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A study of alignment between the Common Core State Standards in math and an introductory college textbook
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A study of alignment between the Common Core State Standards in math and an introductory college textbook
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Content
A STUDY OF ALIGNMENT BETWEEN THE COMMON CORE STATE STANDARDS IN
MATH AND AN INTRODUCTORY COLLEGE TEXTBOOK
by
Julie Ku‘uleialoha Reyes Oda
A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
May 2022
Copyright 2022 Julie Reyes Oda
ii
Dedication
To my family.
iii
Acknowledgements
I want to thank my family for putting up with me writing this, then picking it back up
years later to finish. I appreciate them for attending semesters of my doctoral classes and sitting
outside to keep my infant near to run him to me when he needed his mom. This process was a
family journey!
I also want to thank my dissertation committee: Morgan Polikoff, Larry Picus, and Julie
Slayton. In the second half of this process, Morgan Polikoff became my chair and helped get me
over the finish line with his advice, feedback, references, and encouragement. Larry Picus and
Julie Slayton were so much help as committee members providing constructive feedback. Many
staff at the Doctoral Support Center helped me over the years, with Chris Mattson providing the
finishing touches and encouragement to keep moving.
E lawe i ke a‘o a mālama, a e ‘oi mau ka na‘auao.
He who takes his teachings and applies them increases his knowledge (Pukui, 1983).
iv
Table of Contents
Dedication ....................................................................................................................................... ii
Acknowledgements ........................................................................................................................ iii
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Abstract ........................................................................................................................................ viii
Chapter One: Introduction .............................................................................................................. 1
Background of the Problem ................................................................................................ 2
Statement of the Problem .................................................................................................... 7
Purpose of the Study ........................................................................................................... 8
Significance of the Study .................................................................................................... 9
Limitations and Delimitations ........................................................................................... 10
Definition of Terms ........................................................................................................... 11
Organization of the Study ................................................................................................. 12
Chapter Two: Literature Review .................................................................................................. 13
Remediation ...................................................................................................................... 13
High School Preparation and College Math Readiness Indicators ................................... 20
Standards-Based Education Reform ................................................................................. 34
Measuring Textbook Alignment ....................................................................................... 38
Summary ........................................................................................................................... 41
Chapter Three: Methodology ........................................................................................................ 42
Research Design ................................................................................................................ 43
Textbook Application and Selection ................................................................................. 44
Instrumentation ................................................................................................................. 45
Data Collection. ................................................................................................................ 48
Data Analysis .................................................................................................................... 51
Chapter Four: Results ................................................................................................................... 56
Alignment Indices ............................................................................................................. 56
Summary ........................................................................................................................... 68
Chapter Five: Discussion .............................................................................................................. 70
Summary of Findings ........................................................................................................ 70
Implications for Practice ................................................................................................... 71
Recommendations for Research ....................................................................................... 74
Conclusions ....................................................................................................................... 76
References ..................................................................................................................................... 77
v
Appendix A: List of Topic Cells ................................................................................................... 98
Appendix B: List of Cognitive Demand ..................................................................................... 107
vi
List of Tables
Table 1: Information on the Textbook Reviewed………………………………………… 46
Table 2: Categories of Cognitive Demand……………………………………………….. 53
Table 3: Standards Alignment Indices……………………………………………………. 56
Table 4: Proportional Emphasis on Cognitive Demand Levels in Textbooks and
Standards…………………………………………………………………………………..
59
Table 5: Top 20 Over Emphasized Topics……………………………………………….. 60
Table 6: Top 20 Under Emphasized Topics……………………………………………… 64
Table A1: Codebook Topics……………………………………………………………… 98
Table A2: Level of Emphasis…………………………………………………………….. 104
Table A3: Cognitive Demand……………………………………………………………... 104
Table A4: Extent of Cognitive Demand…………………………………………………... 105
Table A5: Math Practice Codes and Details………………………………………………. 105
Table A6: Math Practices and Processes………………………………………………….. 106
vii
List of Figures
Figure 1: The Readiness Gap by Institutional Sector…………………………………………. 21
Figure 2: Sample Coding Page………………………………………………………………… 48
Figure 3: Sample Textbook Page from Exercise Set 1.1 with Content Codes………………… 50
Figure 4: Sample Proportions from Chapters 8–14 and Content Analyses…………………… 55
viii
Abstract
Despite A Nation at Risk being almost 40 years old with the claims that American schools were
failing and had declining educational standards (National Commission on Excellence in
Education, 1983) the spotlight on failing schools and low educational standards has not gone
away. This study aims to uncover if mathematics content and cognitive demand are consistent
from high school to college. About a third of first- and second-year college undergraduates
reported taking a remedial course in the 2011-2012 year (Skomsvold, 2014). Mathematics was
the remedial course most enrolled in. A content analysis was conducted using the Surveys of
Enacted Curriculum, to determine the level of consistency between the Common Core State
Standards for high school and an introductory college-level (non-STEM) mathematics textbook.
The analysis was to determine which topics and cognitive demand in the book or standards had a
high degree of alignment or low degree of alignment (misalignment). The results showed an
overall lower level of alignment than the Polikoff (2015) study. The top 20 over emphasized
topics made up about half of the book’s content while the standards were 8%. Regarding
cognitive demand, 18 of 20 of the top 20 over emphasized topics focused on procedures that
were consistent with Polikoff (2015). The top 20 under emphasized topics were in 2% of the
textbook and 23% of the standards. Regarding cognitive demand, 14 of 20 of the top 20 under
emphasized topics were focused on procedures. The textbook showed poor alignment to the
standards in content and cognitive demand. The study suggests that the textbook was not
evaluating new content, it was reteaching many high school concepts. While teachers rely
heavily on textbooks to inform curriculum (Chingos & Whitehurst, 2012), further research could
include studying the alignment of instruction, other curriculum materials, and assessments to
include student outcomes and textbook comparisons.
1
Chapter One: Introduction
The Common Core State Standards (CCSS) have been the center of a national
controversy about education reform in recent years. The Common Core State Standards Initiative
began as a state-led effort in 2009 to develop clear, common, and real-world performance goals
to ensure that all students in the United States would be prepared for career, college, and life
after graduation (Common Core State Standards Initiative, n.d.c). Despite numerous reform
efforts to improve student achievement, the CCSS came about as a seemingly simple solution to
this complicated educational problem. In 2019, ACT, Inc. reported that 37% of all ACT-tested
high school graduates met three of the four ACT College Readiness Benchmarks (ACT, Inc.,
2019). On the other hand, 36% of all ACT-tested high school graduates met none of the ACT
College Readiness Benchmarks (ACT, Inc., 2019).
Increasing the number of graduates who have the skills to succeed in college-level
courses has critical implications for future policy and practice. This study focuses on college
readiness by exploring the alignment of the content and student expectations in the transition
period between high school and college. Specifically, the CCSS and an introductory college-
level mathematics textbook will be analyzed using the Surveys of Enacted Curriculum (SEC) to
determine the extent to which the content and student expectations are consistent or to determine
the sources of misalignment in areas of no consistency. The findings could provide information
that would aid policy and practice in Hawai‘i in regards to college readiness standards,
articulation between secondary and postsecondary institutions, and the breadth and depth of
mathematics content and curriculum, which may, in turn, decrease the need for mathematics
remediation for first-year college students in the state.
2
Background of the Problem
On February 24, 2009, President Barack Obama addressed a Joint Session of Congress
and challenged colleges to increase the number of students graduating with a degree or diploma
by 5 million within the next decade (Humphreys, 2012; Kelly & Schneider, 2012; Obama, 2009).
The intention of President Obama’s National Completion Agenda was for the United States to
have the highest concentration of college graduates in the world by 2020 because a good
education is a prerequisite and not an option (Humphreys, 2012; Kelly & Schneider, 2012;
Obama, 2009). This goal not only required financial support but support in human capital.
Individuals and countries benefited from investing in education so long as there was a need for
workers with high-level skills (Organisation for Economic Co-operation and Development,
2012). The United States needs those workers and should invest in education.
The Great Recession, which officially lasted from 2007–2009, began with the burst of the
housing bubble, followed by a loss of wealth, cutbacks in consumer spending, and financial
market chaos (Bivens, 2016). The combination of cutbacks in consumer spending and business
investment resulted in the most extensive job loss (8.6 million) that the United States has seen
since the Great Depression (Carnevale et al., 2013). The Great Recession precipitated President
Obama’s Completion Agenda.
The U.S. Bureau of Labor Statistics (2012) notes that higher unemployment rates are the
most widely recognized recession indicator. At the start of the Great Recession, the national
unemployment rate was 5%, and at the end of the recession, it was 9.5%, which was higher than
most industrialized countries at the time. The unemployment rate climb was not over yet. Four
months after the recession, the unemployment rate peaked at 10% (U.S. Bureau of Labor
Statistics, 2012). The State of Hawai‘i Department of Labor and Industrial Relations reported
unemployment rates from 2007 and 2009 as 2.6% and 6.8%, respectively (State of Hawai‘i
3
Department of Labor and Industrial Relations, 2008; State of Hawai‘i Department of Labor and
Industrial Relations, 2010). Unemployment does not hit all sectors or demographics equally.
However, it disproportionately affects younger Americans (under 24), who are twice as likely to
be unemployed as the entire working population (Carnevale et al., 2013). The recession has
increased the need for employees with postsecondary education to meet the demands of the
changing labor market.
Individuals who have a college degree are most likely to be employed. The demand in the
U.S. labor market for skilled workers has increased, but college graduates have not kept up with
the pace of demand since the late 1970s and early 1980s (Autor, 2011). Carnevale et al. (2010)
maintain that “workers with college degrees had the lowest unemployment rates over the past
three years, thus receiving the best possible shelter from the Great Recession of 2007” (p. 5).
High-skilled and technical positions had no net change in employment gains/losses during 2007–
2009, while the brunt of job losses was felt in middle-skill occupations (Carnevale et al., 2012).
Autor (2011) concludes that while many middle-skill tasks are becoming automated, more
opportunities arise for workers with high educational attainment and analytical skills. Employers
demand a more skilled workforce, and workers must pursue higher education to meet that
demand. With the current state of the global economy moving in the direction of high-skilled and
technical areas, it would seem logical that the trend should be towards more students completing
postsecondary education to become more competitive for jobs. Hawai‘i postsecondary
institutions must produce more graduates to fill these vacancies, or the positions will be filled
outside the state or country.
Society benefits from college graduates in the way of taxes and productivity. Young
graduates benefit by earning 30% more than peers with some college or 40% more than peers
4
with only a high school diploma (U.S. Census Bureau, 2012). Income is only one aspect of
compensation. Even though college graduates work more weeks per year and hours per week,
they receive better fringe benefits like paid sick leave, vacation, health insurance, and retirement
benefits in addition to the higher median earnings. The United States needs graduates in STEM
science, technology, engineering, and math-related fields to compete globally. There are jobs
available, and the benefits are favorable. However, students are not rushing to obtain STEM-
related degrees.
According to the National Center for Education Statistics (2021), 63% of students who
start at 4-year colleges or universities will graduate within 6 years at the same institution. The
graduation rate for 2-year colleges averaged 33% within 3 years. Overall, Hawai‘i fares worse
than the nation’s average, with the University of Hawai‘i Mānoa graduating 62%, the University
of Hawai‘i Hilo graduating 38%, and University of Hawai‘i West O‘ahu graduating 39% of
students within 6 years, and the community colleges averaging 24% within 3 years (University
of Hawai‘i, 2020). Considering the benefits to both society and the individual of a college
degree, these graduation rates are unacceptable. To increase the number of college graduates in
Hawai‘i, Hawai‘i P–20 Partnerships for Education is coordinating a campaign called 55 by ’25.
The partnership is between the Executive Office on Early Learning, the Hawai‘i State
Department of Education (HIDOE), and the University of Hawai‘i System (UH). This campaign
sets a statewide goal of having 55% of working-age adults possess a college degree by 2025.
One of the most significant challenges that both secondary and postsecondary institutions
face in their efforts to increase graduation rates is improving students’ success in mathematics.
Student success in mathematics is of particular importance because the U.S. economy will be
driven by STEM careers in the future (Hagedorn & DuBray, 2010). Young Americans are not
5
flocking to STEM careers despite the potential for well-paid positions in high-skill and technical
areas. Interestingly, the Lemelson-MIT Invention Index (2012) surveyed Americans between the
ages of 16–25 about their perceptions regarding invention and innovation; 60% responded that
there are barriers in the pursuit of schooling and a career in STEM, and 28% of respondents
disclosed that school did not prepare them for STEM. The data clarify that more students are
interested in STEM, but the interest of those students is not retained at the postsecondary level
(Watkins & Mazur, 2013). The retention issues can be due to lack of interest, inadequate high
school preparation, significant changes, or dropping out (Museus et al., 2011).
Many students enroll in college without the fundamental knowledge to succeed in their
first college-level mathematics course and end up needing remediation. Mathematics has the
highest need for all remedial courses in community colleges (Bahr, 2008; Hagedorn & DuBray,
2010). On the one hand, remedial courses are lauded in creating accessibility to students who
would not otherwise have attended college. On the other hand, a critique of remediation
replicates courses that have or should have been taken in high school (Bahr, 2008). Remediation
overall is a challenge with the growing demand for courses coupled with low graduation rates.
However it is vital to provide higher education access (Dowd, 2007) even if those students
choose majors other than STEM.
Mathematics is the gateway to college, and even those students who choose to take the
vocational route are not exempt from core college courses. The Carl D. Perkins Vocational and
Technical Education Act of 2006 requires participating programs of study to align vocational and
academic curricula to ensure that students receive a rigorous, challenging, and relevant education
(Dortch, 2012). In the past, students avoided college-level mathematics requirements by
choosing a vocational program or substituting a mathematics course for logic. University of
6
Hawai‘i at West O‘ahu does not allow students of non-STEM majors to substitute another course
for their single terminal mathematics requirement, instead emphasizing the belief that all
students can complete a college-level mathematics course (M. Ledward, personal
communication, July 1, 2014).
High school should prepare students for the changing demands of a global economy. The
desires of high school students are not being met, as shown by low high school graduation rates
coupled with the amount of remediation in college, especially for low-income, minority, and
female students (Kirst & Bracco, 2004; Long et al., 2009). Mathematics courses in high school
have become a gatekeeper that hinders students’ graduation (Oguntoyinbo, 2012). While college
success has been linked to a student’s race, ethnicity, and affluence (Dowd, 2007), the
demographics of the United States are changing, and minorities will soon outnumber whites in
public schools (Hussar & Bailey, 2014). In public schools across the United States, high-stakes
tests are given in multiple grade levels from elementary to high school. Standardized tests place
low-income and minorities at a disadvantage early (Mathison, 2003). Still, U.S. students are not
motivated, with only 50% interested in learning mathematics (Organisation for Economic Co-
operation and Development, n.d.). Some might argue that K–12 teachers do not inspire students
to love mathematics but instead teach to the test, which may leave students unmotivated and
uninspired to learn mathematics (Oguntoyinbo, 2012).
Improving the pipeline from K–12 to college is a situation in need of further
consideration. There is a need to test for alignment of expectations as one way to streamline
curriculum, content, and expectations for college. This disconnect between high school and
college requirements results in the readiness gap in college-level mathematics. This study
addresses the problem of the college readiness gap by focusing on expectations for mathematics
7
by determining the alignment between high school and college, which could, in turn, reduce the
number of students enrolling in remedial courses and increase the success rates in college-level
mathematics. Higher student achievement would increase persistence, leading to a rise in
graduation rates (Hanushek & Woessmann, 2008).
Statement of the Problem
According to the National Center for Public Policy and Higher Education (2010), 60% of
first-year college students are not prepared for the rigors of college-level courses. The National
Center for Education Statistics (2014) claims 14.2% of students at public 4-year institutions and
20.8% of students at 2-year public institutions reported taking a remedial course in 2011–2012
(Skomsvold, 2014). An increasing number of students enrolled in remedial mathematics courses
at community colleges and universities across the United States because they are underprepared
for college-level coursework (Bahr, 2008; Boyer et al., 2007; Hagedorn et al., 1999; Horn et al.,
2009; Hoyt J., 1999). There is a disconnect between secondary and postsecondary educational
expectations. Students have academic deficiencies that are unknown due to a substantial gap
between high school standards/exams and college placement tests/admissions policies (National
Center for Public Policy and Higher Education, 2008; National Center for Public Policy and
Higher Education, 2010). Previously, the only way to register for college-level math or English
course was to attain the minimum cutoff score on the placement test. Presently, changes have
been made to use factors other than placement test scores to determine what class the student
should be placed in (Kapi‘olani Community College, 2015; University of Hawai‘i West O‘ahu,
2020). The question is whether high school is leaving the student underprepared or if college is
expecting too much. Popular convention tends to blame K–12 for low achievement, with the
government and private citizens coming up with new ways to fix education every year.
8
One of those fixes for college readiness is the CCSS. However, the alignment of
secondary and postsecondary math content expectations is crucial in ensuring that students are
prepared for the rigors of college. The Hawai‘i CCSS claims to “align with college and
workforce expectations” to deliver students rigorous content that uses higher-level thinking
(Hawai‘i State Department of Education, n.d.a). The implications of not being prepared for
college are devastating and result in a large number of remedial students who do not persist in
college or do not graduate (Bettinger & Long, 2009; Bryk & Treisman, 2010; Complete College
America, 2012; Hagedorn et al., 1999; Melguizo et al., 2011).
Articulation between secondary and postsecondary institutions should be a fundamental
element in student success, yet there is a gap in the studies that examine more than just
performance measures. Although research exists on the CCSS and college readiness, more
studies are focused on aligning the standards with assessments. There is also a lack of research
into whether or not the CCSS is aligned to actual college academic content.
Purpose of the Study
A result of low achievement or misalignment can result in remediation. High levels of
remediation in college affect course completion, persistence, and graduation (Bettinger & Long,
2009; Bryk & Treisman, 2010; Complete College America, 2012; Hagedorn et al., 1999;
Melguizo et al., 2011; Hoyt, 1999). A discrepancy model most accurately describes the issue of
remediation. There is a large gap between what is expected in mathematical competency and
what is currently being achieved. It can be hypothesized that there are false assumptions with
creating standards with no support structure that will affect teacher behavior and, ultimately,
student outcomes. The first assumption is that the CCSS (or secondary education) will make all
students college-ready. The second assumption is that a college entrance exam accurately
9
evaluates a student’s college-level readiness. And the final assumption is that if a student
completes the CCSS, that student will pass the college entrance exam, and will be ready for
college-level mathematics or English courses. Based on all of these assumptions, the remediation
rate should be considerably lower. It is not.
