.
5. Did you find the graphing calculator useful? If yes, please explain how/when it was useful.
6. To what extent did you benefit from presenting your work to the class (or watching your
classmates present their work to the class)? Explain.
7. Other comments (room on back)
165
Appendix C
Sample Application Problems
166
Sample Application Problems
Test 1 (Factoring): A construction worker accidently drops a tool from the top of a 256-
foot building. The height h of the tool after t seconds is given by h = -16t2 + 256. When will the
tool hit the ground?
Test 2 (Rational Equations): On an architect?s blueprint, 1 inch corresponds to 4 feet.
Find the length of a wall represented by a line 3 ? inches long on the blueprint. Round to the
nearest tenth if necessary.
Test 3 (Functions): Michelle just purchased a used car from her uncle and agreed to pay
him a certain amount of money at the end of each month. After 3 months, she owed $6700 on the
car. After 7 months, she owed $4300 on the car. Will she be able to pay off her car by the end of
the 12th month? Explain carefully.
Test 4 (Radicals): A CSI Forensic Team found a dead man lying in the road next to a
very tall apartment building. The Forensic Team determined that the man was traveling at least
90 feet per second when he hit the ground. If the formula describes
how fast a person will be falling when they hit the ground based on their initial height off the
ground, how high off the ground was the man when he fell off the building?
Test 5 (Quadratic Equations): The following equation describes the profit, P(x) that a car
dealership makes based on the number of employees, x, that it hires: P(x) = -3x2 + 240x.
A) Find the number of employees that the dealership should hire in order to maximize its profit.
B) What is the maximum profit that the dealership can make?
167
Appendix D
Reformed Teaching Observation Protocol
168
Reformed Teaching Observation Protocol
169
170
171
172
173
Appendix E
Paired Lesson Plans
174
Traditional Lesson Plan for Difference of Squares
Title of Lesson: Factoring Differences of Squares (Traditional course)
Audience: Math 0800
Content Objectives:
Have students see that (x + a)(x ? a) = x2 ? a2
Have students see that adding x feet to the length of a square and then subtracting x feet from the
width of the square reduces the area of the square by x2 feet
Behavioral Objectives:
The student will be able to factor an expression containing a difference of squares
The student will be able to verify that the expression was factored correctly
Prerequisites:
How to multiply binomial expressions: (a + b)(c + d)
How to add like terms
Materials: None
Procedure
Phase Preliminary:
Overview: Ask students if they have any questions on their homework or other concepts already
covered in class.
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will answer homework questions posed by the students
Key Questions: Are there any questions with the homework or other material that we have
covered?
175
Phase 1:
Overview: Demonstrate to students that the product (A + B)(A - B) will always result in an
answer of the form x2 ? a2.
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will present the generalization for the difference of squares: (A+B)(A-B) = A2 ?
B2
?There is a special pattern in math called the ?difference of squares?. The pattern was
given its name because anytime you multiply two terms (A-B)(A+B) [write this on the board],
you will always get an answers that looks like this A2 ? B2 [continue writing on the board = A2 ?
B2]
2) The instructor will then demonstrate this difference of squares pattern with the following
problems:
a) (x+3)(x-3) = x2 ? 9
b) (x+5)(x-5) = x2 ? 25
Key questions:
Suppose you have (Y+Z)(Y-Z). Without working it out, what will the product look like?
Are there any questions?
Phase 2:
Overview: Apply the factoring pattern in Phase 1 to factor several relatively simple difference of
squares expressions.
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
176
1) The instructor will explain that recognizing a difference of squares can enable someone to
identify the product from which it came.
2) The instructor will then state the following rules for factoring a difference of squares:
a) Verify that the expression is indeed a difference of squares by i) observing that the
expression is a difference of two terms and by ii) identifying the square roots of each
term
b) Write the square roots down in their corresponding parentheses
c) Place an addition sign in one parentheses and a subtraction sign in the other
parentheses
3) The instructor will demonstrate how to factor the following problems:
a) x2 ? 49
b) y4 ? 100b2
c) z10 ? 144R6
4) The instructor will then use the previous problems to show students how to check their
answers by multiplying the products out again.
5) The instructor will ask students if they have any questions. Once all questions are answered,
the instructor will ask students to factor the following problem: t4 ? 49z6
Solution: t4 ? 49z6 = (t2 ? 7z3) (t2 + 7z3)
Question: How do you know if you have factored correctly?