While this study does not include student learning, the effectiveness of the CCSS, or
pedagogy, the focus is on the consistency between secondary and postsecondary mathematics
expectations. The basis for the comparison will use the grades 9–12 CCSS for secondary
mathematics expectations and a textbook as the proxy for college-level mathematics
expectations. The content analysis tool used for this study is the SEC which can analyze content
and cognitive demand between the CCSS and the content in the textbook. The level of alignment
of the textbook to the CCSS will show the connection or disconnection between secondary and
postsecondary expectations.
The purpose of the dissertation is to address the problem of mathematics remediation by
using a content analysis tool on an introductory college-level mathematics textbook to
investigate the alignment to standards and levels of cognitive demand. The research questions for
this study are as follows: To what extent is an introductory college-level mathematics textbook
aligned with the CCSS? If there is misalignment, what are the sources of misalignment?
Significance of the Study
The two measures of pipeline progress here are graduation and persistence. The National
Center for Higher Education Management Systems (2010) reports that 73.5% of ninth-graders
graduate from high school, 45.9% enter college, and 31.4% are still enrolled in their sophomore
year of college. The report concluded that the percentage of students who graduate on time, go
directly to college, return for their second year, and graduate within 150% of normal time is
20.8% (The National Center for Higher Education Management Systems, 2010). The decrease in
10
the forward progression of ninth-graders illustrates the loss at each stage of the educational
pipeline toward postsecondary institutions.
While there may be a plethora of reasons why so few students enter college and even less
complete, studies have shown that remediation has become a gatekeeper and hurts persistence
and graduation rates (Bettinger & Long, 2009; Bryk & Treisman, 2010; Complete College
America, 2012; Hagedorn et al., 1999; Hoyt, 1999; Melguizo et al., 2011). Overall, there is a
problem with college preparedness that precedes the problem of remediation. As a result of this
problem, 48 states, two territories, and the District of Columbia banded together to increase
college preparedness in high schools by creating the CCSS in 2009 (Common Core State
Standards Initiative, n.d.d). Why are these students not prepared for the rigors of college? The
lack of college preparedness and rate of remediation suggest that college readiness should be
examined.
HIDOE and UH can identify a common definition of college readiness by examining the
alignment of secondary and postsecondary expectations. As the only statewide school system in
the country, HIDOE can standardize proficiency expectations across the entire state. UH has 10
campuses and can standardize college-level expectations through their faculty committees. As
HIDOE and UH articulate content and expectations, a straightforward trajectory will be drawn
between high school and college mathematics. In reality, by mutually defining college readiness,
high school standards and college standards will become one collection of standards that
progress towards the same goal.
Limitations and Delimitations
There are a few limitations and delimitations that apply to this study. One textbook will
represent introductory college-level mathematics content and therefore cannot be generalized to
all introductory mathematics courses/textbooks in the United States. No student outcomes or
11
content of instruction will be conducted. This study was not expanded to include student
outcomes or instruction. This study uses only one rater to code the content. However, Polikoff et
al. (2015) concluded that using just two raters produced reliable textbook content analyses.
Definition of Terms
• Alignment is the degree of overlap of topic and cognitive demand between the CCSS and
the textbook.
• Cognitive Demand refers to the expectations for students and one of the two variables in
the SEC coding procedure (Wisconsin Center for Education Research, 2009).
• A college-level course includes any course numbered 100 or above and exhibits what the
institution would consider college-level skills or content (University of Hawai‘i,
Institutional Research and Analysis Office, 2013).
• Common Core State Standards refers to a set of Mathematics and English language
arts/literacy standards developed by the National Governors Association and the CCSSO
to bring consistent goals to K–12 to ensure that students graduate career, college, and
ready (Common Core State Standards Initiative, n.d.a).
• Completion Agenda is a call to action in response to the speech on February 24, 2009, by
President Barack Obama where he addressed a Joint Session of Congress and challenged
colleges to increase the number of students graduating with a degree or diploma by 5
million within the next decade (Humphreys, 2012; Kelly & Schneider, 2012; Obama,
2009).
• Content is the “intersection between content topic and cognitive demand” (Porter, 2002).
One of the two variables in the SEC coding procedure (Wisconsin Center for Education
Research, 2009).
12
• Gatekeeper courses are defined as the initial required reading, writing, or math course.
(University of Hawai‘i Community College, 2016).
• Persistence measures student success of those retained or still enrolled at the institution
within a certain period (usually 150% of normal time to completion) (University of
Hawai‘i, Institutional Research and Analysis Office, 2015).
• Remedial Courses include courses numbered below 100 and exhibit what the institution
would not consider college-level skills or content.
• Standard describes what is important and can be used to judge curriculum or evaluation.
• STEM refers to science, technology, engineering, and math.
• Surveys of Enacted Curriculum is a tool to assess the content of what is taught, the
materials used to assist in teaching, and the alignment between the two (Porter, 2002).
Organization of the Study
This dissertation explores college readiness by analyzing the level of alignment of the
CCSS with a college-level mathematics textbook. This chapter provides an overview of the
study. Chapter two will review the literature related to remediation, college readiness, standards-
based reform, and alignment. Chapter three provides the methodology used in this study, which
includes the research design, textbook application and selection, instrumentation, data collection,
and data analysis. The method for this study is qualitative, an in-depth case study of one textbook
using the SEC as the content analysis tool compared to the CCSS. Chapter four describes the
study’s findings, and Chapter five details the overall conclusions and recommendations.
13
Chapter Two: Literature Review
The previous chapter introduced the need to align high school and college expectations
and improve the pipeline between K–12 and college concerning mathematics. This topic is
important because if the United States continues to be competitive globally, workers with high-
skill or technical backgrounds are needed. This chapter will discuss remediation, high school
preparation and college readiness indicators, CCSS, and alignment studies to further understand
this problem and goal. These areas are necessary to learn more about because current graduates
are not meeting the needs of employment demand (Carnevale et al., 2013). This chapter will
further address the differences related to expectations of secondary and postsecondary
institutions regarding students approaching college-level math competency. The next step would
be toward more students completing postsecondary education, especially in STEM fields where
most high-skill and technical jobs are. However, this study focuses on non-STEM college-level
mathematics courses to ascertain the absolute minimum mathematics college-level standards at
the postsecondary level.
Remediation
The Need for Remediation
The problem of mathematics remediation is a critical issue in education and society.
Remedial education refers to courses below college-level, typically for mathematics, reading,
and English. The purpose of remedial education is to provide students with the skills necessary to
achieve college-level proficiency (Bahr, 2010; Bettinger & Long, 2005). Many college students
take at least one remedial course with a substantial portion in math. Low completion rates for
remedial mathematics courses create gatekeeper courses, increased costs for re-enrollment, and
resource challenges (Bahr, 2008). These challenges are predominantly concentrated at the
community colleges that typically have fewer resources than 4-year institutions to serve
14
underprepared students (Dowd & Grant, 2006; Long et al., 2009). With the high number of
students enrolling in remedial courses every year, remediation has become a necessary function
within higher education. Although remediation affects colleges and universities of all types, this
section will primarily focus on community colleges. This study examines the disconnect between
high school and college requirements and as a result of the large readiness gap, remediation
occurs. Mathematics remediation creates an urgent situation nationally and locally in Hawai‘i. In
reviewing a book that is used at schools on the island, it can be determined if there are pockets of
misalignment in content or cognitive demand.
There has been interest in remedial math due to the sheer number of students requiring
remediation. National estimates of the number of students enrolled in remedial coursework have
increased. Parsad and Lewis (2003) estimated that 28% of first-year students enrolled in one or
more remedial courses in Fall 2000. Skomsvold (2014) examined remedial course-taking and
found that in the 2011–2012 academic year, 32.8% of first- and second-year undergraduate
students reported ever taking a remedial course.
Mathematics has the highest enrollment of all remedial courses. In a study by the NCES,
Skomsvold (2014) further reported that more first- and second-year undergraduates took
remedial mathematics (13.1%) than English (8%), writing (5.9%), and reading (5.6%). Crisp and
Delgado (2014) claim that 80% of their nationally representative sample from the Beginning
Postsecondary Students Longitudinal Study (BPS: 04/09) was enrolled in mathematics
remediation as compared to English (25%) and reading (35%). Romano (2011) found more
remedial courses offered in math at the community colleges in New York and lower academic
levels as opposed to reading and writing. Level three represented the lowest level of remediation,
meaning the current course is a prerequisite for two other remedial courses. Of all the level three
15
remedial courses, mathematics (41%) represented the greatest number of courses offered in
comparison to reading (7%) and writing (14%). While the enrollment of students in remediation
is an important topic, remediation in the subject of mathematics has become the issue of greatest
concern.
Who Is Enrolled in Remedial Education?
Remedial students are different from the larger community college population who do not
enroll in remedial courses in gender, race, age, socioeconomic status, and academic preparation.
Remedial students tend to be older than the average student (Calcagno et al., 2007; Ignash,
1997). Ignash (1997) found that almost 27% of first-year students in remedial courses were over
30. Students of the lowest SES were most likely to enroll in remedial courses (Adelman, 2004).
Remedial students come from various segments of the population and may have different
developmental needs than non-remedial students. Females and minorities are overrepresented in
remedial courses (Bettinger & Long, 2005; Crisp & Delgado, 2014; Hagedorn et al., 1999). The
racial gap in math remediation affects Black and Hispanic students in disproportionate numbers
(Adelman, 2004; Attewell et al., 2006; Bahr, 2010; Bettinger & Long, 2005; Ignash, 1997).
However, Hoyt and Sorensen (2001) and Hagedorn et al. (1999) found that sex and ethnicity
were not significant predictors of placement in remedial courses. These conflicting findings may
indicate the need for additional research regarding remediation and the populations most
impacted.
In Hawai‘i, the remediation results show that Native Hawaiians are overrepresented in
remedial or developmental courses in the UH System. Between 2008–2012, Native Hawaiian
students enrolled in the UH System have increased by 57.4% compared to other ethnicities 3.7%
(University of Hawai‘i, Institutional Research and Analysis Office, 2013). In Fall 2012, when
16
comparing the percent of students enrolled in remedial or developmental education to overall
students, it shows Native Hawaiians (14.2%) and non-Hawaiians (7.3%) (Office of Hawaiian
Affairs, 2014). In 2012, Native Hawaiians comprised 37.3% of students enrolled in a remedial or
developmental mathematics course in the UH System (Office of Hawaiian Affairs, 2014) even
though they only make up 23.3% of the UH System enrollment (University of Hawai‘i,
Institutional Research and Analysis Office, 2013). While most of the research focused on
ethnicities that are not common in Hawai‘i (Adelman, 2004; Attewell et al., 2006; Bahr, 2010;
Bettinger & Long, 2005; Ignash, 1997) and other research (Hoyt and Sorensen, 2001; Hagedorn
et al,1999) claimed that sex and ethnicities were not predictors of placement into remedial
courses, there is evidence of Native Hawaiians not being prepared to place into college-level
mathematics courses with the overrepresentation in remedial mathematics.
High school preparation plays a part in remedial course-taking. Math high school
preparation and the grades earned are significant in remedial math placement (Adelman, 2004;
Hagedorn et al., 1999; Hoyt & Sorensen, 2001). Hagedorn et al. (1999) explored relationships
between remedial/non-remedial students and 12 variables. The findings indicated that non-
remedial students who spent more time studying had higher GPAs and scored better on
standardized tests. Two variables were significant for remedial students: GPA and gender.
Enrollment in remedial courses increased with lower GPAs and among women. A 2001
Maryland report described the relationship between high school preparation and remedial course-
taking of students who graduated in 1999 and enrolled in a Maryland college or university in the
subsequent academic year (Keller, 2001). Students who took a college-preparatory course of
study in high school were less likely to enroll in remedial courses in college. When looking only
at math remediation, 26% of students who took the college-preparatory course of study and 38%
17
of students who did not take college-preparatory courses needed remediation in college. 46% of
students who completed the college-preparatory course of study in high school and attended
community college required remedial math assistance.
Students are leaving high school and entering college at a disadvantage leading to math
remediation, even among students who are taking college-preparatory coursework. Additionally,
Hoyt and Sorensen (2001) studied the effects of high school preparation on remedial placement
in an open-admission state college in Utah. They concluded that the best predictors of remedial
math placement were grades and high school math preparation. These studies indicate that
students who lack college-preparatory course-taking in high school with lower GPAs are much
more likely to need remediation.
The Outcomes of Remediation
The goals of all remedial students are not the same, but graduation, persistence, or
transfer to a 4-year college is a way to measure student success. Attewell et al. (2006) found that
community college students who took remedial courses did not have a lower chance of
graduation. The study concluded that the gap in graduation rates was due to a lack of high school
preparation and not the enrollment in remedial courses in college. Bahr (2010) focused on racial
outcomes in math remediation at community colleges in California and concluded that the odds
of successfully remediating a White student were 3.1 times that of a Black student and 1.6 times
that of a Hispanic student. Bettinger and Long (2009) studied 18–20-year-old first-time freshmen
in Ohio who took the ACT and found that students in remedial courses were less likely to drop
out and more likely to complete a degree.
The results are mixed on whether remediation decreases the odds of transfer to a 4-year
college. Crisp and Delgado (2014) found that enrollment in remedial math decreases the odds of
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transfer to a 4-year institution, while Bahr (2008) and Bettinger and Long (2005) found that
remedial math completers were 15% more likely to transfer with a credential than college-level
completers. Crisp and Delgado (2014) suggest that there is no impact on persistence, remediation
reduces the odds for transfer, and conclude that remediation is not beneficial. Moreover, Bahr
(2008) conceded that while most students who enrolled in remedial math failed, he
acknowledged the benefits for students who completed the course.
Critiques of Remediation
The sizable number of students who need remediation coupled with low completion rates
brings about the issue of cost to the government, taxpayers, postsecondary institutions, and
students (Levin & Calcagno, 2008). Many authors use Breneman and Haarlow (1998) cost
estimates, which confirm earlier estimates from Breneman et al. (1998) that remedial education
costs American public colleges $1 billion annually.
Other costs should be considered. It can be argued that remedial education is a repeat of
courses taught in high school and taxpayers are paying twice (Crisp & Delgado, 2014; Hoyt &
Sorensen, 2001; Long et al., 2009; Merisotis & Phipps, 2000). Remediation extends the time to
graduate, raising the cost of tuition for those additional credits (Barbatis, 2010), and decreasing
the odds of completion (Calcagno et al., 2007). With an increasing demand for college graduates,
colleges must find ways to balance costs and benefits to educate the most underprepared
students.
Critics argue that remediation lowers academic standards (Bahr, 2008). The estimates of
remediation are believed to be higher because hidden remediation happens when the instructor
deems the class not up to standards, teaches at a lower level, but exists as a college-level course
(Grubb, 1999). Courses may also be partially remedial with added labs or class time (Romano,
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2011). Proponents for access and equity claim that remediation allows higher education to give
opportunity and access to all and functions in a larger social context. Bahr (2010) focused on
racial outcomes in math remediation at community colleges in California and concluded that the
odds of successfully remediating a White student were 3.1 times that of a Black student and 1.6
times that of a Hispanic student. Bahr (2008) argued that “remediation is, by definition, a
‘remedy’ intended to restore opportunity to those who otherwise may be relegated to meager
wages, poor working conditions, and other consequences of socioeconomic marginalization” (p.
422). Remedial mathematics becomes a tremendous barrier for many students with nearly 40%
of remedial students in community colleges never completing the remedial sequence (Complete
College America, 2012). Bahr (2008) also found that the typical remedial non-completer has an
83% chance of not completing a credential or transferring.
Remedial education provides access and opportunities to an underserved and
underprepared population seeking higher education but still provides a core function in higher
education. Math has the largest enrollment of all remedial courses. The literature suggests that
demographics and academic preparation influence remedial math placement. Remedial students
exhibit different demographic and academic factors than their non-remedial equivalents. Studies
that illustrate high school preparation have been shown to play a significant role in remedial
placement (Adelman, 2004; Hagedorn et al., 1999; Hoyt & Sorensen, 2001). If high schools are
not consistent with expectations in college, remediation will continue to flourish. Remediation
will be better addressed with alignment studies comparing standards in high school with the
expectations in college. The following section will provide additional background on college
readiness using previous research on high school preparation and tests that colleges use to
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determine their college readiness standard. This varied interpretation of what college-ready
means has affected student enrollment and progress in college.
High School Preparation and College Math Readiness Indicators
A Nation at Risk (National Commission on Excellence in Education, 1983) sparked
research on student achievement during high school and readiness for college. National test
scores were declining, only one-third of 17-year-olds could solve a multi-step mathematical
problem, and one-quarter of mathematics courses taught at public 4-year universities were
remedial. Students are not graduating high school with the minimum skills or requirements
needed for college especially in mathematics (Achieve, 2008). Uncovering the reasons for the
decline in mathematics outcomes is required to increase achievement in the future.
College admissions policies and placement tests dictate who can attend college and what
constitutes college readiness (Norman et al., 2011). Inevitably, student outcomes such as
enrollment, choice of major, persistence, and graduation rates are negatively affected by the issue
of college readiness (Bettinger & Long, 2009). When students are not college-ready, remediation
occurs, which increases the length of time spent in college (Barbatis, 2010) and decreases
completion odds (Calcagno et al., 2007). In all likelihood, most students who enroll in remedial
mathematics failed the course (Bahr, 2008). The National Center for Public Policy and Higher
Education (2010) maintains a gap between being eligible to attend a college and the expectation
of college-level work, as illustrated in Figure 1. Highly selective 4-year institutions have the
smallest readiness gap but serve 10% of all students. Less selective 4-year institutions have a
mid-sized gap and serve 30% of all students, while nonselective 2-year institutions have the
largest college readiness gap and serve the remaining 60% of all entering freshmen.
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Figure 1
The Readiness Gap by Institutional Sector
Note. From Beyond the Rhetoric: Improving college readiness through coherent state policy, by
the National Center for Public Policy and Higher Education (2010).
This disconnect between high school and college requirements results in the readiness
gap. To examine the mathematics readiness gap, an exploration of the relationship between high
school preparation and college readiness indicators is necessary. Colleges set the standards for
college readiness with required entrance exams (Allen, 2013; Fitchett et al., 2011; Norman et al.
2011) or placement tests (Parsad & Lewis, 2003). The high school responded by increasing
graduation course requirements (Clune & White, 1992; Schiller & Muller, 2003; Teitelbaum,
2003) and changing course-taking patterns (Allensworth et al., 2009; Nomi, 2012). Furthermore,
to gain additional understanding of this study, information was provided to demonstrate the lack
22
of mathematics preparation for college and how high school policy has changed to meet the
needs of colleges.
High School Preparation
Signaling theory describes behavior as “when two parties (individuals or organizations)
have access to different information” (Connelly et al., 2011, p. 39). Signaling theory can be
applied to explain the relationship between high school preparation and college expectations.