Key Questions:
If you have a difference of squares, how do you find its factors?
Are there any questions?
Phase 3:
177
Overview: Apply the factoring pattern in Phase 1 to factor other more complicated variations of
difference of squares problems
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will then present and solve a variety of factoring problems that embody
different ways in which the difference of squares pattern can emerge:
a) 3x2 ? 12 (factor out the 3 first in order to reveal a difference of squares)
b) -9x2 + 100 (rearrange the terms in order to reveal a difference of squares)
c) P8 ? 16 (perform the difference of squares twice to fully factor the expression)
2) The instructor will then present the factoring problem ?x2 + 9? and emphasize that problems
possessing the ?sum of squares? pattern cannot be factored
3) The instructor will then ask students if they have any questions. After questions are answered,
the instructor will ask students to factor the following problems:
a) x4 ? 81
Solution: x4 ? 81 = (x2 ? 9)(x2 + 9) = (x + 3)(x ? 3)(x2 + 9)
Questions: Can x2 ? 9 be factored down further? How about x2+ 9?
b) 162 ? 8y4
Solution: 162 ? 8y4 = 2(81 ? 4y4) = 2(9 ? 2y2)(9 + 2y2)
Question: What did you have to do to see that you had a difference of squares?
Question: How many times did you have to factor a difference of squares?
Key Questions:
What are signs that you might have a difference of squares pattern?
What might you need to do in order to see a difference of squares pattern?
178
Can you factor a sum of squares?
Are there any questions?
Phase 4:
Overview: Apply the difference of squares factoring pattern to real estate
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will present the following scenario:
?A city developer normally sells square lots (S x S). However, he offers a special deal to a
newlywed couple. For the same price as the square lot, the developer will turn the square lot into
a rectangle by adding B feet to the length and then by subtracting B feet from the width. Should
the couple accept the developer?s offer??
2) The instructor first will solve the scenario abstractly in terms of S and B.
Solution: The original lot has an area of (S)(S) = S2. The modified lot will have an area of
(S+B)(S-B) = S2 ?B2. So the modified lot will have B2 less area. Therefore the couple should not
accept the offer.
3) The instructor will then present a solution to the scenario by letting S = 10 and B = 3.
Solution: The original lot has an area of (10)(10) = 100 square feet. The modified lot will
have an area of (10 + 3)(10 ? 3)= (100 + 30 ? 30 ? 9) = (100 ? 9) = 91 square feet.
Key Questions: Are there any questions?
179
Reform-oriented Lesson Plan for Difference of Squares
Title of Lesson: Factoring Differences of Squares (Reform-oriented course)
Audience: Math 0800
Content Objectives:
Have students see that (x + a)(x ? a) = x2 ? a2
Have students see that adding x feet to the length of a square and then subtracting x feet from the
width of the square reduces the area of the square by x2 feet
Behavioral Objectives:
The student will be able to factor an expression containing a difference of squares
The student will be able to verify that the expression was factored correctly
Prerequisites:
How to multiply binomial expressions: (a + b)(c + d)
How to add like terms
Materials: None
Procedure
Phase 1:
Overview: Present a situation where a developer modifies square lots by adding a particular
distance to the length of the square and then subtracting the same distance from the width of the
square (to form a rectangle).
Grouping: Students will sit in groups that are heterogeneous in mathematical ability
Tasks:
1) Determine if the developer?s modifications alters the area of the original lot
180
2) Determine the effect that adding/subtracting x feet to a square lot has on the area of the
original lot
3) When given the area of the new/modified lot, determine what changes the developer made.
4) Generalize the factoring pattern for x2 ? c2
Key questions:
How could you go about figuring out what happens to the area of the lot when the developer
implements his changes?
What happens to the area of the lot when the developer adds/subtracts the same amount to each
side?
How much does the area of the lot change when the developer adds/subtracts x feet to each side?
What strategies (ex. pictures, tables) helped you discover the effect that occurred when altering
the original lots?
Phase 2:
Overview: Apply the factoring pattern in Phase 1 to factor various expressions.
Grouping: Students will sit in groups that are heterogeneous in mathematical ability
Tasks:
1) Factor: x4 ? 81
Solution: x4 ? 81 = (x2 ? 9)(x2 + 9) = (x + 3)(x ? 3)(x2 + 9)
Questions: Can x2 ? 9 be factored down further? How about x2+ 9?
2) Factor: 162 ? 8y4
Solution: 162 ? 8y4 = 8(81 ? y4) = 8(9 ? y2)(9 + y2) = 8(3 + y)(3 ? y)(9 + y2)
Question: What did you have to do to see that you had a difference of squares?
Question: How many times did you have to factor a difference of squares?
181
Key questions:
How do you verify that you have factored a problem correctly?
How do you factor a sum of squares (like x2 + 9)?
Handout Given to Students
Recall that a city developer wanted to change the boring square house lots of a neighborhood
into more creative rectangle lots. To spice things up, he added, say, 2 meters to the length of the
square and then subtracted the same amount (in this case 2 meters) from the depth. So if the
developer adds some amount to the length but immediately subtracts that same amount from the
depth, what happens to the area of the lot?
For each of the changes that were made to the square lots, complete these tasks:
*Make and label a sketch of the original square lot, using the variable X to represent the length
of the original square
* Make and label a sketch of the new lot, using the variable X to represent the length of the
original square
*Write an expression for the area of the new lot as a product of its length and width
*Write an expression without parentheses for the area of the new lot as a sum of smaller areas.
Use your sketch to explain this expression.
1) Suppose that the developer wanted to increase one side of the square lot by 5 meters but
decrease the other side by 5 meters. Will the area of the new lot be the same as the area of the
original lot?
2) Suppose that the developer wanted to increase one side of the square lot by 3 meters but
decrease the other side by 3 meters. Will the area of the new lot be the same as the area of the
original lot?
182
3) So when the developer increases/decreases the lot
by some amount, B, what happens to the area of the
lot?
4) If you see a lot with an area of x2 ? 16, what were
the original dimensions?
5) If you see something like x2 ? c2, how do you factor that?
Factor: x4 ? 81
Factor: 162 ? 8y4
Dimensions of new
lot
Area
183
Traditional Lesson for Shifting of Graphs
Title of Lesson: Shifting of Graphs
Audience: Math 0800
Content Objectives:
Have students recognize 4 families of graphs (linear, quadratic, radical, absolute value)
Have students see that y = (x + a)2 will shift the graph horizontally ?-a? units
Have students see that y = (x)2 + a will shift the graph vertically ?a? units
Have students see that y = -(x)2 will flip the graph
Behavioral Objectives:
The student will be able to graph quadratic, radical, and absolute value functions
Prerequisites:
How to create and use an x/y table
How to compute absolute value expressions
Materials: None
Procedure
Phase Preliminary:
Overview: Ask students if they have any question on their homework or other concepts already
covered in class
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will answer homework questions posed by the students
Key Questions: Are there any questions with the homework or other material that we have
covered?
184
Phase 1:
Overview: Provide four common functions and their corresponding x/y tables and graphs
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will write the following functions on the board and then explain how to use an
x/y table to graph each of the functions
y = x
y = x2
y =
y = | x |
The instructor will explain that the above four functions are common in mathematics courses and
that it is important to become familiar with each of their shapes
?These 4 functions are ?mother functions? because they are the simplest versions of each family
of functions, and from each of them come all of the other functions that we will see in this
course. Note that the four basic shapes are lines, U?s, cursive r?s, and v?s.?
Key Questions:
If you ever forget what the shape of a graph is, what can you do? (use an x/y table)
Phase 2:
Overview: Demonstrate the change in the y = x2 function?s graph by systematically modifying
the original function
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
185
1) The instructor will state that for the y = x2 family of functions, y = x2 + a will shift the graph
?a? spaces along the y-axis, y = (x + a)2 will shift the graph ?-a? spaces along the x-axis, and y =
-x2 will flip the graph
2) The instructor will modify y = x2 and provide x/y tables to support the creation of the newly
adjusted graphs
y = x2
y = x2 + 1
y = x2 ? 1
y = (x + 1)2
y = (x ? 1)2
y = - x2
3) The instructor will provide other tips:
a) Notice that these types of graphs have a natural symmetry
b) It helps to focus on the vertex of a graph when shifting it
Key Questions:
What is the basic shape to a y = x2 graph?
How does adding/subtracting within the squaring mechanism affect the graph compared to
adding/subtracting outside the squaring mechanism?
What happens to the original graph when you put a negative sign on the x2?