Brown and Conley (2007) demonstrated the adaptation of signaling theory to the high school-
college transition period as high schools are being signaled to what is important to teach/learn
through standards, assessments, and college admissions requirements. High schools cannot
conform to unclear or contradictory signals from colleges and may reject or misinterpret them.
The signals that colleges send to high schools have prompted policy changes of graduation credit
requirements (GCR). This section reviews GCR and its effects on course-taking patterns and
student outcomes.
State governments have the responsibility of educating citizens and regulating education
through mandates. In 1983, A Nation at Risk prompted many states to increase graduation
requirements by recommending three credits of mathematics for all high school graduates
(National Commission on Excellence in Education, 1983; Plunk et al., 2014). In response, 41
states increased their graduation requirements within a year of the report (Teitelbaum, 2003) to
raise the academic performance of all students (Porter et al., 1998). Math credit requirements
have changed from 2001–2013. In 2001, 45 states reported state graduation rates, and of those
45, 24 required three credits of math (National Center for Education Statistics, 2001). By 2013,
42 states required at least three credits of math (Buddin & Croft, 2014). The underlying
assumption of increasing GCR is that course content is a predictor of learning; therefore,
23
requiring students to enroll in additional courses will increase learning and improve student
achievement (Porter et al., 1998).
Instituting policy that increased GCR was a straightforward and economical method for
reform (Plunk et al., 2014; Teitelbaum, 2003). States could mandate higher GCR without
providing additional funding to schools, with schools adjusting their course offerings. The
assumptions were simple; raising the number of required courses and proficiency will increase
(Carlson & Planty, 2012; Schiller & Muller, 2003). The effects of policy changes to GCR
differed based on the outcome being measured. Increasing GCR had a positive effect on the
number and level of high school math courses completed (Teitelbaum, 2003) even though the
number of students dropping out of high school went up (Hoffer, 1997; Lillard & DeCicca, 2001;
Plunk et al., 2014). High school completion rates were not found to be related to GCR (Porter et
al., 1998). However, some students have been found to benefit from the additional coursework
by increasing their chances of going to college (Aughinbaugh, 2012; Gamoran et al., 1997; Long
et al., 2009). In sum, the results of increasing GCR have been mixed, and whether the policies
effectively improve educational outcomes and college readiness should be asked.
The rise in the number of mathematics credits required for graduation intended to
increase knowledge in the subject with students taking higher-level courses. The completion of
advanced math courses would indicate more knowledge. Research by Clune and White (1992)
explored the effects of increased GCR on lower-achieving students in the 1980s using a sample
from four states. The findings indicated that although the number of math credits earned
increased, there was no change in the level of courses taken. The gains in math were
concentrated on middle-level courses. More students enrolled in beginning college preparatory
courses like Pre-Algebra, Geometry, and Algebra 1, but enrollment declined to start with
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Algebra 2. Courses above Algebra 2 have been found to improve college readiness (Adelman,
2006; Buddin & Croft, 2014) and retention in advanced math courses is imperative in increasing
student achievement. However, Adelman (2006) maintains that Algebra 2 is not enough to
satisfy math requirements for any major. The need is for advanced quantitative literacy.
A subsequent study by Teitelbaum (2003) calculated the effect of increasing GCR in
mathematics on credits earned, course-taking patterns, and test scores. The study used NELS
data and analyzed high school transcript and test scores from 5,586 seniors in 1992. The study
found that schools with higher GCR had students more likely to complete advanced math than
students at schools with lower GCR. Nearly 35% of students at schools with higher GCR took
advanced math compared with 31% of students at schools with lower GCR. There is a notable
difference in student samples between Clune and White (1992), who used schools in the lowest
quartile of achievement, and Teitelbaum (2003), who used NELS data and controlled for student
and school characteristics. Due to the differences in conclusions by Clune and White (1992) and
Teitelbaum (2003), it seems that the difference in population sample could be a factor in
concluding negative effects of GCR on specific populations.
Some patterns have emerged in regards to courses taken in high school. The entry point
in the math sequence influenced the trajectory of the math curriculum in high school. Several
studies explored GCR on course-taking patterns (Schiller & Muller, 2003) and the benefits of
taking Algebra 1 in high school (Allensworth et al., 2009; Gamoran & Hannigan, 2000; Nomi,
2012). Schiller and Muller (2003) analyzed National Educational Longitudinal (NELS) data from
1988–1992 that included a sample of 10,046 students from 50 states, including the District of
Columbia. In states with higher GCR, students tended to be placed in a higher-level math course
as a freshman but earned an average of 1.6 credits of advanced math, which were fewer
25
advanced math credits than peers in lower GCR states. The authors speculated that higher GCR
might impede advanced math course-taking, considering credits also increased in other subject
areas to understand that phenomenon. While states with higher GCR had students initially placed
in higher-level courses as freshmen, they earned less advanced math credits by the end of high
school.
Allensworth et al. (2009) and Nomi (2012) studied a 1997 Chicago Public School
mandate requiring Algebra 1 for all ninth graders. The two studies examined the consequences of
the Algebra 1 requirement in grade nine and after graduation for low-and average-ability
(Allensworth et al., 2009) and high-ability students (Nomi, 2012). Allensworth et al. (2009)
found no benefits for students taking Algebra 1 versus remedial math in ninth grade. While more
students overall earned credit, low-ability students’ failure rates rose, and of those who passed,
grades declined. Math test scores and graduation rates remained unchanged, and low-ability
students were less likely to complete courses above Algebra 2. Requiring Algebra 1 in ninth
grade negatively affected low-ability students because average and high-ability students did not
change their enrollment after the mandate.
Additional research by Nomi (2012) on the effects of the mandate for high-ability
students showed that the consequences for high-skill students were a reduction in peer skill level
due to the inclusion of low-ability students in Algebra 1 as opposed to remedial math. Before the
Algebra 1 mandate, schools were likely to track students. After the mandate, schools detracked
students and created more mixed-ability classes due to the lack of remedial courses.
Subsequently, the test scores for high-ability students went down. As a result, Allensworth et al.
(2009) and Nomi (2012) concluded that mandating GCR alone is not enough to promote higher
student achievement.
26
While some benefit from the increase in rigorous coursework, the unintended
consequences of state mandates affect the most disadvantaged students (Porter et al., 1998).
Raising the GCR in mathematics also increased the number of students who dropped out of
school. While Porter et al. (1998) argued that there was no relationship between GCR and high
school completion, high school completion had been negatively affected by GCR for particular
demographic groups. Plunk et al. (2014) investigated the intended and unintended consequences
of increasing GCR policies. The authors found that dropouts increased across the sample and
resulted in an overall decrease in college enrollment. Blacks and Hispanics dropped out at twice
the rate of Whites and thus, were less likely to attend college. The study recommended that
additional supports for at-risk students be paired when GCR policies are enacted.
Increasing the credit requirements for mathematics has been found to have a positive
effect on college outcomes. The chances of attending college increase with a more rigorous and
comprehensive mathematics curriculum (ACT, Inc., 2015; Aughinbaugh, 2012; Gamoran et al.,
1997; Long et al., 2009). It is the level and not merely the number of credits earned that affects
outcomes. Aughinbaugh (2012) suggested that taking an advanced mathematics curriculum
increased the probability of attending college by 17%, and with a 20% increased likelihood of
enrolling at a 4-year college by age 21.
In a similar study of GCR policies using different outcome measures, Plunk et al. (2014)
studied three outcomes of raising the course graduation requirements for mathematics. The result
of this analysis did not show any broad, meaningful benefit to increasing GCR for college
attainment. However, it implied that student exposure to the more rigorous coursework benefited
the individual. The results for students who attended college with or without completing a degree
did not show any significant associations with increased GCR.
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High schools have interpreted the signal for college readiness by increasing GCR but
increasing GCR has not been proven to be an effective method for increasing student outcomes
(Clune & White, 1992; Hoffer, 1997; Lillard & DeCicca, 2001; Plunk et al., 2014) and has led to
dropouts that adversely affected Black and Hispanic students. The math course taken in ninth
grade influences the trajectory of math curriculum completion (Allensworth et al., 2009; Nomi,
2012; Schiller & Muller, 2003), with coursework after Algebra 2 increasing the odds of college
readiness (Adelman, 2006; Buddin & Croft, 2014). A rigorous high school curriculum has
improved student success in college (Adelman, 1999; Adelman, 2006). The importance of high
schools correctly deciphering the signals for college expectations and a method to assess that is
critical for future research. Colleges must also be clear about the type of signals that are sent with
the requirements of incoming students.
College Math Readiness Indicators
There is a gap between what high schools require for graduation and the minimum
requirements for college at admission. This gap is more pronounced at public universities than at
community colleges (Adelman, 2006). The term “college-ready” does not have a common
definition across institutions (Greene & Forster, 2003; Merisotis & Phipps, 2000; Oudenhoven,
2002; Porter & Polikoff, 2012). College readiness is defined institutionally and reinforced
through admission policies. A student’s major or type of institution may also affect the standard
for readiness (Porter & Polikoff, 2012). For high schools to prepare students who can enroll in
college-level courses without remediation, colleges must be explicit in their requirements for
first-year students. This section will focus on college academic readiness indicators in
mathematics using entrance exams and placement tests. By examining the alignment of the
CCSS and the college-level mathematics textbook there may be hope to bring secondary and
28
postsecondary mathematics expectations closer and find a more common definition of college
readiness in Hawai‘i.
Admission policies are used to identify and accept students who are anticipated to
succeed (Berdie, 1960; Sawyer, 2013). The National Association for College Admission
Counseling report states that the top factors in college admissions are grades in college prep
courses, grades in all courses, rigor of curriculum, and admission test scores (Clinedinst, 2020).
Mathematics is a critical college prep subject. In 2018, 46% of colleges gave performance on
admission tests the highest rating, making it the fourth most important factor in admissions
(Clinedinst, 2020). Admission tests play no role in attendance for students who apply to open-
enrollment colleges.
Colleges use entrance exams as an objective measure of achievement or potential of
applicants across the nation. The two leading college admission tests are the Scholastic Aptitude
Test (SAT) and American College Test (ACT). The SAT, redesigned in March 2016, claims to
“focus on the knowledge, skills, and understandings that research has identified as most
important for college and career readiness and success,” while the pre-March 2016 SAT tested
general reasoning skills (Collegeboard, n.d.a). The ACT is an achievement test in English,
mathematics, reading, and science (Allen, 2013).
College entrance exams are used as indicators in determining which students will have a
high probability of success in college. ACT, Inc. indicated that a score of 22 (median
benchmark) gave students a 75% chance of earning a C or 50% chance of earning a B in College
Algebra (Allen, 2013). A study by Medhanie et al. (2012) claimed the ACT score of students
who began with college-level coursework was 22.39 and was consistent with the median
benchmark. ACT Research Division (1998) reported the average test scores of students who
29
completed Algebra 1, Algebra 2, or Geometry (18), Trigonometry (21), Precalculus (23), or
Calculus (25) as their highest courses. It becomes clear that high school students would have to
take Precalculus in high school to be prepared for college-level mathematics.
ACT math scores were significant in determining university Calculus placement. Norman
et al. (2011) found that students with higher ACT scores were 1.54 times more likely to be
recommended to Calculus than students with lower ACT scores. Additionally, strong math skills
were indicated by high school math GPA and high school math levels. Interestingly, the study
found no relationship between high school math curricula and Calculus placement.
Hoyt and Sorensen (2001) examined remedial placement rates after students completed
college preparatory coursework in high school at an open-admission state college in Utah. ACT
scores determined course placement. Enrollment in College Algebra required an ACT score of
24 or higher, higher than the ACT median of 22. Hoyt and Sorensen (2001) found that the only
students to meet the requirement of 24 were those who took Precalculus or above in high school.
This finding was consistent with ACT, Inc (n.d.a) that showed the national class of 2020’s only
math course patterns that neared a score of 24 ended in Calculus or other advanced math.
Fitchett et al. (2011) analyzed the effectiveness of the mathematics placement process at
a public university in Colorado. Before making course recommendations, advisors reviewed a
student’s ACT scores, high school GPA (HSGPA), last math course and grade, and other
contextual factors. Students who had ACT scores of 16 or lower had a success rate of 58% in
their first college math course. Students with scores of 17 or higher had success rates of 80%.
Advisors relied on ACT scores when deciding between an easier or more challenging course, but
when ACT scores were 17 or higher, advisors used the student’s HSGPA instead. When
considering other factors like HSGPA in the placement process, the success rate in the following
30
course improved. Using multiple factors to determine placement may be possible to yield better
results.
College entrance exams have not been valid predictors of success for all students. The
lack of alignment between what is taught in high school and what is expected in college could be
a reason specifically in mathematics courses. Moreover, Wainer and Steinberg (1992) found
statistical bias against women in the mathematics portion of the SAT. Using bidirectional
(retrospective and prospective) analysis, women scored less than men on the SAT who were in
the same class and earned the same grades. Female bias on entrance exams is far-reaching,
considering females have a higher college enrollment rate (Conger & Long, 2013). The bias of
these tests contributes to the inability of entrance exams to predict success in minority and
female students and challenge the overall validity of the exams.
For the class of 2020 in Hawai‘i, although enrollment held at 32% at UH, there was a
decline in the Native Hawaiian college-going rate, which dropped from 42% to 35% (Hawai‘i P–
20, n.d.). The ACT is given in 11
th
grade for all public school students and of the 9,451 (82%)
who took the ACT, only 18% scored at or above a 22 which is considered college-ready by P–20
(Hawai‘i P–20, n.d.). According to Hawai‘i P–20 (n.d.), the class of 2020 had the lowest
mathematics ACT score when looking as far back as 2015.
Placement tests also serve as indicators of college preparedness or readiness. Once a
student is accepted into a college, placement test scores may be used to determine a student’s
initial math course. More than half of all postsecondary institutions (61%) use placement tests
for entering students in mathematics (Parsad & Lewis, 2003), with some colleges using the
results to determine course enrollment and others using the results as a means to make a
recommendation as to course enrollment (Medhanie et al., 2012). The two most commonly used
31
placement tests are ACT’s Computer Adaptive Placement Assessment and Support System
(COMPASS) and College Board’s ACCUPLACER. Both the COMPASS and ACCUPLACER
can be used as placement and diagnostic tests. When using these tests for placement, COMPASS
tests up to five mathematics subject areas: Pre-Algebra, Algebra, College Algebra, Geometry,
and Trigonometry (ACT, Inc., n.d.b). The ACCUPLACER categorizes its math areas into
arithmetic, college-level math, and elementary Algebra, which test up to trigonometry
(CollegeBoard, n.d.b). Both tests state online that there is no “passing score” but defer the
interpretation of the results to the applicant’s college.
Each college determines the cutoff score used on the placement test. Faculty will consider
vendor recommendations when setting cut scores. The validity of the placement test depends on
the interpretation of the results and how cutoff scores are determined (Belfield & Crosta, 2012).
Donovan and Wheland (2008) investigated ACT and COMPASS scores in predicting success in
Intermediate Algebra at a public, open-enrollment university in Ohio. Incoming students who
scored 20 or less on the ACT were required to take the COMPASS placement test and were
placed in either a remedial, Intermediate Algebra or college-level course. The Intermediate
Algebra course was considered a bridge course between remedial and college-level. In fall
semesters, students who earned an A or B in the Intermediate Algebra class had ACT and
COMPASS scores that were significantly different from students who earned a C, D, or F. There
was a strong relationship between ACT and COMPASS scores and student success in the
Intermediate Algebra course.
Cutoff scores that are used for placement decisions are not on a continuum. A score one
or 10 points above the cutoff register as an identical score, while a score one point above or
below the margin could mean the difference between college-level or remedial. Melguizo et al.
32
(2011) examined the effectiveness of placement decisions within the remedial sequence at the
largest community college district in California using the completion of the following math
course as the outcome measure. The study results concluded that the effectiveness of placement
decisions varied by level. Using the regression discontinuity design, students at the margin
should be placed in the higher course when the coefficients are negative and placed in the lower
course when the coefficients are large and positive. Scott-Clayton and Rodriguez (2015) studied
placement policies and compared students right above and below the cutoff score at a large
community college system. The findings concluded that 25% of students at the margin could
have successfully completed a college-level course instead. Cutoff scores should not be firm
from these perspectives, and placement policies may be diversionary at the margin.
The accuracy of placement tests has been studied using course grades as the outcome.
James (2006) argues that ACCUPLACER was a good predictor of success when the arithmetic
and elementary Algebra tests were combined; 77.3% of students would be placed correctly and
noted that the placement test better predicted upper-level success. Mattern and Packman (2009)
reported a correct placement rate for students scoring a B or higher (66.5%) or C or higher
(75.1%) using the college-level math section of ACCUPLACER. When combining the sections
of college-level math and elementary Algebra, the rate of correct placement rose to 81% for both
outcomes. The findings of James (2006) and Mattern and Packman (2009) suggest that the rate
of correct placement increases when more than one math section of the ACCUPLACER is used.
The validity of placement tests questions the negative consequences of placement
decisions. Placement tests may extend the college path for some students with the addition of
remedial courses and slow progress towards completion (Belfield & Crosta, 2012). Females tend
to score lower on math placement tests than males but still earn higher course grades (Donovan
33
& Wheland, 2008). The suggestion is that COMPASS is a better predictor for males. While there
are gender consequences in using admissions policies based on test scores, including the HSGPA
does not solve the gender issues (Conger, 2015).
There have been stronger correlations between other variables (ACT score, HSGPA,
number or type of high school math courses) when compared to first college math course grade
than with math placement tests (Fitchett et al., 2011; Harwell et al., 2009; Hoyt & Sorensen,
2001). If placement tests contribute little after using scores from entrance exams, colleges should
consider using ACT scores for placement (Medhanie et al., 2012). Nevertheless, once a student is
placed into the initial math course, no further placement testing is performed, so this first
assessment becomes critical as the only judge of mathematics competence in college (Donovan
& Wheland, 2008). With more than half of colleges foreseeing increased recruitment of transfer
students in the next 5 years (Clinedinst, 2015) and many students coming from open-enrollment
colleges, entrance and placement exams that are comparable at all levels of selectivity are
fundamental to maintaining consistent levels of mathematics proficiency.
Reviewing high school course-taking patterns and GCR brought to light the barriers that
interfere with postsecondary success in mathematics. The current system of GCR and control
that high schools have over the courses that students take sends mixed signals as to what is
essential for students to know. Policies are disconnected, promote confusion over what is
essential, and provide inadequate college preparation. One method of addressing this confusion
is to look at high school standards to determine if the CCSS can serve as a benchmark to an
actual instructional artifact used in college. This will help understand the degree to which
expectations are aligned or not aligned to secondary and postsecondary requirements.