Phase 3:
Overview: Demonstrate the change in the y = function?s graph by systematically modifying
the original function
Grouping: Students sit in desks by themselves facing the instructor
186
Tasks:
1) The instructor will state that for the y = family of functions, y = + a will shift the
graph ?a? spaces along the y-axis, y = will shift the graph ?-a? spaces along the x-axis,
and y = will flip the graph
2) The instructor will modify y = and provide x/y tables to support the creation of newly
adjusted graphs
y =
y = + 1
y = - 1
y = (students try)
y =
y =
3) The instructor will provide other tips:
a) Notice that the graph does not go infinitely in both directions; it has a ?starting point?
b) The starting point exists because negative numbers are not allowed in the square roots
c) The graph starts off rather quickly and then grows very slowly
Key Questions:
What is the basic shape to a y = graph?
How does adding/subtracting within the squaring mechanism affect the graph compared to
adding/subtracting outside the squaring mechanism?
What happens to the original graph when you put a negative sign on the ?
Phase 4
187
Overview: Demonstrate the change in the y = | x | function?s graph by systematically modifying
the original function
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will state that for the y = | x | family of functions, y = | x | + a will shift the
graph ?a? spaces along the y-axis, y = | x + a | will shift the graph ?-a? spaces along the x-axis,
and y = - | x | will flip the graph
2) The instructor will modify y = | x | and provide x/y tables to support the creation of newly
adjusted graphs
y = | x |
y = | x | + 1 (students try)
y = | x | - 1
y = | x + 1 |
y = | x ? 1 |
y = - | x |
Key Questions:
What is the basic shape of a y = | x | graph?
How does adding/subtracting within the squaring mechanism affect the graph compared to
adding/subtracting outside the squaring mechanism?
What happens to the original graph when you put a negative sign in front of the |x|?
Phase 5:
Overview: Demonstrate the change in each function?s graph by modifying the original function
in multiple ways
188
Grouping: Students sit in desks by themselves facing the instructor
Tasks:
1) The instructor will state that the above changes are cumulative; modifying a function in 2
ways will cause the graph to change in those 2 respective ways. For example:
y = (x + 2)2 ? 1 will cause the vertex (and thus the rest of the graph) to shift left 2 spaces and
down 1 space. The instructor will show the change in the graph by using both the rules as well as
using an x/y table.
y = will cause the vertex (and thus the rest of the graph) to shift right 2 spaces and
up three spaces; the negative sign in front of the square root will cause the graph to flip. The
instructor will show the change in the graph by using both the rules as well as using an x/y table.
2) Ask students to graph several functions
a) Ask students to graph y = -|x + 4| - 2 on their own. After a few minutes, ask them to
describe the graph of the function.
b) Ask students to graph y = on their own. After a few minutes, ask them to
describe the graph of the function.
c) Ask students to graph y = (x ? 3)2 ? 1 on their own. After a few minutes, ask them to
describe the graph of the function.
Key Questions:
How can you look at a graph and determine its shape, orientation, and location?
If you forget which way a graph is supposed to shift, what can you do to figure out the graph?s
correct orientation? (use an x/y table)
What range of x-values should you use when creating an x/y table to graph a function? (at least -
6 to 6)
189
190
Reform-oriented Lesson for Shifting of Graphs
Title of Lesson: Shifting of Graphs (Reform-oriented course)
Audience: Math 0800
Content Objectives:
Have students recognize 4 families of graphs (linear, quadratic, radical, absolute value)
Have students see that y = (x + a)2 will shift the graph horizontally ?-a? units
Have students see that y = (x)2 + a will shift the graph vertically ?a? units
Have students see that y = -(x)2 will flip the graph
Behavioral Objectives:
The student will be able to graph quadratic, radical, and absolute value functions
Prerequisites:
How to create and use an x/y table
How to compute absolute value expressions
Materials: Graphing Calculators
Procedure
Phase 1:
Overview: Present graphs of 4 functions (absolute value, quadratic, radical, and linear) and ask
the students to match those graphs to their corresponding equations
Grouping: Students will sit in preselected groups that are heterogeneous in mathematical ability
Tasks:
1) Students match the four graphs presented to their corresponding equations by using either an
x/y table or the graphing calculator
Key questions:
191
What is the basic shape of the following graphs: quadratic, radical, absolute value, linear
How are the linear and absolute value graphs related?
How are the quadratic and absolute value graphs similar and different?
How is the radical graph different from the other graphs?
How could you convince someone that the graph of a quadratic function (for example) is shaped
like a U?
If you use an x/y table to graph the absolute value function, and you only use the points
associated with x = 0, 1, 2, what type of graph might you create?
If you use an x/y table to graph your functions, which x-values and how many x-values should
you use?
Phase 2:
Overview: Students figure out what happens to the original graphs when the equations of those
graphs are modified
Grouping: Students will sit in preselected groups that are heterogeneous in mathematical ability
Tasks:
1) Students graph 6 carefully selected versions of y = x2 and determine if they can make any
generalizations regarding the way the original graph is changed
y = x2
y = -x2
y = (x+2)2
y = (x-2)2
y = x2 + 2
y = x2 - 2
192
2) Students confirm within their groups that their graphs are correct, and then the class as a
whole will verify that the graphs are correct. Students will also describe how they graphed the
functions.
3) Students will discuss generalizations within their groups and then discuss with the rest of the
class their generalizations
Key questions:
What generalizations/patterns do you notice?
Do you think that the patterns you see hold for other types of functions?
What did you notice when you added 2 inside the squaring mechanism versus when you
subtracted 2 within the squaring mechanism?
What happened when you added 2 outside the squaring mechanism versus when you subtracted 2
outside the squaring mechanism?
How does adding/subtracting within the squaring mechanism affect the graph compared to
adding/subtracting outside the squaring mechanism?
Phase 3:
Overview: Students determine if the generalizations made in Phase 2 hold for absolute value
equations
Grouping: Students will sit in preselected groups that are heterogeneous in mathematical ability
Tasks:
1) Students graph 6 carefully selected versions of y = |x|
y = |x|
y= -|x|
y = |x+1|
193
y = |x - 1|
y = |x| + 1
y = |x| - 1
2) Students confirm within their groups that their graphs are correct, and then the class as a
whole will verify that the graphs are correct and discuss how they graphed the functions.
3) Students will discuss within their groups if their generalizations from quadratic functions also
hold for absolute value functions
Key questions:
How many of the generalizations you made for quadratic functions held for absolute value
functions?
Do you think these generalizations hold for radical functions also?
What are the pros and cons of using a graphing calculator to graph functions?
Phase 4:
Overview: Students use the previous generalizations to graph functions with combinations of
shifting [ex. y = -(x + 2)2 ? 4]
Grouping: Students will sit in groups that are heterogeneous in mathematical ability
Tasks:
1) Students will consider the graphs below and declare how they think the graph will look.
f(x) = (x - 3)2 + 2,
f(x) = -|x - 2| - 4,
2) Students will then verify that their declarations are correct
Key questions:
194
How can you look at a graph and determine its shape, orientation, and location?
If you are ever unsure of how a graph should look, what can you do?
195
Handout Given to Students
Families of Functions
Match the following functions to their corresponding graphs and x/y tables:
Table
of
values
Graphs
196
Quick question: Are the above graphs functions? How do you know?
2. The above functions are sometimes called ?parent? functions because they are written as
simply as possible. But what happens when you start changing these parent functions one piece
at a time?
Let?s first look at the parent function f(x) = x2. How do you think the picture associated with
f(x) = x2 will change as you modify different parts of the function?
Graph the following functions (LABEL EACH GRID) and see if you can detect a pattern. (Use
as many of the following grids as you like to keep your graphs from getting too cluttered.)
(Hints: 1. Divide the work among your teammates, 2. your graphing calculator can save you
bunches of time)
x f(x)
-3 3
-2 2
-1 1
0 0
1 1
2 2
3 3
x f(x)
-3 -3
-2 -2
-1 -1
0 0
1 1
2 2
3 3
x f(x)
-3 undefined
-2 undefined
-1 undefined
0 0
1 1
4 2
9 3
x f(x)
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
197
f(x) = x2 f(x) = (x + 2)2 f(x) = (x - 2)2 f(x) = x2 + 2 f(x) = x2 - 2 f(x) = -x2
Can you make any generalizations about how the picture will change based on how the function
is changed?