34
Standards-Based Education Reform
Despite numerous reform efforts, the United States still lags behind other developed
countries in mathematics, performing below average (31 of 38 countries) on the PISA in 2018
(Organisation for Economic Co-operation and Development, 2021). Marzano et al. (2005) add
that “one of the constants within K–12 education is that someone is always trying to change it”
(p. 65). College readiness for mathematics cannot be separated from the powerful influence of
reform and mandates on elementary and secondary education has had over the past few decades.
The inadequacy of mathematics achievement by U.S. students has become devastating to the
future of the economy and the ability of the United States to compete internationally. Due to
persistently low academic achievement, a few critical changes in goals and philosophies have
influenced standards-based reform.
Background
In 1983, the U.S. Secretary of Education formed the National Commission on Excellence
in Education and tasked the group with reporting the condition of American education to the
Secretary and the Nation (National Commission on Excellence in Education, 1983). The report
was titled A Nation at Risk and was released in 1983. A Nation at Risk looked at public and
private secondary and postsecondary institutions to generate reform to raise the commitment to
improving the quality of education in the United States (National Commission on Excellence in
Education, 1983). In 1981, teens were graduating with a high school diploma unprepared for the
rigors of college or work (National Commission on Excellence in Education, 1983). While this
report was not specifically aimed at mathematics, the influence of this report has changed
mathematics education and sparked the federally driven standards-based reform movement.
In 1986, the National Council of Teachers of Mathematics (NCTM) formed the
Commission on Standards for School Mathematics. The Commission wanted to assist in
35
bettering mathematics education in the United States. It tasked the group with creating a vision
of what a mathematically literate student looks like in an evolving, technologically advancing
21st-century society and the standards to navigate towards that newly defined vision (Center for
the Study of Mathematics Curriculum, 2004). The outcome of this document showed a
consensus of an organization of mathematics education professionals on what students should
learn during their K–12 years and strong support for a new vision of elementary and secondary
mathematics.
In 1994, standards-based reform was written into federal law with the reauthorization of
the Elementary and Secondary Schools Act of 1994 (ESEA) in response to the need for students
to meet the changing demands of a diverse economy (Riley, 1995). While flexibility was built
into the reauthorization, schools and school districts would be held accountable for student
achievement results against those high standards. Challenging state standards would become the
benchmark for all educational components to converge. Educational components included school
improvement, accountability, curriculum and instruction, school leadership, and professional
development (Riley, 1995).
No Child Left Behind Act of 2001 (NCLB), the most ambitious reauthorization of ESEA,
focused on improving academic achievement for all students to meet the competence standards
set forth by that student’s state. The academic subjects covered by NCLB were English/language
arts, mathematics, and science. Each state would maintain the autonomy in deciding what those
standards would be, but the mandate was that there would be articulated standards. The federal
government would not dictate the content of those standards. The result of creating individual
state standards was that 50 states had 50 sets of standards. In complying with the accountability
feature, each state designed assessments to measure the effectiveness in reaching those standards.
36
NCLB raised the bar for standards and accountability but added to earlier policies and was not
the start of the standards-based reform movement (McDonnell, 2005).
Most recently, the Every Student Succeeds Act of 2015 (ESSA) was a reauthorization of
ESEA, focused on an equal opportunity so that regardless of additional support the student
needed, schools focused on college and career success (U.S. Department of Education, 2020).
ESSA kept the accountability portion of NCLB and still required students to meet state standards
measured by assessments (U.S. Department of Education, n.d.).
The Common Standards Era
Standards-based reform set a new agenda for school districts, administrators, teachers,
and students. The commonality between reform from A Nation at Risk to NCLB was that there
was a need for U.S. students to increase their educational attainment and meet higher
expectations than what was currently expected to remain competitive internationally. The result
of standards-based reform was a demand for accountability using standards and coding those
requirements into law. The problem was that all states maintained independence in determining
the standards that would be measured. The next step in standards-based reform was to work
toward a common set of standards.
Toward the end of the 2000s, the standards-based reform movement moved towards
establishing a common set of standards due to federal requirements under NCLB. The CCSS
were developed by the National Governors Association and the Council of Chief State School
Officers (CCSSO) for mathematics and English language arts/literacy and adopted by over 40
states and the District of Columbia (Common Core State Standards Initiative, n.d.a). The
endorsement of the CCSS has generated a national debate over the effectiveness and acceptance
of the new standards. In support of educational innovation and reform, which included
37
developing and adopting common standards, President Obama established Race to the Top and
allocated $4.35 billion as a competitive grant program (U.S. Department of Education, 2009).
Hawai‘i was awarded $75 million in the second round of the Race to the Top
competition. In its application, Hawai‘i required all public schools to implement the Hawai‘i
Common Core, which included the CCSS. In the application timeline, HIDOE started phasing in
the CCSS in certain grade levels with full implementation in all public schools by 2014–2015
(Hawai‘i State Department of Education, n.d.b.). The CCSS replaced mathematics and English
Language Arts in the Hawai‘i Content and Performance Standards (HCPS). The CCSS were new
rigorous standards intended to produce globally competitive students (Hawai‘i State Department
of Education, 2012).
Standards do not come with a prescribed curriculum but describe what students should
have learned at the end of each grade level (Hawai‘i State Department of Education, n.d.a).
Students who are competent in the CCSS subjects are supposed to be college, career, and life
ready and enroll in college-level courses upon graduation (Common Core State Standards
Initiative, n.d.a). Neither the Hawai‘i Common Core nor the CCSS set out standards-based on
individual courses. Instead, after the eighth grade, the CCSS changes from grade level to a single
set of high school standards lumped together as grades 9–12. The CCSS also did not specify the
sequence or scope of courses but lists the high school standards as number and quantity, algebra,
functions, modeling, geometry, statistics, and probability (Common Core State Standards
Initiative, n.d.b). The ability to measure the curriculum alignment to standards is imperative to
improve student achievement. Finding the right tool, SEC, to test the level of alignment with
standards is essential in gathering reliable data about whether there is an ordinal progression
between high school and college expectations.
38
Measuring Textbook Alignment
College preparedness for mathematics indicates the type of elementary and secondary
education received. The large number of college students who need remediation shows a lack of
alignment between high school and college. The goal seems to be to get students from high
school into college, but neither side knows what the other is doing (Conley, 2005). High schools
are required to use a specific set of standards, for example, the CCSS, while colleges retain the
autonomy of determining their standards. Colleges demonstrate their expectations in admissions,
placement, and curriculum. Once a student is admitted and placed, the curriculum becomes the
reigning standard. Textbooks are a principal tool of instruction and impact student learning. High
schools and colleges need to establish clear and coherent expectations (Conley, 2005).
Ascertaining the current level of alignment provides a baseline when attempting to remedy
problems that originate from misaligned expectations.
“Alignment is the core idea in systemic, standards-based reform” (Porter, 2002, p. 5;
Smith & O’Day, 1991). Martone and Sireci (2009) believe that students will learn what is being
taught and display their achievement if testing, standards, and instruction are aligned. For the
purposes of this study, content alignment is the degree to which the CCSS and college textbook
is the same. Should the expectations for high school math and an introductory college-level
textbook align, then the CCSS would guide students towards what colleges expect them to know.
The three prevailing methods for evaluating alignment are Achieve, Webb, and SEC (La
Marca et al., 2000; Martone & Sireci, 2009; National Assessment Governing Board, 2009). Each
method evaluates alignment using various markers/artifacts/standards, with the results providing
different types of information. The Webb and Achieve methodologies identify alignment
between standards and assessments (National Assessment Governing Board, 2009; Webb, 1997;
Webb, 2007). The SEC method allows the use of standards, assessments, instruction, and
39
curriculum materials for study (Porter, 2002). While the Webb method is most prevalent in
analyzing assessments and standards (Porter, 2006), this study will analyze standards with an
artifact, a textbook, so the only method appropriate for this undertaking is the SEC.
Textbooks merit scrutiny since a teacher’s use of textbooks impacts curriculum and
instruction. Textbooks play a prominent role in all school subjects, but none so much as in
mathematics classes (Fan et al., 2013). Because of the reliance on textbooks, it would be
important that the textbooks cover the curriculum in line with the standards. Polikoff (2015)
states that for a textbook to be considered aligned, all content must be covered in the
corresponding grade levels with no content coverage outside of the standards. Extending on that
notion, it has become essential to determine if the first mathematics textbook in college is
aligned with what high school standards require to ascertain if the college readiness message is
consistent between secondary and postsecondary institutions.
Kane et al. (2016) conducted a study to determine how teachers and principals were
implementing the CCSS and what resources helped them or helped increase student success like
professional development days, textbooks, online resources, and classroom observations.
Participants included 1,498 teachers from elementary and middle schools and 142 principals
across five states: Delaware, Maryland, Massachusetts, New Mexico, and Nevada (Kane et al.
2016). One part of the study explored whether specific textbooks impacted students’ test scores
on the Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter
Balanced Assessment (SBA). In the sample, 31% did not use textbooks, but of the teachers who
used textbooks, 30 or more teachers used five particular textbooks. Those five textbooks made
up the study. The study's implications showed that there were no differences in student
achievement in three of the textbooks. However, the other two textbooks did show statistical
40
significance towards student achievement on the PARCC and SBA. Students who used GO
Math! (Houghton Mifflin Harcourt) primarily scored 0.1 standard deviations higher, while
students who used the other textbook (name not released) scored 0.15 standard deviations lower
in comparison to other textbooks or no textbook at all (Kane et al., 2016). Kane et al. (2016)
concluded that for the two textbooks which gave significant results, the effect on student
achievement was sizable. Therefore schools should consider textbook selection a critical
decision-making component.
Floden et al. (1981) conducted a study where 66 student teachers were given hypothetical
situations and asked how six factors would affect the content of fourth-grade mathematics
classes. The two most powerful pressures were standardized tests and district objectives.
Textbooks came in last of the six factors. On the other hand, textbooks were the weakest pressure
that affected what teachers used to decide what to teach.
These studies illustrate textbook use in the elementary and secondary levels. K–12
teachers rely heavily on textbooks to inform curriculum (Chingos & Whitehurst, 2012), and
should topics not be in a textbook, and it will likely not be included as a topic of instruction
(Stein et al., 2007). Polikoff (2015) conducted an alignment study of four popular fourth-grade
mathematics textbooks to the CCSS using the SEC. The study found that 90% of CCSS topics
were covered in the textbooks on average. Of the three textbooks that were claimed to be CCSS
aligned, the three sources of misalignment were a difference in cognitive demand from the CCSS
to the requirements in the textbooks, discrepancy between the emphasis on content between each
book and the CCSS, and does not cover between 17% to 25% of the SEC cells in the CCSS. If
textbook publishers tout their textbooks as CCSS aligned and teachers or districts rely on that
claim to inform curriculum purchases, the goal of teaching the CCSS is not realized. Polikoff
41
(2015) concluded that the lack of textbook alignment will hamper implementation. However,
those independent researchers will have to continue to supply data in hopes of informing schools
to purchase textbooks and for textbook publishers to improve the alignment of the content.
There has been textbook alignment research done in K–12, but none using a textbook as
the proxy for college requirements. The SEC is the only alignment tool of the three popular
alignment methods to analyze standards with a textbook. Analyzing the first college-level
mathematics textbook against the CCSS would make sense to ensure that K–12 and
postsecondary education are vertically aligned with respect to mathematics and that the
definition of college-ready is proven with standards and curriculum materials. The textbook
would be assumed to be closely aligned to instruction or curriculum in the class. It would be used
to determine the connection between what high school expects for graduation and a range of
proficiency colleges expect.
Summary
With the adoption of the CCSS, Hawai‘i aimed to raise the level of achievement of
students and increase college readiness in English language and mathematics. The research in
this literature review shows that many high school students are not prepared for the rigors of
college-level mathematics courses. The substantial number of college students who enroll in
remedial mathematics courses every year shows this.
Using the SEC will provide local educators and policymakers within Hawai‘i insight into
the problems of college readiness and remediation. Through this study, the relationship between
high school math standards and a college-level textbook will identify areas of alignment and
misalignment to strengthen articulation between secondary and postsecondary education and
reduce the number of students who require mathematics remediation in college.
42
Chapter Three: Methodology
This study centered around addressing the gap between secondary and postsecondary
expectations in mathematics. This study tested the alignment of the CCSS to an introductory
college-level mathematics textbook to determine levels of alignment and misalignment. The
disparities between expectations for completion of high school and expectations upon entering
college served as the basis for identifying a common definition of college readiness. This study
aimed to provide HIDOE and colleges in Hawai‘i with an accurate picture of the level of
alignment of mathematics curriculum and expectations to further articulation between K–12 and
higher education in Hawai‘i. This study focused on the textbook’s content and alignment to
CCSS and is not directly concerned with student learning, student achievement, the effectiveness
of the CCSS, or pedagogy. The main reason for analyzing the alignment between the CCSS and
a college-level textbook was to determine if those two items target the same content and
cognitive demand and if the CCSS did prepare students for college-level academics.
The research questions that guided this study are as follows: To what extent is an
introductory college-level mathematics textbook (for non-STEM majors) aligned with the
CCSS? If there is misalignment, what are the sources of misalignment? The framework used for
the content analysis was the SEC, which the CCSSO developed in partnership with Andrew
Porter and John Smithson of the Wisconsin Center for Education Research (Smithson, 2009).
The SEC is a two-dimensional taxonomy using content or subject topic and cognitive demand to
test for levels of alignment or misalignment (Porter, 2002; Smithson, 2009). The SEC was used
as the data analysis tool in examining the content and cognitive demand between the CCSS and
the textbook to compare the relationship between the two. The data collected from the CCSS and
43
textbook were summarized into an Excel spreadsheet that highlighted and emphasized the
pertinent subject topics within the CCSS and textbook.
This chapter will describe the research design, textbook application and selection,
instrumentation, data collection, and data analysis for this study. The research design will
provide an overview of the methodology used in the study. The textbook application and
selection will give a brief overview of the textbook and the reason for the inclusion of this
textbook. The instrumentation section will describe the textbook and SEC framework as it relates
to the research questions. The data collection will outline the SEC method in detail and the
procedures used to collect data. Finally, data analysis describes the technique that will be used to
analyze the data.
Research Design
Because the research questions seek to explore the extent of alignment, a rigorous and
systematic approach to analyzing the content was chosen. The SEC is not the only alignment
taxonomy in existence. There are three prevailing approaches in testing for alignment: the SEC,
the Webb process, and the Achieve system. The SEC was chosen because it is the only tool to
rate alignment between standards and curriculum materials (Martone & Sireci, 2009). Moreover,
the SEC can analyze instruction and assessments in addition to standards and curriculum
materials, not only between standards and assessments like Webb and Achieve.
This study utilized the SEC framework in mathematics to execute a content analysis
between high school standards and a college textbook to determine the level of alignment.
Textbooks that meet or exceed the CCSS do not necessarily mean that the content was intended
to align with the CCSS. Content is defined as the intersection of content topic and cognitive
demand (Porter, 2004). The CCSS was used as the standards component for high school, and
Thinking Mathematically by Blitzer (2015) represented the introductory college-level
44
mathematics textbook for non-STEM majors. The data from the SEC framework has already
been used in 30 states and has resulted in reliable content analyses that can be replicated (Porter
et al., 2008). Using the SEC will provide local educators and policymakers within Hawai‘i
insight into the problems of college readiness and remediation. Studies like this help identify the
potential mismatch of textbooks with the intended student learning outcomes. The selection of
instructional material should be a thoughtful process, and this study will inform current and
future higher education textbook selection. College readiness should not be the sole
responsibility of K–12 educators. Higher education should care about college readiness in the
interests of mutual benefit if all students are to enter college fully prepared. Students will feel
more confident in navigating the college curriculum if they have the necessary prior knowledge
for the course.
Textbook Application and Selection
Textbooks are an integral part of the curriculum and are the principal tools of instruction.
Considering the role that textbooks play in instruction, it becomes an important factor impacting
student learning. In 2014, two textbooks were obtained, one for Survey of Mathematics and the
other for College Algebra. This textbook was selected by contacting the textbook's publisher,
Pearson, and requesting a copy of a MATH 100 or 103 textbook. The textbook chosen for this
study was for a Survey of Mathematics course to ascertain the minimum required to earn credit
in an introductory college-level mathematics course. This study analyzed Thinking
Mathematically by Blitzer (2015), the best-selling textbook on the market for Survey of
Mathematics courses in the United States (J. Hamad, personal communication, March 6, 2014).
When this work was completed, the textbook was current, although in 2018, a 7th edition was
published and is the most current. This textbook has been adopted at 281 schools in the United
States, including six in Hawai‘i (J. Hamad, personal communication, March 6, 2014). Of the six
45
Hawai‘i schools that used Blitzer (2015), four schools that utilized this textbook are part of the
University of Hawai‘i system. The course is labeled MATH 100 “Survey of Mathematics” at the
University of Hawai‘i at Hilo, the University of Hawai‘i at West O‘ahu, Windward Community
College, and Kaua‘i Community College. Hawai‘i’s two remaining colleges are private
universities and label the courses MATH 1115 “Survey of Mathematics” at Hawai‘i Pacific
University and MA 100 “Quantitative Reasoning and Mathematical Skills” at Chaminade
University. Although the course numbering is different, the intent of the course being terminal
and not a prerequisite for another math course is the same at all Hawai‘i institutions.
Blitzer (2015) envisioned the sixth edition of the textbook for non-math majors and tried
to engage students in meaningful real-world applications. The topics in the textbook were not
sequential as each chapter did not build on the next, making this book appropriate for “a one- or
two-term course in liberal arts mathematics, quantitative reasoning, finite mathematics, as well
as for courses specifically designed to meet state-mandated requirements in mathematics”
(Blitzer, 2015, p. vii).
Instrumentation
This study assumed that a content alignment study could find the answers to the sources
of alignment or misalignment for content and cognitive demand between the CCSS and textbook.
In this study, the instrument that was used to collect data was the SEC. First, the textbook will be
described. Next, the background and validity of the SEC will be detailed to give an
understanding of how the data was collected.
Description of the Textbook
The textbook under review for this study was Thinking Mathematically by Blitzer
(2015). Thinking Mathematically was in its sixth edition, containing 14 chapters and 78 sections
for 939 pages. Blitzer (2015) states in the preface that the sixth edition has four primary goals: to
46
help students understand the fundamentals of mathematics, to show students how to solve real
problems that they may encounter in their lives, to enable students to have quantitative reasoning
skills that they may confront in college, career, and life, and to engage students in developing
problem-solving and critical thinking skills. The improvement in this edition includes 366
worked-out examples based on real-world examples and data, 653 short answer exercises that
focus on concepts and vocabulary, optional enrichment essays, brief reviews, sample homework
assignments, learning guides, and a more extensive prerequisite review.
Table 1
Information on the Textbook Reviewed
Title Publisher
Publication
date Pages Sections Chapters
Thinking
Mathematically Pearson 2015 939 78 14
47
Description of the Framework
The framework for content analysis was the SEC. The SEC is a two-dimensional
taxonomy for the content analysis and coding of standards, assessments, and other curriculum
materials by subject topic or content and cognitive demand (Council of Chief State School
Officers, n.d.). The SEC framework was used as the data analysis tool to examine the content
and cognitive demand between the CCSS and the textbook to determine the consistency between
the two. This study used the framework proposed by Polikoff et al. (2020), which recommended
four revisions to the existing SEC content languages to suit the context of the study, which
measured against the CCSS—like this study.
Instrumentation used in this study was the SEC topic cells (Appendix A) and the list of
cognitive demand categories (Appendix B). Polikoff et al. (2020) recommended four types of
revisions to suit mathematics content in the CCSS: (1) the addition of 40 coarse-grain topics that
focused on operations, (2) clarification of the meaning of certain coarse- and fine-grained topics,
combining topic sections, or simplifying the instrument for organization, (3) changing the
number of cognitive demand levels from five to six, and (4) including CCSS Mathematical
Practice questions at the end. For purposes of this study, there were no Mathematical Practice
questions administered or measured.
There were 228 content topic cells plus “999” used for “out of subject area” for a total of
229 topic cells. The list of content topic cells can be found in Appendix A.
The Polikoff et al. (2020) revision added one more cognitive demand level from the
original to make it six. Seven cognitive demand categories were used for this study:
memorize/recall, perform procedures, demonstrate/communicate understanding, justify/evaluate,
generalize, apply to real-world problems, and non-specific cognitive demand. The difference
48
from the six to seven categories was the inclusion of “non-specific cognitive demand.” The list
of cognitive demand categories can be found in Appendix B.
Data Collection
This study examined the alignment of an introductory college-level textbook with the
CCSS. The data from this study was collected from the review of the textbook using the Polikoff
et al. (2020) revised SEC framework. Data for this study was drawn from Thinking
Mathematically by Blitzer (2015).
The SEC procedure for coding was to have a content and cognitive demand for each item.
Data was notated on a worksheet on Excel, and the analysis was also completed on Excel. Figure
2 is an example of the first exercise set of the book. Each chapter had its own Excel sheet,
separated by sections so that each row could hold up to six pairs of topic and cognitive demand
per problem.
Figure 2
Sample Coding Page
49
The textbook was broken up into chapters and sections within chapters. Every fifth
problem in each section was rated on two dimensions: content and cognitive demand based on
the SEC taxonomy. Using every fifth problem does not reduce reliability in results (Polikoff et
al., 2015), however, using every fifth problem still gave this study 1,294 problems to be coded.
Figure 3 is a sample of one chunk of problems from Figure 2. The sample contains one set of
practice exercises with two coded items.
50
Figure 3
Sample Textbook Page from Exercise Set 1.1 with Content Codes
Item Rating
1 Logic, reasoning, and proof X justify/evaluate
5 Equivalent and non-equivalent fractions X
memorize/recall; Logic, reasoning, and proof
X justify/evaluate
Note. From Thinking Mathematically (p. 11) by Blitzer (2015).
51
Alignment was calculated in the typical way following the procedures in Polikoff (2015).
Trained raters had already analyzed the CCSS for grades 9–12, and those results were used for
the analysis. Once all the problems were rated, alignment was determined by calculating the
proportion of total textbook/standards content in each SEC cell and applying the formulas from
Polikoff (2015). Each textbook item was weighted equally. Some items had more than one cell,
involving multiple topics or cognitive demand.
To describe areas of alignment and misalignment, the textbook and standards were
analyzed in the following ways. First, each problem was broken down by topic and cognitive
demand level, following the procedures in Polikoff (2015). Second, I examined the most over-
and under-represented content in the textbook relative to the standards by calculating the
difference between the emphasis of the textbook and standards in each cell and sorting those
cells from high to low. The results will discuss the top 20 over- and under emphasized cells. I
reviewed the dataset for completion, consistency, reasonableness, and missing values.
Data Analysis
Following data collection, the focus of this analysis was to identify what topics were
emphasized by the CCSS and the textbook and determine those topics, which had a high degree
of alignment. Alternately, to determine what topics had a low level of alignment. The analysis
sought to determine if differences exist in content as well as in the emphasis of cognitive
demand. This analysis will answer the research questions: To what extent is an introductory
college-level mathematics textbook (for non-STEM majors) aligned with the CCSS? If there is
misalignment, what are the sources of misalignment?
The study’s data was calculated and analyzed by me, then organized by research
questions. Research question 1 asked if the content was aligned with the standards. Using the
52
main alignment index and the alternative alignment index as the tool, analyses were conducted to
determine the level of alignment and misalignment.
Once all the data from the textbook was collected, the analysis began using the two
formulas and proportions on Excel to determine the level of alignment. There were two formulas
used to calculate alignment. The first was main alignment, and the formula was as follows:
Main Alignment Index = 1−2
|4
!
−5
!
|
2
!
According to Polikoff (2015), “the main alignment index requires exact proportional
agreement” (p. 1199) and does not account for any additional content outside of the exact
proportion. In this case, 4
!
is the proportion of content in the CCSS and 5
!
is the proportion of
content in the textbook in cell i of document y. The main alignment index is the preferred index
because of the proportional agreement at the cell level and “best predicts student achievement
gains” (Gamoran et al., 1997, as cited in Polikoff, 2015, p. 1199).
The alternative index does not require exact proportional agreement, so if the content is
in the standards, the content topic value in that cell will be accepted in full (Polikoff, 2015). The
values are reported in decimals between 0 and 1, which can be converted to percentages. The
second was alternative alignment, and the formula was as follows:
Alternative Alignment Index = 2 5
!
!|#
!
$%
For research question 2, the data were separated into cognitive demand and content.
Proportional analysis on cognitive demand was simply the proportion of the book in the
cognitive demand category. There are seven categories for cognitive demand coded as B–G and
Z. The seven categories for cognitive demand are listed in Table 2. The values are reported in
53
decimals between 0 and 1. For one row, all seven cognitive demand cells can add up to 1. In this
case, due to rounding, it added up to .997. The categories are further described in Appendix B.
Table 2
Categories of Cognitive Demand
B C D E F G Z
Memorize/
recall
Perform
procedures
Demonstrate/
communicate
understanding
Justify/
evaluate
Generalize Apply to
real-world
problems
Non-
specific
cognitive
demand
54
Research question 2 also analyzed over emphasized and under emphasized content with
the corresponding cognitive demand. As illustrated in Figure 4, column R “difference” was used
to put the topics from greatest to least (positive to negative numbers). This gave a list of over to
under emphasized content. A perfect balance will be indicated by a 0.00, while a positive number
represents that the book emphasized the topic more and a negative number represents that the
CCSS emphasized the topic more. The data was aggregated into the top 20 lists for over and
under emphasized content for analysis. After the data was grouped, the topics were more closely
looked at to give more insight into the type of topics and cognitive demands on the top 20 lists.
55
Figure 4
Sample Proportions from Chapters 8–14 and Content Analyses
56
Chapter Four: Results
The purpose of this study was to address remediation by conducting an analysis between
an introductory college-level mathematics textbook and the CCSS to determine the levels of
alignment and sources of misalignment. The proportions were calculated and compared to
determine what topics and cognitive demand were being over- or under emphasized in the
textbook and the CCSS. Chapter Four presents the findings for the following research questions:
To what extent is an introductory college-level mathematics textbook aligned with the CCSS? If
there is misalignment, what are the sources of misalignment?
This chapter will report the data that answer the research questions using both the main
and alternative alignment indices and describe the content in the cells. The findings will analyze
the areas of alignment and misalignment and discuss explanations.
Alignment Indices
The main and alternative alignment indices will be presented for the textbook. There
were 1,603 topics and cognitive demand combinations that were content analyzed using both
indices. The results of the analyses are provided in Table 3.
Table 3
Standards Alignment Indices
Textbook Main alignment index Alternate alignment index
Thinking Mathematically .286 .543
57
Main Alignment Index
The main alignment index considers each cell, indicating what proportion of the textbook
content is in exact agreement with the standards (Polikoff, 2015). The main alignment index for
the textbook was .286. This means that 29% of the textbook is in exact agreement with the
CCSS.
The main alignment index was lower than a 2015 study by Polikoff, where he used the
SEC to determine the degree to which four popular fourth-grade textbooks (three of which
claimed to be CCSS aligned) were aligned with the standards. While the textbook in this study
has a value of .286, Polikoff (2015) found values generally higher than that. In Polikoff (2015),
three CCSS books used in the study had values of .29, .36, and .40; only Saxon was slightly
lower at .282. Saxon did not claim to be CCSS aligned. Thus, the book studied here is somewhat
less aligned to standards on index 1 than previous mathematics books that have been studied
using the same methods. However, the SEC mathematics framework used in this study contains
more topics and cognitive demand levels than the framework used in Polikoff (2015), and as
noted by Polikoff and Fulmer (2013), a greater number of SEC cells generally results in lower
estimates of alignment. Thus, the raw values may not be directly comparable.
Alternative Alignment Index
The alternative alignment index also considers each cell and looks at the proportions to
agree with the standards but is not as strict as the main alignment index as “it does not require
exact proportional agreement” (Polikoff, 2015, p. 1201). The alternative alignment index for the
textbook was .543. This means that 54% of the textbook is included in the CCSS content.
The textbook value of .543 is much lower than the values Polikoff (2015) found; this
book would be much lower than his range of .647 to .796.
58
The results from Polikoff (2015) are comparable when looking at the spread between the
main and alternative indices. The spread between this study’s indices was .257, while the
Polikoff (2015) textbooks range from .365–.406. This study was below the lower end of
Polikoff’s range. While 29% of the material in the book is in exact proportional agreement with
the standards, an additional 26% agrees in terms of topic/cognitive demand but not in terms of
proportion. The SEC approach treats the standards equally—they are all weighted the same—and
this is an untestable assumption about the relative importance of the standards. Using the
alternative index removes this assumption because it does not require that exact proportional
agreement. Still, with the removal of the assumption of the weighted importance of standards,
46% of the material in the book is not aligned with standards.
Cognitive Demand
Table 4 presents the proportions for cognitive demand for the CCSS and textbook.
Procedures are primarily emphasized in the CCSS (53%) and textbook (79%). In comparison,
Polikoff (2015) reviewed seven textbooks, and between 58% and 76% of the content emphasized
procedures. In other words, this book is more procedural than any of the fourth-grade textbooks
reviewed in that study.
The cognitive demands that are emphasized in the textbook more than the CCSS are
procedures and non-specific cognitive demand (.8%). Moreover, other cognitive demand levels
are emphasized in the CCSS more than in the textbook, such as memorize (5%), demonstrate
understanding (13%), conjecture/analyze (3%), solve non routine problems (.8%), and apply to
real-world problems (5%). When the textbook emphasizes procedures 26% more than in the
CCSS, other cognitive demand levels are lowered.
Table 4
Proportional Emphasis on Cognitive-Demand Levels in Textbooks and Standards
Standards/Textbook Memorize
Perform
Procedures
Demonstrate
Understanding Conjecture/Analyze
Solve Non-
Routine
Problems
Apply to
Real-
World
Problems
Non-
Specific
Cognitive
Demand
Common Core
Standards .082 .528 .221 .097 .014 .053 .002
Thinking
Mathematically .030 .787 .089 .069 .006 .006 .010
59
60
Over Emphasized Content
Table 5 presents the top 20 SEC cells that the book emphasizes more than the CCSS.
When taken together, the top 20 over emphasized cells in the textbook make up 51% of the total
content in the book, while in the CCSS those same cells account for just 8%. The CCSS only
addresses 9 of these 20 cells when tallying numbers and does not mention 11 of 20. When
narrowing the search to the top 3 topics, the book emphasizes 12%, while the CCSS addresses
none.
Table 5
Top 20 Over Emphasized Topics
Number Topic Cognitive demand Total
book
CCSS Difference
1 Logic, reasoning, and
proof
Procedures
0.046 0.000 0.046
2 Rounding Procedures 0.037 0.000 0.037
3 Sets Procedures 0.037 0.000 0.037
4 Out of subject area Procedures 0.046 0.019 0.027
5 Base-ten and non-
base-ten systems
Procedures
0.028 0.000 0.028
6 Add/subtract whole
numbers and
integers
Procedures
0.027 0.002 0.025
7 Computing with
exponents and
radicals
Procedures
0.035 0.010 0.025
8 Logic, reasoning, and
proof
Demonstrate
Understanding
0.023 0.000 0.023
61
Number Topic Cognitive demand Total
book
CCSS Difference
9 Formulas,
expressions, and
equations
Procedures
0.041 0.019 0.022
10 Conversions Procedures 0.021 0.000 0.021
11 Multiply fractions Procedures 0.022 0.004 0.018
12 Simple probability Procedures 0.026 0.008 0.018
13 Multi-step equations Procedures 0.030 0.014 0.016
14 Classification and
Venn diagrams
Procedures
0.020 0.004 0.016
15 Equivalence of
decimals, fractions,
and percents
Procedures
0.014 0.000 0.014
16 Logic, reasoning, and
proof
Conjecture/Analyze
0.014 0.000 0.014
17 Combinatorics Procedures 0.017 0.004 0.013
18 Personal financial
literacy
Procedures
0.012 0.000 0.012
19 Multiply decimals Procedures 0.010 0.000 0.010
20 Factors, multiples,
and divisibility
Procedures
0.010 0.000 0.010
When looking at cognitive demand specifically, 18 of 20 topics focus on procedures. As
shown in the prior analysis, the textbook over emphasized procedures relative to the standards.
Polikoff (2015) agrees in his findings that procedures are also over emphasized.
In terms of the topics themselves, of the top five over emphasized topics, only one has
any content in the CCSS. This fourth topic was “out of subject area” (1.9% in standards). The
62
other topics that round out the top five where the book covers more of the content are “logic,
reasoning, and proof” (4.6%), “rounding” (3.7%), “sets” (3.7%), and “base-ten and non-base-ten
systems” (2.8%). All five topics have cognitive demand, that is, procedures.
The topics that are not covered in the CCSS are not covered in the grades 9–12 standards.
For example, the second topic, “rounding,” is in the CCSS in grades 3, 4, and 5. The book asks
the question, use e to approximate to the nearest hundred-thousandth (p. 26). Specifically, this
addresses the grade 5 standards Number and Operations in Base Ten 5.NBT “use place value
understanding to round decimals to any place” (Common Core State Standards Initiative, n.d.b,
p. 35). This specific math problem was located in chapter 1.2 which is the second section of the
book. While the chapter was not noted as a review, it was titled Problem Solving and Critical
Thinking and the cells were of lower-level mathematics concepts that would be considered
below high school. The first chapter seemed like a review of basic math in preparation for the
rest of the book.
Another example, “sets” shows up in the CCSS in grades K, 6, 7, and high school. In this
problem, the student is asked to “express each set using the roster method…the set of natural
numbers less than or equal to 6” (p. 58). This section is located in chapter 2.1 under set theory
and explores three methods that represent sets. This is an example of one of them: the roster
method. Sets show up as a special topic in the SEC Taxonomy, while in the CCSS the word sets
presents in a few ways, nothing is related to this use of set theory. This is also below high school
level math concepts.
In the next sets section, 2.2 subsets show up in the CCSS under High School Statistics
and Probability S-CP to “describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
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other events” (p. 82). One problem asks to determine whether Í , Ì , both, or neither can be used
to form a true statement: {F, I, N} ____ {F, I, N, K}. This problem is simply using basic
identification illustrating whether the left is a subset of the right and the difficulty is in
understanding the definitions of the terms and symbols rather than the actual calculations.
A final example, the fifth topic, “base-ten and non-base-ten systems,” is covered in an
entire chapter titled number representation and calculation. In section 4.2, the problem asks to
convert a base ten numeral into a numeral in a given base: 63 to base 2. The CCSS only
references “base-ten” and not “non-base-ten systems.” The base-ten system is assumed to have
been taught before middle school, so this is not considered high school content. The CCSS
focuses solely on base-ten and nothing else. Although the SEC Taxonomy would include this
entire chapter, the standards do not exist at any grade level due to remaining silent on non-base-
ten systems.
Under Emphasized Content
Table 6 presents the top 20 SEC cells that the CCSS emphasizes more than the textbook. Overall,
the top 20 under emphasized cells in the textbook make up 2% of the total content in the book,
while in the CCSS those same cells account for 23%. All 20 topics are in the CCSS. The book
does not cover half of the top 20 under emphasized cells. Of the top 5 under emphasized topics,
the CCSS emphasizes 7%, while the textbook addressed none.
64
Table 6
Top 20 Under Emphasized Topics
Number Topic Cognitive
demand
Total
book
CCSS Difference
1 Complex numbers Procedures 0.000 0.006 -0.006
2 Rigid transformations Demonstrate
understanding 0.000 0.006 -0.006
3 Compound probability Procedures 0.001 0.008 -0.007
4 Vectors Procedures 0.001 0.008 -0.007
5 Matrices and determinants Procedures 0.000 0.008 -0.008
6 Formulas, expressions, and
equations
Demonstrate
understanding 0.000 0.008 -0.008
7 Exponential Procedures 0.001 0.010 -0.009
8 Data in a table or graph Procedures 0.001 0.010 -0.009
9 Completing the square Procedures 0.000 0.010 -0.010
10 Factoring polynomials Procedures 0.000 0.010 -0.010
11 Data in a table or graph Demonstrate
understanding 0.004 0.014 -0.010
12 Functions to model
data/phenomena
Procedures
0.000 0.010 -0.010
13 Standard algorithm for
subtraction
Procedures
0.012 0.023 -0.011
14 Use of graphing calculators Procedures 0.001 0.012 -0.011
15 Vectors Demonstrate
understanding 0.001 0.012 -0.011
16 Geometric constructions Procedures 0.000 0.012 -0.012
17 Standard algorithm for
addition
Procedures
0.000 0.012 -0.012
65
Number Topic Cognitive
demand
Total
book
CCSS Difference
18 Conditional probability Procedures 0.000 0.012 -0.012
19 Empirical/experimental
probability
Apply to real-
world problems 0.000 0.012 -0.012
20 Conditional probability Apply to real-
world problems 0.000 0.017 -0.017
When comparing the number of cognitive demand in under emphasized topics (14 of 20)
to the over emphasized topics (18 of 20), there are 4 fewer procedures focused topics. Under
emphasized topics also include the application of real-world problems.
The top five under emphasized topics will be explained in more detail in terms of the
topics themselves. First, none of the top five under emphasized have any content in the book.
Next, the most under emphasized topic is “complex numbers” (1.7%) found in Number and
Quantity N-CN and Algebra A-REI. Finally, the following four topics have the same emphasis in
the standards (1.2% each):
Rigid transformations are found in Geometry G-CO of the CCSS. The book does not
attempt to cover this topic at all. Examples of rigid transformations in the CCSS are as follows:
• CCSS.MATH.CONTENT.HSG.CO.B.6
o Use geometric descriptions of rigid motions to transform figures and to predict the
effect of a given rigid motion on a given figure; given two figures, use the
definition of congruence in terms of rigid motions to decide if they are congruent.