198
3. In the previous section, you may have developed an idea about how the graph will change
when you modify different parts of the function. Let?s see if your suspicions hold true for the
next set of functions.
f(x) = |x| f(x) = |x + 2| f(x) = |x - 2| f(x) = |x| + 2 f(x) = |x| - 2 f(x) = -|x|
199
What generalizations can you make now?
4. Based on your observations above, what do you think will happen to the graphs of the
following functions?
FIRST WRITE DOWN WHAT YOU THINK WILL HAPPEN. After you?ve done that, then see
if you were right.
f(x) = (x - 3)2 + 2 f(x) = -|x - 2| - 4
If you are ever unsure of how a graph should look (and you don?t have your graphing calculator),
what can you do?
200
Appendix F
Responses to Open-ended Student Surveys
201
Student Responses from the Control Group for Question 1
Question 1: ?How does this math class compare to other math classes that you have had? Explain.?
Category # of
comments
Representative Student Quotes
Comments about the
teacher
Positive comments about
the instructor?s
explanations
11 HH. This class has been very helpful. Mr. Smith taught us in detail the
steps of a problem.
KK. It is different because in my other math classes I was lost because I
didn?t have my instructor break the work down.
LL. The teacher explained things better in this class
TT. This Math 0800 class is very easy because our instructor simplifies the
problems, so everyone can learn it and grasp on to the concept.
VV. [The instructor] explains basic concepts better than other instructors I
have had.
Positive comments about
the instructor in general
7 AA. The only good thing about [the course] was the professor. He made me
not dread coming to class.
BB The teacher spent more time covering the material and actually helping
students
EE. Awesome teacher
RR. The teacher provides an in-depth learning experience.
SS. I enjoyed this math class [because] my teacher was very funny
Neutral comments about
the instructor in general
1 JJ. I have never had a teacher have their back towards the class.
Miscellaneous comments
about the course
Positive miscellaneous
comments about the course
5 QQ. This math class is much easier to follow and understand.
QQ. The pace that is set is extremely acceptable
WW. [The course] was pretty cool.
FF. [The course was] more hands on.
MM. It actually breaks down the material for my understanding
Neutral miscellaneous
comments about the course
2 II. This class is more in depth.
OO. [The course] was a review because I knew most of the material
Negative miscellaneous
comments about the course
1 PP. [The course] was very difficult. I had a hard time in this class.
202
Student Responses from the Experimental Group for Question 1
Question 1: ?How does this math class compare to other math classes that you have had? Explain.?
Category # of
comments
Representative Student Quotes
Conceptual Understanding
Positive comments about
improving conceptual
understanding
8 C. This math class was easier to understand. The pictures and graphs
helped me to visualize the problems and the concepts. This made a big
difference for me.
K. We actually learned WHY things in math are the way they are. And we
were asked why does a graph do this and how does the equation give
certain values. Other classes told us ?this is the answer and that?s it.?
(quotes added)
S. Concepts taught by using logical explanations as opposed to algebraic.
P. [The class] was different, but I learned MUCH more. I feel like I
learned ?math?, not just 0800 stuff. I feel like I have a lot more tools now
to use in my next math course. I am more confident with numbers now.
Group work/student
interactions
Positive comments about
group work/student
interactions
4 O. It was more open and interactive.
R. Never worked in groups before?it was fun.
S. Sitting in groups is different. More discussion than lecture.
Negative comments about
group work/student
interactions
1 J. It was harder to learn in [this class] because we had to ask group
members for answers/explanations rather than [the] teacher
Opportunities for students
to learn the material
themselves
Positive comments about
students having opportunities
to learn the material
themselves
1 E. Lots different in the sense ?we teach ourselves? by trial and error, but
it?s a good different!
Neutral comments about
students having opportunities
to learn the material
themselves
1 G. It is different, almost taught by class with some instruction by teacher.
Negative comments about
students having opportunities
to learn the material
themselves
1 T. I don?t deal well with ?figure it out yourself? methods. I need to be told
how to do something or else I will never get it.
Real World Applications
Real World Applications
(neutral comment)
1 A. [This class] applied more to [the] real world
Miscellaneous comments
about the course
Positive miscellaneous
comments about the course
3 B. Glad there was homework unlike my other class
Q. One of the best [classes]
U. [This class was] much better than previous classes. I enjoyed the
method used to teach this class.