• CCSS.MATH.CONTENT.HSG.CO.B.7
66
o Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
• CCSS.MATH.CONTENT.HSG.CO.B.8
o Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Compound probability found in Statistics and Probability S-CP. This book has
probability and statistics and covers some of the standards in the CCSS in chapter 11.6 pages 725
to 733 as follows:
• CCSS.MATH.CONTENT.HSS.CP.B.6
o Find the conditional probability of A given B as the fraction of B's outcomes that
also belong to A, and interpret the answer in terms of the model.
• CCSS.MATH.CONTENT.HSS.CP.B.7
o Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the
answer in terms of the model.
• CCSS.MATH.CONTENT.HSS.CP.B.8
o (+) Apply the general Multiplication Rule in a uniform probability model, P(A
and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the
model.
Vectors are found in Number and Quantity N-VM. The book does not attempt to cover
this topic at all. Examples of the CCSS for vectors are as follows:
• CCSS.MATH.CONTENT.HSN.VM.A.1
67
o (+) Recognize vector quantities as having both magnitude and direction.
Represent vector quantities by directed line segments and use appropriate symbols
for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
• CCSS.MATH.CONTENT.HSN.VM.A.2
o (+) Find the components of a vector by subtracting the coordinates of an initial
point from the coordinates of a terminal point.
• CCSS.MATH.CONTENT.HSN.VM.A.3
o (+) Solve problems involving velocity and other quantities that can be represented
by vectors.
Matrices and Determinants” are found in Number and Quantity N-VM and Algebra A-
REI. The book does not attempt to cover this topic at all. Examples of matrices and determinants
in the CCSS are as follows:
• CCSS.MATH.CONTENT.HSN.VM.C.6
o (+) Use matrices to represent and manipulate data, e.g., to represent
payoffs or incidence relationships in a network.
• CCSS.MATH.CONTENT.HSN.VM.C.7
o (+) Multiply matrices by scalars to produce new matrices, e.g., as when all
of the payoffs in a game are doubled.
• CCSS.MATH.CONTENT.HSN.VM.C.8
o (+) Add, subtract, and multiply matrices of appropriate dimensions.
• CCSS.MATH.CONTENT.HSN.VM.C.10
o (+) Understand that the zero and identity matrices play a role in matrix
addition and multiplication similar to the role of 0 and 1 in the real
68
numbers. The determinant of a square matrix is nonzero if and only if the
matrix has a multiplicative inverse.
The cognitive demand were all procedures and the total content in the CCSS for all five
under emphasized topics was nearly 7%.
Summary
The analyses identified the level of alignment of this introductory college-level
mathematics textbook with the CCSS. The study revealed that using main and alternative indices
to measure alignment results in two markedly different conclusions, with the alternative showing
a significantly higher level of alignment over the main index.
The level of exact proportional agreement was the primary factor in the difference
between the main and alternative alignment indices. At best, using the alternative index, with the
removal of the weighted standards, there was still 46% not aligned with the standards. At worst,
71% of the book was not aligned with the standards using the main alignment index.
Over and under emphasized content yielded similar results when comparing the opposite
ends of the continuum. They were either only in the book or only in the standards on both
extremes. It was further identified that some items in the SEC Taxonomy were not in the CCSS.
Regarding cognitive demand, the results followed what previous studies have found, textbooks
focus more on procedures. This book turned out to focus much more on procedures than those in
the Polikoff (2015) study. In return, all the other cognitive demand levels have been reduced as a
result.
The Top 20 over emphasized cells made up a little more than half the content in the book,
yet only reaching 8% of the CCSS. The standards touched 9 of 20 cells, and the remaining were
primarily due to concepts being below high school level or that it was a review of basic
69
mathematical concepts. Polikoff et al. (2020) note that difficulty is another dimension that is
separate and distinct from cognitive demand, and the SEC does not measure or emphasize the
level of difficulty. When narrowing the scope to the Top 3 over emphasized cells, the book
covered 12%, while the CCSS did not emphasize those topics at all.
The Top 20 under emphasized cells made up 2% of the content in the book while
reaching nearly a quarter of the CCSS. The standards touched all 20 cells. When reviewing only
the Top 5 under emphasized cells, the book covered nothing while the CCSS covered 7%. In
terms of cognitive demand, under emphasized cells had four fewer procedures, but included
“apply to real-world problems” twice and “demonstrate understanding” four times that the over
emphasized list did not include. Over emphasized cognitive demand included one “demonstrate
understanding” and one “conjecture/analyze” in addition to the 18 procedural ones.
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Chapter Five: Discussion
The purpose of this study was to address remediation by conducting an analysis between
an introductory college-level mathematics textbook and the CCSS to determine levels of
alignment and sources of misalignment. Chapter four presented the findings for the following
research questions: To what extent is an introductory college-level mathematics textbook aligned
with the CCSS? If there is misalignment, what are the sources of misalignment? Chapter five
will summarize the findings, discuss the implications for practice, provide recommendations for
research, and provide the conclusion of the dissertation.
Summary of Findings
The research questions focused on determining if the book was aligned with the
standards. The data from the alignment indices showed that when using the main index, only
29% was aligned while using the alternative index, 54% was aligned with the standards. The
findings confirmed that 46%–71% of the book was not aligned to standards. The alignment
values were slightly lower than Polikoff’s (2015) findings in his study of four fourth-grade CCSS
aligned textbooks. Because the SEC mathematics framework used in this study contained more
topic and cognitive demand cells than the Polikoff (2015) study, the raw values may not be
comparable (Polikoff & Fulmer, 2013).
The second research question was triggered only if there was misalignment, and there
was. The question asked what the sources of the misalignment were. As stated previously, 46%–
71% of the book was not aligned to standards depending on the alignment index, and the sources
of misalignment were both in content and cognitive demand. The data from the analyses showed
that the top 20 over emphasized content in the book accounted for half of the content while only
accounting for 8% of standards. A majority of the top 20 over emphasized topics were lower-
level mathematics concepts than high school standards. The data from the analysis of the top 20
71
under emphasized content in the book made up only 2% of the book, but nearly a quarter of the
high school standards.
Additionally, all the under emphasized topics were in the CCSS. Regarding cognitive
demand, this book emphasized procedures slightly higher than the upper limit in Polikoff’s
(2015) range, which was a quarter more than the CCSS. All of this is consistent with Polikoff
(2015) who found that the three primary sources of misalignment are (1) textbooks over
emphasized procedures and memorization, (2) textbooks emphasize content that is not the same
that is assumed for the standards, and (3) there are SEC cells that textbooks do not cover which
accounts for one fourth to one sixth of the SEC cells.
To summarize, the findings of this study confirmed that using both the main and
alternative alignment index of the SEC produced poor alignment to the CCSS. The findings on
misalignment showed that there was misalignment on content and cognitive demand. There was
an overemphasis of lower-level content and an under emphasis on content that met a quarter of
the standards for the high school level. Some topics were not in the book that were in the CCSS
and there were topics in the CCSS that were not in the book. Finally, cognitive demand focused
primarily on procedures. These were consistent with Polikoff’s (2015) textbook content analysis
alignment study.
Implications for Practice
The findings of this study revealed several implications for practice because there was
much content in the book below the high school level. The implications are split between
whether the book should evaluate students on high school content or whether the book was even
aligned to high school content. In looking at the issues with alignment to high school content,
where is the book aligned and where is it not aligned? Why does alignment even matter? These
questions about the textbook gave rise to further inquiry for practice.
72
The study found that there was a lot of content emphasized below high school standards.
The Survey of Mathematics course was a college-level mathematics course and required
placement into the course for many schools. Since the validity of the placement test depends on
the interpretation of the results and the cutoff score (Belfield & Crosta, 2012), if the content of
this course will focus on standards lower than high school, the colleges should consider making
Survey of Mathematics an open enrollment course with no placement requirement. If the purpose
of remediation is to provide the student the skills needed to achieve college-level proficiency
(Bahr, 2010; Bettinger & Long, 2005) and this course is terminal, the knowledge needed is only
for this course because the student is on the non-STEM track and may become a gatekeeper if
there is no relevance to the content or course after completion.
The study suggested that the textbook was not evaluating students on what they should
have learned in high school because it was not well aligned to high school standards. Teachers
rely heavily on textbooks to inform curriculum (Chingos & Whitehurst, 2012). The book spent a
lot of time reteaching concepts prior to and from high school instead of covering new content.
The literature in chapter two discusses standards-based education reform and that reform only
covers elementary and secondary schools, excluding the course in which this book was used. The
motivating factor in the reform is to create a student prepared for college or work. However,
specifically for mathematics, the CCSS are supposed to create college, career, and life ready
students (Common Core State Standards Initiative, n.d.a). It is unclear how this course fits into
each student’s program since it is a terminal course. Some may argue that this course is not
necessary at all. The literature supports aligning content to standards, but the standards expected
of the college student for this course is unclear. The college needs to be clear about how this
course fits into the program of study and the intended student outcomes.
73
Findings from this study also indicated that the single area of the textbook that was
aligned (nearest to zero) where the content coverage and CCSS coverage met closest was for
mean, median, and mode, however on overall content, this was consistent where the content was
below high school standard material. The area of the textbook not well aligned was in the
number of emphasized topics, higher-level content, and cognitive demand. Polikoff (2015) found
a disparity between the main and alternative indices and concluded that the content in the book
was not evenly spread out as it is assumed to be in the standards. This is likely the case. When
looking at the top 100 over emphasized SEC cells in the study, only 25 of 100 were covered in
the CCSS. That means 75% of the top 100 over emphasized cells were not covered at all in the
standards. The emphasis in those cells on procedures, consistent with Polikoff (2015) and the
findings in chapter four, misalign the cognitive demand for the book. This implication for
practice clarifies the need to determine the content standards for the course by understanding
how the student outcomes for mathematics are relevant to the program of study.
There is no national standard for comparing textbooks and because books are a main
source of curriculum, testing for content and cognitive demand alignment is critical. Textbooks
in mathematics play a larger role than in other subjects (Fan et al., 2013). Students will learn
what is taught if testing, standards, and instruction are aligned (Martone & Sireci, 2009). Since
the findings from the study showed poor alignment to the CCSS, it would lead to the question of
what is considered college-ready?
Students spend three to four years in high school math with curriculum and instruction
based on CCSS that is supposed to create students prepared for career, college, and life after
graduation (Common Core State Standards Initiative, n.d.c). However, if the student enrolls in
their first math class in college and the content is completely out of alignment with the math that
74
was taught over the past four years, it would be absurd. The lack of alignment leaves a potential
for remediation which has become an intellectual death sentence with 40% of community college
students never completing the remedial sequence (Complete College America, 2012). In
addition, the average remedial non-completer has an 83% chance of not graduating or
transferring to another institution (Bahr, 2008). When considering the importance of defining
what “college-ready” is or is not for mathematics, there may be unintended consequences to the
students that can affect their entire career trajectory. Determining alignment to understand what
students should know for a course and defining those standards or requirements are important for
both student and teacher.
Recommendations for Research
This study was limited to one mathematics book and one rater. There are many
opportunities for research after this study. Further research could include alignment studies on
instruction, other curriculum materials, and assessments. There could be a comparison study
against other widely used textbooks to determine if the content and cognitive demand alignment
are similar or dissimilar. Then, effectiveness can be studied by looking at student outcomes.
First, it is recommended that future studies include the alignment of instruction, other
curriculum materials, and assessments to gain a complete look at what is taught in the classroom.
The SEC allows instruction, curriculum materials, and assessments for study (Porter, 2002). The
analyses of these three items will give a more comprehensive look into what the teacher teaches
in the classroom. While textbooks have played a large role in mathematics classes (Fan et al.,
2013), topics that are not in a textbook have a high likelihood of not being a topic of instruction
(Stein et al., 2007), therefore studying the alignment of instruction, other curriculum materials,
and assessments will provide a complete picture of the actual alignment of the class and its
outcome effectiveness.
75
Second, this is a Survey of Mathematics course meant to be a terminal math course and is
common around the United States. It would be interesting to see how other textbooks used for
this class align with the CCSS, but how those books align with each other. There is no set of
national standards for postsecondary education and no common definition of “college readiness”
(Greene & Forster, 2003; Merisotis & Phipps, 2000; Oudenhoven, 2002; Porter & Polikoff,
2012). Each college or university has its standard for what it considers “college-ready” and uses
admissions policies to reinforce that. In addition, each college or university has its placement
process for determining what math course the student is allowed to register for, and most
importantly, if that requires remediation or not. Comparing the textbooks would allow future
research in combination with placement tests and successful outcomes to determine if, for
purposes of non-STEM majors, the current method for determining college-level mathematics
placement is appropriate. This could also assist in the future selection of textbooks for the
course.
Last, research needs to be done to look at student outcomes. This is where vertical
alignment comes in. The colleges need to look at the desired outcomes for the student and course
and determine if it is appropriate for the major. This could be a legacy course that has always
been included in the program of study and is no longer relevant. The success of the student in
this course and its contribution to the overall student or student’s major need to be ascertained. A
study can be done to explore whether a Survey of Mathematics course is required for student
success in non-STEM majors. Outcomes should also be looked at if the course is a barrier. In
order to promote a strong vertical alignment from K–12 to higher education in Hawai‘i, the
recommendation is to allow the colleges to work on the outcomes first, due to the content of the
textbook being out of alignment and below high school grade level and out of standards.
76
Conclusions
There is a need to align high school and college expectations to improve student success
from K–12 and college in mathematics. College students are not meeting the employers’ needs,
and there is an increasing demand for high-skill and technical employees (Carnevale et al.,
2013). Considering that not all students are in school for STEM majors, this study focused on a
non-STEM college-level mathematics course textbook to seek out the absolute minimum
college-level mathematics standards.
Achieving national and local goals for college readiness in mathematics requires
articulation between secondary and postsecondary institutions, that the “college-ready” standard
is clearly articulated, and students have the preparation in content and practice to be successful in
the courses. Otherwise, students may end up on the remediation track that has shown to fail
many students who enroll (Bahr, 2008). Student outcomes have got to be at the forefront of
educational decision-making and the use of standards is a reform effort to improve performance
and strengthen accountability. Alignment studies measure the consistency with those standards
and the ability to quantify that is important for accountability purposes.
This study analyzed the alignment of one introductory college-level mathematics
textbook against the CCSS to determine the level of alignment or misalignment. The study also
identified sources of misalignment, which indicated that more research is needed to determine if
it is important to be aligned with K–12 CCSS standards or another reason that the book is not
aligned. Future studies should investigate the alignment of all course materials, instruction, and
assessments and determine if the outcome for this course is meeting the stated objectives to
determine if student outcomes are being met and unintended consequences are not occurring.
77
References
Achieve. (2008). The building blocks of success: Higher-level math for all students.
http://www.achieve.org/files/BuildingBlocksofSuccess.pdf
ACT, Inc. (n.d.a). The ACT profile report–national.
https://www.act.org/content/dam/act/unsecured/documents/2020/2020-National-ACT-
Profile-Report.pdf
ACT, Inc. (n.d.b). Compass course placement service interpretive guide: Setting the right course
for college success. https://www.act.org/content/dam/act/unsecured/documents/Compass-
CPS_Guide.pdf
ACT, Inc. (2015). The condition of college & career readiness 2015: National.
https://www.act.org/content/dam/act/unsecured/documents/Condition-of-College-and-
Career-Readiness-Report-2015-United-States.pdf
ACT, Inc. (2019). The condition of college & career readiness 2019: National.
https://www.act.org/content/dam/act/unsecured/documents/National-CCCR-2019.pdf
Adelman, C. (1999). Answers in the tool box: Academic intensity, attendance patterns, and
bachelor's degree attainment. U.S. Department of Education.
https://files.eric.ed.gov/fulltext/ED431363.pdf
Adelman, C. (2004). Principal indicators of student academic histories in postsecondary
education, 1972–2000. U.S. Department of Education.
https://files.eric.ed.gov/fulltext/ED483154.pdf
Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through
college. U.S. Department of Education.
http://www2.ed.gov/rschstat/research/pubs/toolboxrevisit/toolbox.pdf
78
Allen, J. (2013). Updating the ACT college readiness benchmarks. ACT, Inc.
https://www.act.org/content/dam/act/unsecured/documents/ACT_RR2013-6.pdf
Allensworth, E., Nomi, T., & Montgomery, N. (2009). College preparatory curriculum for all:
Academic consequences of requiring Algebra and English I for ninth graders in Chicago.
Educational Evaluation and Policy Analysis, 31(4), 367–391.
https://doi.org/10.3102/0162373709343471
Attewell, P., Lavin, D., Domina, T., & Levey, T. (2006). New evidence on college remediation.
The Journal of Higher Education, 77(5), 886–924. https://doi.org/10.1353/jhe.2006.0037
Aughinbaugh, A. (2012). The effects of high school math curriculum on college attendance:
Evidence from the NLSY97. Economics of Education Review, 31, 861–870.
https://doi.org/10.1016/j.econedurev.2012.06.004
Autor, D. (2011). The polarization of job opportunities in the U.S. labor market: Implications for
employment and earnings. Community Investments, 23(2), 11–16.
http://www.researchgate.net/profile/David_Autor/publication/227437438_The_polarizati
on_of_job_opportunities_in_the_U.S._labor_market_implications_for_employment_and
_earnings/links/00b7d520285b657fd8000000.pdf
Bahr, P. R. (2008). Does mathematics remediation work?: A comparative analysis of academic
attainment among community college students. Research in Higher Education, 49(5),
420–450. https://doi.org/10.1007/s11162-008-9089-4
Bahr, P. R. (2010). Preparing the underprepared: An Analysis of racial disparities in
postsecondary mathematics remediation. The Journal of Higher Education, 81(2), 209–
237. https://doi.org/10.1080/00221546.2010.11779049
79
Barbatis, P. (2010). Underprepared, ethnically diverse community college students: Factors
contributing to persistence. Journal of Developmental Education, 33(3), 14–24.
Belfield, C. R., & Crosta, P. M. (2012). Predicting success in college: The importance of
placement tests and high school transcripts (CCRC Working Paper No. 42). Community
College Research Center at Columbia University, Teachers College.
https://ccrc.tc.columbia.edu/media/k2/attachments/predicting-success-placement-tests-
transcripts.pdf
Berdie, R. F. (1960). Some principles and problems of selective college admissions: Should
entrance requirements be raised? The Journal of Higher Education (Columbus), 31(4),
191–199. https://doi.org/10.2307/1977403
Bettinger, E. P. & Long, B. T. (2005). Remediation at the community college: Student
participation and outcomes. New Directions for Community Colleges, 129, 17–26.
https://doi.org/10.1002/cc.182
Bettinger, E. P. & Long, B. T. (2009). Addressing the needs of underprepared students in higher
education: Does college remediation work? The Journal of Human Resources, 44(3),
736–771. https://doi.org/10.1353/jhr.2009.0033
Bivens, J. (2016). Why is recovery taking so long—and who is to blame? Economic Policy
Institute. https://www.epi.org/publication/why-is-recovery-taking-so-long-and-who-is-to-
blame/
Blitzer, R. (2015). Thinking mathematically (6th ed.). Pearson Education, Inc.
Boyer, P. G., Butner, B. K., & Smith, D. (2007). A portrait of remedial instruction: Faculty
workload and assessment techniques. Higher Education, 54(4), 605–613.
https://doi.org/10.1007/s10734-006-9030-8
80
Breneman, D. W., Abraham Jr., A. A., & Hoxby, C. M. (1998). Remediation in higher education:
Its extent and cost. Brookings Papers on Educational Policy, 359–383.
https://www.jstor.org/stable/20067201
Breneman, D. W., & Haarlow, W. N. (1998). Remediation education: Costs and consequences.
Thomas B. Fordham Foundation. https://files.eric.ed.gov/fulltext/ED422770.pdf
Brown, R. S. & Conley, D. T. (2007). Comparing state high school assessments to standards for
success in entry-level university courses. Educational Assessment, 12(2), 137–160.
https://doi.org/10.1080/10627190701232811
Bryk, A., & Treisman, U. (2010). Make math a gateway, not a gatekeeper. The Chronicle of
Higher Education, 56(32).
Buddin, R., & Croft, M. (2014). Do stricter high school graduation requirements improve
college readiness? ACT Inc. https://www.act.org/research/papers/pdf/wp-2014-1.pdf
Calcagno, J. C., Crosta, P., Bailey, T., & Jenkins, D. (2007). Does age of entrance affect
community college completion probabilities? Evidence from a discrete-time hazard
model. Educational Evaluation and Policy Analysis, 29(3), 218–235.
https://doi.org/10.3102/0162373707306026
Carlson, D., & Planty, M. (2012). The ineffectiveness of high school graduation credit
requirement reforms: A story of implementation and enforcement? Educational Policy,
26(4), 592–626. https://doi.org/10.1177/0895904811417582
Carnevale, A. P., Jayasundera, T., & Cheah, B. (2012). The college advantage: Weathering the
economic storm. Georgetown University Center on Education and the Workforce.
https://files.eric.ed.gov/fulltext/ED534454.pdf
81
Carnevale, A. P., Smith, N., & Strohl, J. (2010). Help wanted: Projections of jobs and education
requirements through 2018. Center on Education and the Workforce at Georgetown
University. https://cew.georgetown.edu/wp-
content/uploads/2014/12/HelpWanted.ExecutiveSummary.pdf
Carnevale, A. P., Smith, N., & Strohl, J. (2013). Recovery: Job growth and education
requirements through 2020. Center on Education and the Workforce at Georgetown
University.
Center for the Study of Mathematics Curriculum. (2004). Curriculum and evaluation standards
for school mathematics.
http://www.mathcurriculumcenter.org/PDFS/CCM/summaries/standards_summary.pdf
Chingos, M. M., & Whitehurst, G. J. (2012). Choosing blindly: Instructional materials, teacher
effectiveness, and the common core. Brown Center on Education Policy at Brookings.
https://www.brookings.edu/wp-
content/uploads/2016/06/0410_curriculum_chingos_whitehurst.pdf
Clinedinst, M. (2015). State of college admission. National Association of College Admission
Counseling. http://www.nxtbook.com/ygsreprints/NACAC/2014SoCA_nxtbk/#/26
Clinedinst. M. (2020). 2019 State of college admission report. Journal of College Admission,
246, 15.
Clune, W. H., & White, P. A. (1992). Education reform in the trenches: Increased academic
course taking in high schools with lower achieving students in states with higher
graduation requirements. Educational Evaluation and Policy Analysis, 14(1), 2–20.
https://doi.org/10.3102/01623737014001002
82
Collegeboard. (n.d.a). Compare SAT specifications.
https://collegereadiness.collegeboard.org/sat/inside-the-test/compare-old-new-
specifications
CollegeBoard. (n.d.b). Practice for ACCUPLACER.
https://accuplacer.collegeboard.org/students/accuplacer-tests
Common Core State Standards Initiative. (n.d.a). About the Standards.
http://www.corestandards.org/about-the-standards/
Common Core State Standards Initiative. (n.d.b). Common Core State Standards for
Mathematics. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf
Common Core State Standards Initiative. (n.d.c). Development Process.
http://www.corestandards.org/about-the-standards/development-process/
Common Core State Standards Initiative. (n.d.d). Mathematics Standards.
http://www.corestandards.org/Math/
Complete College America. (2012). Remediation: Higher education's bridge to nowhere.
https://completecollege.org/wp-content/uploads/2017/11/CCA-Remediation-final.pdf
Conger, D. (2015). High school grades, admissions policies, and the gender gap in college
enrollment. Economics of Education Review, 46, 144–147. https://doi.org/
10.1016/j.econedurev.2015.03.003
Conger, D., & Long, M. C. (2013). Gender gaps in college enrollment: The role of gender
sorting across public high schools. Educational Researcher, 42(7), 371–380.
https://doi.org/10.3102/0013189X13503983
Conley, D. T. (2005). Align high school with college for greater student success. The Education
Digest, 71(2), 4–12.
83
Connelly, B. L., Certo, S. T., Ireland, R. D., & Reutzel, C. R. (2011). Signaling theory: A review
and assessment. Journal of Management, 37(1), 39–67.
https://doi.org/10.1177/0149206310388419
Council of Chief State School Officers. (n.d.). Content Analysis. http://secpd.org/about-sec-
sidebar/content-analysis/
Crisp, G., & Delgado, C. (2014). The impact of developmental education on community college
persistence and vertical transfer. Community College Review, 42(2), 99–117.
https://doi.org/10.1177/0091552113516488
Donovan, W. J., & Wheland, E. R. (2008). Placement tools for developmental mathematics and
intermediate algebra. Journal of Developmental Education, 32(2), 2–11.
Dortch, C. (2012). Carl D. Perkins career and technical education act of 2006: Implementation
issues. Congressional Research Service.
https://dpi.wi.gov/sites/default/files/imce/cte/pdf/cpaimpissues.pdf
Dowd, A. C. (2007). Community colleges as gateways and gatekeepers: Moving beyond the
access "saga" toward outcome equity. Harvard Educational Review, 77(4), 407–419.
https://doi.org/10.17763/haer.77.4.1233g31741157227
Dowd, A. C., & Grant, J. L. (2006). Equity and efficiency of community college appropriations:
The role of local financing. Review of Higher Education, 29(2), 167–194.
https://doi.org/10.1353/rhe.2005.0081
Economic Policy Institute. (n.d.). The great recession. The state of working America:
http://stateofworkingamerica.org/great-recession/
84
Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics in education:
Development status and directions. ZDM Mathematics Education, 45, 633–646.
https://doi.org/10.1007/s11858-013-0539-x
Fitchett, S., King, K., & Champion, J. (2011). Outcomes of mathematics placement: An analysis
of advising and enrollment data. PRIMUS, 21(7), 577–591.
https://doi.org/10.1080/10511970903515323
Floden, R. E., Porter, A. C., Schmidt, W. H., Freeman, D. J., & Schwille, J. R. (1981). Responses
to curriculum pressures: A policy-capturing study of teacher decisions about
content. Journal of Educational Psychology, 73(2), 129–141.
http://dx.doi.org/10.1037/0022-0663.73.2.129
Gamoran, A., & Hannigan, E. C. (2000). Algebra for everyone? Benefits of college-preparatory
mathematics for students with diverse abilities in early secondary school. Educational
Evaluation and Policy Analysis, 22(3), 241–254.
https://doi.org/10.3102/01623737022003241
Gamoran, A., Porter, A. C., Smithson, J., & White, P. A. (1997). Upgrading high school
mathematics instruction: Improving learning opportunities for low-achieving, low-
income youth. Educational Evaluation and Policy Analysis, 19(4), 325–338.
https://doi.org/10.3102/01623737019004325
Greene, J. P., & Forster, G. (2003). Public high school graduation and college readiness rates in
the United States. Center for Civic Innovation at the Manhattan Institute.
http://www.manhattan-institute.org/pdf/ewp_03.pdf
Grubb, W. N. (1999). Honored but invisible: An inside look at teaching in community colleges.
Routledge. https://doi.org/10.4324/9780203900857
85
Hagedorn, L. S., & DuBray, D. D. (2010). Math and science success and nonsuccess: Journeys
within the community college. Journal of Women and Minorities in Science and
Engineering, 16(1), 31–50. https://doi.org/10.1615/JWomenMinorScienEng.v16.i1.30
Hagedorn, L. S., Siadat, M. V., Fogel, S. F., Nora, A., & Pascarella, E. T. (1999). Success in
college mathematics: Comparisons between remedial and nonremedial first-year college
students. Research in Higher Education, 40(3), 261–284.
https://doi.org/10.1023/A:1018794916011
Hanushek, E. A., & Woessmann, L. (2008). The role of cognitive skills in economic
development. Journal of economic literature, 46(3), 607–668.
https://doi.org/10.1257/jel.46.3.607
Harwell, M., Post, T. R., Cutler, A., Maeda, Y., Anderson, E., Norman, K. W., & Medhanie, A.
(2009). The preparation of students from national science foundation-funded and
commercially developed high school mathematics curricula for their first university
mathematics course. American Educational Research Journal, 46(1), 203–231.
doi:10.3102/0002831208323368
Hawai‘i P–20. (n.d.). Career and college readiness indicators: Class of 2020.
http://hawaiidxp.org/files/ccri_pdfs/2020/Statewide20.pdf
Hawai‘i State Department of Education. (n.d.a). Hawai‘i Common Core Standards.
http://www.hawaiipublicschools.org/TeachingAndLearning/StudentLearning/CommonCo
reStateStandards/Pages/home.aspx
Hawai‘i State Department of Education. (n.d.b). RTTT Application.
https://www.hawaiipublicschools.org/VisionForSuccess/AdvancingEducation/milestones/Ra
ceToTheTop/Pages/home.aspx
86
Hawai‘i State Department of Education. (2012). Hawai‘i race to the top: Exectutive summary.
http://www.hawaiipublicschools.org/DOE%20Forms/Advancing%20Education/RTTTSu
mmary.pdf
Hoffer, T. B. (1997). High school graduation requirements: Effects on dropping out and student
achievement. Teachers College Record, 98(4), 584–607.
Horn, C. M., McCoy, Z., Campbell, L., & Brock, C. (2009). Remedial testing and placement in
community colleges. Community College Journal of Research and Practice, 33, 510–
526. https://doi.org/10.1080/10668920802662412
Hoyt, J. E. (1999). Remedial education and student attrition. Community College Review, 27(2),
51–72. https://doi.org/10.1177/009155219902700203
Hoyt, J. E., & Sorensen, C. T. (2001). High school preparation, placement testing, and college
remediation. Journal of Developmental Education, 25(2), 26–34.
Humphreys, D. (2012). What's wrong with the completion agenda--and what we can do about it.
Liberal Education, 98(1), 8.
Hussar, W. J., & Bailey, T. M. (2014). Projections of education statistics to 2022. U.S.
Department of Education, National Center for Education Statistics.
http://nces.ed.gov/pubs2014/2014051.pdf
Ignash, J. M. (1997). Who should provide postsecondary remedial/developmental education?
New Directions for Community Colleges 1997(100), 5–20.
https://doi.org/10.1002/cc.10001
James. C. L. (2006). ACCUPLACER Online: Accurate placement tool for developmental
programs? Journal of Developmental Education, 30(2), 2–8.
87
Kane, T. J., Owens, A. M., Marinell, W. H., Thal, D. R. C., & Staiger, D. O. (2016). Teaching
higher: Educators' perspectives on common core implementation. Harvard University.
https://cepr.harvard.edu/files/cepr/files/teaching-higher-report.pdf
Kapi‘olani Community College. (2015). Alternative placement.
http://www2.hawaii.edu/~kcctest/alternativeplacement.html
Keller, M. (2001). College performance of new Maryland high school graduates: Student
outcome and achievement report. Maryland Higher Education Commission.
https://files.eric.ed.gov/fulltext/ED494293.pdf
Kelly, A. P., & Schneider, M. (2012). Introduction. In A. P. Kelly, & M. Schneider (Eds.),
Getting to graduation: The completion agenda in higher education. Johns Hopkins
University Press.
Kirst, M. W., & Bracco, K. R. (2004). Bridging the great divide: How the K–12 and
postsecondary split hurts students, and what can be done about it. In M. W. Kirst, & A.
Venezia (Eds.), From high school to college: Improving opportunities for success in
postsecondary education (pp. 1–30). Jossey-Bass.
La Marca, P. M., Redfield, D., Winter, P. C., & Despriet, L. (2000). State standards and state
assessment systems: A guide to alignment. Council of Chief State School Officers.
https://www.govinfo.gov/content/pkg/ERIC-ED466497/pdf/ERIC-ED466497.pdf
Lemelson-MIT. (2012). Young americans recognize the impact of innovation on U.S. economy.
https://lemelson.mit.edu/news-events/news/young-americans-recognize-impact-
innovation-us-economy
88
Levin, H. M., & Calcagno, J. C. (2008). Remediation in the community college: An evaluator's
perspective. Community College Review, 35(3), 181–207.
https://doi.org/10.1177/0091552107310118
Lillard, D. R., & DeCicca, P. P. (2001). Higher standards, more dropouts? Evidence within and
across time. Economics of Education Review, 20(5), 459–473.
https://doi.org/10.1016/S0272-7757(00)00022-4
Long, M. C., Iatarola, P., & Conger, D. (2009). Explaining gaps in readiness for college-level
math: The role of high school courses. Education Finance and Policy, 4(1), 1–33.
https://doi.org/10.1162/edfp.2009.4.1.1
Martone, A., & Sireci, S. G. (2009). Evaluating alignment between curriculum, assessment, and
instruction. Review of Educational Research, 79(4), 1332–1361.
https://doi.org/10.3102/0034654309341375
Marzano, R. J., Waters, T., & McNulty, B. A. (2005). School leadership that works: From
research to results. Association for Supervision and Curriculum Development.
Mathison, S. (2003). The accumulation of disadvantage: The role of educational testing in the
school career of minority children. Workplace, 10, 32–43.
https://louisville.edu/journal/workplace/issue5p2/mathison.html
Mattern, K. D. & Packman, S. (2009). Predictive validity of ACCUPLACER scores for course
placement: A meta-analysis. Collegeboard.
https://files.eric.ed.gov/fulltext/ED561046.pdf
McDonnell, L.M. (2005). No Child Left Behind and the federal role in education: Evolution or
revolution? Peabody Journal of Education, 80, 2, 19–38.
89
Medhanie, A. G., Dupuis, D. N., LeBeau, B., Harwell, M. R., & Post, T. R. (2012). The role of
the ACCUPLACER mathematics placement test on a student's first college mathematics
course. Educational and Psychological Measurement, 72(2), 332–351.
https://doi.org/10.1177/0013164411417620
Melguizo, T., Bos, J., & Prather, G. (2011). Is developmental education helping community
college students persist? A critical review of the literature. The American Behavioral
Scientist (Beverly Hills), 55(2), 173–184. https://doi.org/10.1177/0002764210381873
Merisotis, J. P., & Phipps, R. A. (2000). Remedial education in colleges and universities: What's
really going on? Review of Higher Education, 24(1), 67–85.
https://doi.org/10.1353/rhe.2000.0023
Museus, S. D., Palmer, R. T., Davis, R. J., & Maramba, D. C. (2011). Racial and ethnic minority
students' success in STEM education. ASHE Higher Education Report, 36(6), 1-140.
https://doi.org/10.1002/aehe.3606
National Assessment Governing Board. (2009). Design of content alignment studies in
mathematics and reading for 12th grade NAEP and other assessments to be used in
preparedness research studies.
https://www.nagb.org/content/nagb/assets/documents/publications/design-document-
final.pdf
National Center for Education Statistics, U. S. Department of Education. (2001). State
requirements for high school graduation, in Carnegie units: 2001.
https://nces.ed.gov/programs/digest/d01/dt153.asp
National Center for Education Statistics, U. S. Department of Education. (2014). High school
coursetaking. https://nces.ed.gov/programs/coe/indicator_cod.asp
90
National Center for Education Statistics, U. S. Department of Education. (2021). Undergraduate
retention and graduation rates. https://nces.ed.gov/programs/coe/indicator/ctr
National Center for Higher Education Management Systems. (2010). Student pipeline–Transition
and completion rates from 9th grade to college.
http://www.higheredinfo.org/dbrowser/index.php?submeasure=119&year=2010&level=n
ation&mode=data&state=0
National Center for Public Policy and Higher Education. (2008). Mixed signals in California: A
mismatch between high schools and community colleges.
https://vtechworks.lib.vt.edu/bitstream/handle/10919/83321/CaliforniaMismatchCommun
ityColleges.pdf?sequence=1&isAllowed=y
National Center for Public Policy and Higher Education. (2010). Beyond the Rhetoric: Improving
college readiness through coherent state policy.
https://vtechworks.lib.vt.edu/bitstream/handle/10919/83318/CollegeReadinessStatePolicy
.pdf?sequence=1&isAllowed=y
National Commission on Excellence in Education. (1983). A nation at risk: Imperative for
educational reform. https://edreform.com/wp-
content/uploads/2013/02/A_Nation_At_Risk_1983.pdf
Nomi, T. (2012). The unintended consequences of an Algebra-for-all policy on high-skill
students: Effects on instructional organization and students' academic outcomes.
Educational Evaluation and Policy Analysis, 34(4), 489–505.
https://doi.org/10.3102/0162373712453869
Norman, K. W., Medhanie, A. G., Harwell, M. R., Anderson, E., & Post, T. R. (2011). High
school mathematics curricula, university mathematics placement recommendations, and
91
student university mathematics performance. PRIMUS, 21(5), 434–455.
https://doi.org/10.1080/10511970903261902
Obama, B. (2009). Remarks of President Barack Obama--Address to joint session of congress.
https://obamawhitehouse.archives.gov/the-press-office/remarks-president-barack-obama-
address-joint-session-congress
Office of Hawaiian Affairs, Research Division. (2014). A native Hawaiian focus on post-
secondary education within the University of Hawaiʻi System. https://www.oha.org/wp-
content/uploads/FactSheet.A-Native-Hawaiian-Focus-on-Post-Secondary-
Education...2014.pdf
Oguntoyinbo, L. (2012). Math problem. Diverse Issues in Higher Education, 29(13), 18–19.