Neutral miscellaneous
comments about the course
2 H. It was harder, but still able to be learned.
L. It was a little more challenging.
Negative miscellaneous
comments about the course
1 I. In previous classes, when [a student] was asking a question, the
instructor NEVER answered back with a question. There is no point to
[answering back with a question] when a [student] asks a question; [the
student] needs help, not being asked another question and confusing them
203
even more. This is far worse than the Pakistan teacher I had in high
school.
204
Students Responses from the Control Group for Questions 2-4
Category # of
comments
Representative Student Quotes
Student Learning
Positive classroom
environment
1 SS. [The instructor] actually cared about our opinions on the lessons.
Comments about the
Teaching Method
Positive comments about
the teaching method
10 AA. I enjoyed the lecture.
HH. The teacher knows what he is doing and tried his best in giving us easy
ways to solve the problems
JJ. [The instructor] does not make math boring.
RR. The teaching style of [the instructor] is very good.
SS. I liked that [the instructor] made math as fun as he knew how
QQ. [The instructor] is my favorite thing about this course, because even
though he?s brilliant, he can explain things in a VERY easy to understand
way.
Negative comments about
the teaching method
2 JJ. [The instructor?s] back towards the class
JJ. The use of the word ?like?. Math is a science, not an art, so it is ?is? not
?like??so I have been told by past teachers.
Comments about the
teacher
Positive comments about
the teacher
9 DD. The professor was very helpful
MM. The enthusiasm from the teacher
SS. I really enjoyed my instructor. I think he is a wonderful teacher. I wish
I could take him for all of my math courses. I?m going to miss him.
OO. Professor Smith made sure everyone understood.
Comments about the
course in general
Positive comments about
the course in general
3 GG. [The course] was easy.
LL. The material wasn?t as hard as I thought.
PP. [The course is] preparing me for [the subsequent math course].
Negative comments about
the course in general
5 GG. Some tests were hard.
II. [There was] not enough time
NN. We could only retake 1 test
TT. [I did not like that students] couldn?t use calculators on the [final
exam] (I needed [the calculator] at times.)
Comments about
homework
Positive comments about
homework
4 UU. I liked the homework. It wasn?t overwhelming but it helped a lot.
WW. [I liked the] homework.
Negative comments about
homework
2 TT. [I did not like] all the homework problems
Other comments
Positive comments about
Math Lab
1 VV. Math Lab provides a lot of extra practice?very helpful.
Negative comments about
the Math Lab
1 AA. The math lab was mind-numbing. Busy work isn?t for me. Also, I feel
that it hindered me in making [me] better.
Outside tutoring 1 TT. [I liked getting help from] tutor[s] from the Instructional Support Lab
Negative comments about
the class?s meeting time
5 CC. I hate that I chose a morning class
VV. 8am. Blah!
Students? positive
responses when asked
about any negatives in the
course
10 BB. Nothing [was bad]. Everything was great.
MM. N/A?[Nothing was bad]
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Student Responses from the Experimental Group for Questions 2-7
Category # of
comments
Student Quotes
Student Learning
Learning with
Understanding
4 K. Everything was broken down and nothing was harder than it had to be.
My favorite part was learning how to find the square root of a number with
a calculator and how the process of using monsters and certain level prisons
were used.
N. [I liked the] VISUAL LEARNING.
P. [There were] no huge ?math terms?
Positive classroom
environment
1 T. [I liked the] freedom to express ideas.
Student Presentations
Positive general comments
about student presentations
in general
4 O. [I liked] the way everyone was kinda forced to get involved and talk in
front of class.
L. It made me nervous, but it pushed me to make sure I had the right
answers.
Mixed comments about
student presentations in
general
2 G. [They helped me] A lot, except when they were wrong, then it confused
me even more
T. Seeing different ways of doing something both confused and helped me.
Negative comments about
student presentation in
general
3 J. [I] would rather have had [the] teacher explain it versus another student
[explaining it].
S. Not applicable; more scrutiny/arguing took up class time that could have
been spent on more examples and material.
Student presentations
improved student learning
6 K. It gave us a chance to not only compare answers but to see HOW we got
the answer or if we found an easier way to do a problem.
M. [Watching classmates present their work] helped me to understand what
I had done in simple terms.
N. Explaining it to others helps you learn.
U. [Student presentations helped] a lot. It helped me retain what I learned in
class.
Student presentations
helped students to learn
multiple ways to solve a
problem/view a concept
7 B. You could see how people worked problems different ways.
K. [Student presentations helped us see] if we found an easier way to do a
problem.
Student presentations
improved students?
confidence in their math
abilities/public speaking
skills
4 C. When presenting my work to the class I gained more confidence in the
way I was solving my problems. Others were also able to point out flaws in
my work as I was [able to point out flaws in] theirs. It basically made the
whole class a big group.
E. [They helped me] not to be afraid of being wrong.
N. [It] helped me to see that I can do things right.
Group Work
Positive comments about
group work
20 A. Yes, [it was] fun to see how everybody had a different thought process.
C. It was fun to argue over whose answers were wrong and right and then
find out why, and some students had great ways of explaining things too.
M. [I] enjoyed the interaction. [We] supported each other.
O. It made class more fun and more interesting. You make friends easier.
P. [I] LOVED it! It was VERY helpful to me and I learned A LOT from my
classmates. Sometimes on the test I knew how to do a problem because I
remembered something a classmate said.
U. [Working with classmates helped] a lot. It brought up different views
and opinions of problems that were beneficial to knowing problems inside
and out.
Negative comments about 5 C. Sometimes the people in the groups could be a little bit distracting.
206
group work
J. [I did not like] Working in groups and depending on your table for
correct answers and ways to solve problems.
F. Not really. I like the teacher to teach.
J. [I] would prefer to work with classmates like once a week instead of
every day
T. I like talking to them but not sharing work.
Graphing Calculators
Positive comments about
graphing calculators in
general
8 O. Yes, I had never used one since this class, and those things can
practically solve the problem for you.
P. Yes [I liked them], but I knew I couldn?t use it on the final so I felt I
couldn?t ?depend? on it.
T. Yes. The use (once I got a handle on using it) of the table function and
graphing equations was helpful.
Graphing calculators helped
me to understand problems
2 G. [They helped with] actually seeing how problems were worked.
S. Yes, graphs helped comprehension.
Graphing calculators helped
me check my answers
4 L. Yes, it helped to verify my answers
N. VERY! [It was] awesome to learn ways to check.
Graphing calculators made
graphing easier
8 C. Yes. When my mind went blank on figuring out how to graph an
equation, I remembered the graphing calculator way which saved me a
couple of times.
J. Yes, [they] helped with graphs, tables, and square roots
K. Yes, when you want to have a visual of the vertex or look at the x-
intercepts or see how the graph shifts when you have y = x2 vs. y = (x + 1)2
? 4
Graphing calculators helped
me to solve problems
4 A. Yes, [I liked graphing calculators because] basic math [took] less time to
figure out
R. Yes, as soon as I learned how to use [the calculator], the problems
became easier.
U. Yes, [calculators] helped me solve problems using graphs and tables. I
liked that I didn?t have to solve everything mathematically.
Comments about the
Teaching Method
Positive comments about
the teaching method
7 E. The professor didn?t stand in front of class and lecture boringly every
day.
N. [I] LOVED [the course]. All of the course was helpful.
G. Math was fun this semester.
K. The class was great. I haven?t had a decent math teacher since 8th grade,
and I didn?t hate waking up in the morning for math for a change after the
teachers in high school.
Negative comments about
the teaching method
11 F. Too much time trying to figure things out on my own. I like a teacher
that teaches the whole time. Example?example?example.
O. Need to focus a little more on working a few more problems that are
going to be on test.
S. Too much discussion; [it] takes longer to get through material.
H. I would rather have learned the regular way of teaching with just
problems instead of stories.
Comments about the
teacher
Positive comments about
the teacher
7 N. Very enthusiastic teacher.
P. [The teacher] does not make us feel stupid; you do not talk down to us
V. The teacher is an excellent teacher, tutor, a lot of fun, and very
knowledgeable and helpful.
Other comments
Conflicts with the Math
Lab
3 G. The class [way] did not correspond with the math lab way.
P. Sometimes [the teacher] did not match what we [students] did in [Math]
207
lab and that was hard. Sometimes I needed more practice than just one
worksheet, because it was different than the lab work.
S. Examples in class were not [the] same as [Math] lab or book (as in-
depth).
Positive general comments 7 A. [I liked the] Handouts
B. [I liked] the homework
N. Thank you for EVERYTHING!
O. Keep up the good work Mr. Luke
Negative comment about
the class?s meeting time
1 U. [The class] was at 8am.