Organisation for Economic Co-operation and Development. (n.d.). Programme for international
student assessment (PISA) results from PISA 2012.
http://www.oecd.org/unitedstates/PISA-2012-results-US.pdf
Organisation for Economic Co-operation and Development. (2012). What are the returns on
higher education for individuals and countries? http://www.oecd.org/edu/skills-beyond-
school/Education%20Indicators%20in%20Focus%206%20June%202012.pdf
Organisation for Economic Co-operation and Development. (2021). Student performance in
maths (mean score). http://gpseducation.oecd.org
Oudenhoven, B. (2002). Remediation at the community college: Pressing issues, uncertain
solutions. New Directions for Community Colleges, 117, 35–44.
https://doi.org/10.1002/cc.51
92
Parsad, B., & Lewis, L. (2003). Remedial education at degree–granting postsecondary
institutions in fall 2000. U.S. Department of Education, National Center for Education
Statistics, http://nces.ed.gov/pubs2004/2004010.pdf
Plunk, A. D., Tate, W. F., Bierut, L. J., & Grucza, R. A. (2014). Intended and unintended effects
of state-mandated high school science and mathematics course graduation requirements
on educational attainment. Educational Researcher, 43(5), 230–241.
https://doi.org/10.3102/0013189X14540207
Polikoff, M. S. (2015). How well aligned are textbooks to the common core standards in
mathematics? American Educational Research Journal, 52(6), 1185–1211.
https://doi.org/10.3102/0002831215584435
Polikoff, M. S. & Fulmer, G. W. (2013). Refining methods for estimating critical values for an
alignment index. Journal of Research on Educational Effectiveness, 6(4), 380–395.
https://doi.org/10.1080/19345747.2012.755593
Polikoff, M. S., Gasparian, H., Korn, S., Gamboa, M., Porter, A. C., Smith, T., & Garet, M.S.
(2020). Flexibly using the surveys of enacted curriculum to study alignment. Educational
Measurement, Issues and Practice. 39(2), 38–47. https://doi.org/10.1111/emip.12292
Polikoff, M. S., Zhou, N., & Campbell, S. E. (2015). Methodological choices in the content
analysis of textbooks for measuring alignment with standards. Educational Measurement,
Issues and Practice, 34(3), 10–17. https://doi.org/10.1111/emip.12065
Porter, A. C. (2002). Measuring the content of instruction: Uses in research and practice.
Educational Researcher, 31(7), 3–14. https://doi.org/10.3102/0013189X031007003
93
Porter, A. C. (2004, June). Curriculum Assessment. Vanderbilt University.
https://www.yumpu.com/en/document/read/38404436/curriculum-assessment-2004-by-
andrew-porter-data-center
Porter, A. C. (2006). Measuring alignment. National Council on Measurement in Education
Newsletter, 14(4).
Porter, A. C., Floden, R., & Fuhrman, S. (1998). The effects of upgrading policies on high school
mathematics and science. Brookings Papers on Educational Policy, 1, 123–172.
https://www.jstor.org/stable/20067196
Porter, A. C., & Polikoff, M. S. (2012). Measuring academic readiness for college. Educational
Policy, 26(3), 394–417. https://doi.org/10.1177/0895904811400410
Porter, A. C., Polikoff, M. S., Zeidner, T., & Smithson, J. (2008). The quality of content analyses
of state student achievement tests and content standards. Educational Measurement:
Issues and Practice, 27(4), 2–14. https://doi.org/10.1111/j.1745-3992.2008.00134.x
Pukui, M. K. (1983). ‘Olelo no‘eau: Hawaiian proverbs and poetical sayings. Bishop Museum
Press.
Riley, R. W. (1995). The improving America's school act of 1994: Reauthorization of the
elementary and secondary education act. U.S. Department of Education.
https://books.google.com/books?id=d89ToGQZFQwC&pg=PA1&lpg=PA1&dq=The+im
proving+america%27s+school+act+of+1994:+Reauthorization+of+the+elementary+and+
secondary+education+act+riley+1995&source=bl&ots=saIKnBu60K&sig=ACfU3U0g9x
jA_aEiifP6tAmqJycBRffmaA&hl=en&sa=X&ved=2ahUKEwjXt9-
256b1AhVGUt8KHUftDjEQ6AF6BAggEAM#v=onepage&q=The%20improving%20a
94
merica's%20school%20act%20of%201994%3A%20Reauthorization%20of%20the%20el
ementary%20and%20secondary%20education%20act%20riley%201995&f=false
Romano, R. M. (2011). Remedial educational at community colleges in the state university of
New York. Community College Journal of Research and Practice, 35(12), 973–992.
https://doi.org/10.1080/10668920902718007
Sawyer, R. (2013). Beyond correlations: Usefulness of high school GPA and test scores in
making college admissions decisions. Applied Measurement in Education, 26(2), 89–112.
https://doi.org/10.1080/08957347.2013.765433
Schiller, K. S., & Muller, C. (2003). Raising the bar and equity? Effects of state high school
graduation requirements and accountability policies on students' mathematics course
taking. Educational Evaluation and Policy Analysis, 25(3), 299–318.
https://doi.org/10.3102/01623737025003299
Scott-Clayton, J. & Rodriguez, O. (2015). Development, discouragement, or diversion? New
evidence on the effects of college remediation policy, Education Finance and Policy,
10(1), 4–45. https://doi.org/10.1162/EDFP_a_00150
Skomsvold, P. (2014). Profile of undergraduate students: 2011–2012 (NCES 2015-167).
National Center for Education Statistics.
https://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2015167
Smith, M. S., & O'Day, J. (1991). Systemic school reform. In S. H. Fuhrman, & B. Malen (Eds.),
The politics of curriculum and testing: The 1990 yearbook of the politics of education
association (pp. 233–267). Taylor & Francis.
Smithson, J. (2009). Describing the academic content of PISA mathematics and science item
pools. PISA Conference.
95
https://edsurveys.rti.org/PISA/documents/SmithsonNCES_PISA_Conference_Paper_FIN
AL.pdf
State of Hawai‘i, Department of Labor and Industrial Relations. (2008). Annual evaluation of the
Hawaii unemployment compensation fund. https://labor.hawaii.gov/wp-
content/uploads/2012/12/UTF-Eval-FY2007-08.pdf
State of Hawai‘i, Department of Labor and Industrial Relations. (2010). Annual evaluation of the
Hawaii unemployment compensation fund. https://labor.hawaii.gov/wp-
content/uploads/2012/12/UTF-Eval-FY2009-10.pdf
Stein, M., Remillard, J., & Smith, M. (2007). How curriculum influences students’ learning. In
F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp.
557–628). Information Age.
Teitelbaum, P. (2003). The influence of high school graduation requirement policies in
mathematics and science on student course-taking patterns and achievement. Educational
Evaluation and Policy Analysis, 25(1), 31–57.
https://doi.org/10.3102/01623737025001031
U.S. Bureau of Labor Statistics. (2012). The recession of 2007–2009.
http://www.bls.gov/spotlight/2012/recession/pdf/recession_bls_spotlight.pdf
U.S. Census Bureau. (2012). Statistical abstract of the United States: 2012.
https://www2.census.gov/library/publications/2011/compendia/statab/131ed/2012-
statab.pdf
U.S. Department of Education. (n.d.). Every student succeeds act (ESSA).
https://www.ed.gov/essa?src=ft
96
U.S. Department of Education. (2009). Race to the top: Executive summary.
https://files.eric.ed.gov/fulltext/ED557422.pdf
U.S. Department of Education. (2020). Every student succeeds act (ESSA).
https://www2.ed.gov/policy/elsec/leg/essa/index.html
University of Hawai‘i Community Colleges. (2016). Gatekeeper
Success. https://uhcc.hawaii.edu/ovpcc/initiatives/achievingdream/gatekeeper
University of Hawai‘i, Institutional Research and Analysis Office. (2013). June 2012 Hawai`i
public high school graduates enrolled in English and mathematics classes at the
university of Hawai‘i in fall 2012. http://www.hawaii.edu/cgi-
bin/iro/maps?RDCCF12.pdf
University of Hawai‘i, Institutional Research and Analysis Office. (2015). Graduation and
persistence rates, fall cohorts. http://www.hawaii.edu/iro/srtk/srtkuh14.pdf
University of Hawai‘i, Institutional Research and Analysis Office. (2020). Student right-to-know.
http://hawaii.edu/iro/srtk.php?reportId=SRTK
University of Hawai‘i West O‘ahu. (2020). Math and English course placement.
https://westoahu.hawaii.edu/noeaucenter/wp-content/uploads/2020/07/2020-Math-And-
English-Course-PlacementREV3.pdf
Wainer, H. & Steinberg, L. S. (1992). Sex-differences in performance on the mathematics section
of the scholastic aptitude-test–-A bidirectional validity study. Harvard Educational
Review, 62(3), 323–336. https://doi.org/10.17763/haer.62.3.1p1555011301r133
Watkins, J., & Mazur, E. (2013). Retaining students in science, technology, engineering, and
mathematics (stem) majors. Journal of College Science Teaching, 42(5), 36–41.
http://www.jstor.org/stable/43631580
97
Webb, N. L. (1997). Criteria for alignment of expectations and assessments in mathematics and
science education. National Institute for Science Education and Council of Chief State
School Officers. https://files.eric.ed.gov/fulltext/ED414305.pdf
Webb, N. L. (2007). Issues related to judging the alignment of curriculum standards and
assessments. Applied Measurement in Education, 20(1), 7–25.
https://doi.org/10.1207/s15324818ame2001_2
Wisconsin Center for Education Research. (2009). Coding procedures for curriculum content
analyses. Center for Curriculum.
https://secure.wceruw.org/seconline/Reference/CodingProcedures2008.pdf
98
Appendix A: List of Topic Cells
Table A1
Codebook Topics
ID Topic
138 Add/subtract whole numbers and integers
139 Meaning of addition
140 Strategies and models for addition
141 Place value algorithm for addition
142 Standard algorithm for addition
143 Number facts for addition
144 Meaning of subtraction
145 Strategies and models for subtraction
146 Place value algorithm for subtraction
147 Standard algorithm for subtraction
148 Number facts for subtraction
149 Multiply whole numbers and integers
150 Meaning of multiplication
151 Strategies and models for multiplication
152 Place value algorithm for multiplication
153 Standard algorithm for multiplication
154 Number facts for multiplication
155 Divide whole numbers and integers
156 Meaning of division
157 Strategies and models for division
158 Place value algorithm for division
159 Standard algorithm for division
160 Number facts for division
161 Combinations of operations on whole numbers or integers (e.g., 14+25*7)
162 Order of operations
163 Equivalent and non-equivalent fractions
164 Add/subtract fractions
165 Multiply fractions
166 Divide fractions
167 Combinations of operations on fractions (e.g., ½* ¾ + ¼)
168 Ratio and proportion
169 Representations of fractions
170 Equivalence of decimals, fractions, and percents
171 Add/subtract decimals
172 Multiply decimals
99
ID Topic
173 Divide decimals
174 Combinations of operations on decimals
175 Computing with percents
176 Computing with exponents and radicals (e.g., square roots)
177 Absolute value
178 Place value
179 Whole numbers and integers
180 Fractions as a number
181 Decimals
182 Percents
183 Real and/or rational numbers
184 Exponents and scientific notation
185 Factors, multiples, and divisibility
186 Odd/even/prime/composite/square numbers
187 Estimation
188 Rounding
189 Comparisons of two or more whole numbers and/or integers
190 Number comparisons of fractions and/or decimals
191 Opposites, reciprocals, identities
192 Compose and decompose whole numbers
193 Compose and decompose fractions
194 Compose and decompose decimals
195 Relationships between operations (e.g., addition is inverse of subtraction)
196 Base-ten and non-base-ten systems
197 Mathematical properties (e.g., distr. property)
198 Use of measuring instruments
199 Measurement theory (e.g., arbitrariness of measures, standard units, unit size)
200 Conversions
201 Metric (SI) system
202 Length and perimeter
203 Area and volume
204 Surface area
205 Direction, location, and navigation
206 Angle measure
207 Circles (e.g.,
pi
, radius, and area)
208 Mass (weight)
209 Time
210 Temperature
211 Money
212 Derived measures (e.g., rate and speed)
213 Calendar
100
ID Topic
214 Accuracy and precision in measurement
215 Simple interest
216 Compound interest
217 Rates (e.g., discount and commission)
218 Spreadsheets
219 Personal financial literacy
220 Use of variables
221 Formulas, expressions, and equations
222 One-step equations (e.g., 5x=10)
223 Coordinate planes
224 Arithmetic or geometric patterns
225 Multi-step equations (e.g., 5x+2 = 12)
226 Inequalities
227 Linear and non-linear relations
228 Functions (e.g., meaning, functions as objects)
229 Operations on functions
230 Rate of change/slope
231 Polynomials (e.g., meaning, functions as objects)
232 Operations on polynomials
233 Factoring polynomials
234 Operations on radicals
235 Rational expressions
236 Completing the square
237 Quadratic formula
238 Functions to model data/phenomena
239 Computational algebra
240 Quadratic equations
241 Systems of equations
242 Systems of inequalities
243 Compound inequalities
244 Matrices and determinants
245 Conic sections
246 Rational, negative exponents, or radicals
247 Rules for exponents
248 Complex numbers
249 Binomial theorem
250 Factor/remainder theorem
251 Field properties of real number system
252 Multiple representations
253 Basic terminology
254 Precise definitions of geometric objects and properties
101
ID Topic
255 Logic, reasoning, and proof
256 Points, lines, rays, segments, and planes
257 Angles
258 Vectors
259 Rigid transformations (i.e. slides/translations, flips/reflections, turns/rotations)
260 Dilations
261 Defining congruence in terms of transformations
262 Congruence of triangles
263 Congruence of other figures
264 Defining similarity in terms of transformations
265 Similarity of triangles
266 Similarity of other figures
267 Parallel lines
268 Classifying polygons (triangles, quadrilaterals, etc.)
269 Triangles
270 Quadrilaterals
271 Other polygons
272 Pythagorean theorem
273 Right triangles
274 Circles (arc length and area)
275 Circles (chords, tangents, and secants)
276 3-D relationships
277 Classifying and describing 3-D figures
278 Structure of 3-D figures (e.g., nets, cross-section)
279 Polyhedra
280 Cylinders, cones, and spheres
281 Geometric constructions
282 Loci
283 Analytic or coordinate geometry
284 Symmetry
285 Geometric modeling
286 Geometric patterns (e.g., tessellations)
287 Non-Euclidian geometry
288 Topology
289 Data in a table or graph
290 Bar graphs and histograms
291 Pie charts and circle graphs
292 Pictographs
293 Line graphs
294 Stem and leaf plots
295 Scatter plots
102
ID Topic
296 Box plots/box and whisper plots
297 Line plots/dot plots
298 Classification and Venn diagrams
299 Tree diagrams
300 Mean, median, and mode
301 Variability, standard deviation, and range
302 Line of best fit
303 Quartiles and percentiles
304 Bivariate distribution
305 Confidence intervals
306 Correlation
307 Hypothesis testing
308 Data transformation
309 Central Limit Theorem
310 Statistical/Empirical questions
311 Experimental studies
312 Observational studies
313 Simple probability
314 Compound probability
315 Conditional probability
316 Empirical/experimental probability
317 Sampling and sample spaces
318 Independent vs. dependent events
319 Expected value
320 Binomial distribution
321 Normal curve
322 Simulations (Monte Carlo methods)
323 Sequences and series
324 Limits
325 Continuity
326 Rates of change
327 Maxima, minima, and range
328 Differentiation
329 Integration
330 Basic ratios
331 Radian measure
332 Right-triangle trigonometry
333 Law of Sines and Cosines
334 Identities
335 Trigonometric equations
336 Polar coordinates
103
ID Topic
337 Periodicity
338 Amplitude
339 Sets
340 Logic
341 Mathematical induction
342 Linear programming
343 Networks
344 Iteration and recursion
345 Combinatorics (permutations, combinations, fundamental counting principles)
346 Fractals
347 History of math
348 Notation
349 Relations
350 Linear
351 Quadratic
352 Polynomial
353 Rational
354 Logarithmic
355 Exponential
356 Trigonometric and circular
357 Inverse
358 Composition
359 Domain and range functions
360 Use of calculators
361 Use of graphing calculators
362 Use of computer and the internet
363 Computer programming
364 Use of spreadsheets
365 Use of software/programs/applications
104
Table A2
Level of Emphasis
Level of emphasis Code
Level of emphasis a
Values 0–10
Table A3
Cognitive Demand
Cognitive Demand Code
Memorize/recall b
Perform procedures c
Demonstrate/communicate understanding d
Justify/evaluate e
Generalize f
Apply to real-world problems g
105
Table A4
Extent of Cognitive Demand
Extent of cognitive demand Value
No focus 0
Minor focus 1
Moderate focus 2
Major focus 3
Table A5
Math Practice Codes and Details
Math Practices/Process (MATH ONLY) code value
For how many days during this log period did your
instruction provide opportunities for students to …
a 0-10
For how many days during this log period did your
instruction provide opportunities for students to …
b text
106
Table A6
Math Practices and Processes
Process Code
Select and use tools and techniques strategically mpp101
Attend to precision mpp102
Problem solve mpp103
Use mathematics to model real-world situations mpp104
Reason mathematically mpp105
Prove, justify and critique mpp106
Represent and make connections within mathematics mpp107
Persevere in mathematics mpp108
107
Appendix B: List of Cognitive Demand
Memorize/Recall: Level B
• Recite basic math facts
• Recall mathematics terms and definitions
• Recall formulas and computational procedures
Perform Procedures : Level C
• Use numbers to count, order, or denote
• Do computation procedures or algorithms
• Follow the steps in mathematical procedure or apply a formula
• Solve equations and routine word problems
• Follow procedures to organize or display data
• Read or produce graphs and tables
• Execute geometric constructions
Demonstrate/Communicate Understanding: Level D
• Communicate understanding of mathematical concepts
• Use representations to model mathematical concepts, relationships, and/or operations
• Explain findings and results from data analysis
• Develop and explain relationships between concepts and/or operations
• Show or explain relationships between models, diagrams, and/or other representations
Justify/Evaluate: Level E
• Determine the truth of a mathematical proposition
• Justify a mathematical solution, conclusion, or claim
• Write formal or informal proofs
• Identify faulty arguments or misrepresentations of data
• Make and investigate mathematical conjectures
• Reason inductively or deductively
• Determine the reasonableness of mathematical solutions (e.g. by using mental strategies
or estimation)
Generalize: Level F
• Recognize, generate, or create patterns
• Find a mathematical rule to generate a pattern or number sequence
• Apply and extend mathematical properties to new contexts (e.g. extend understanding of
whole-number addition to decimal addition; extend understanding of the distributive
property in whole-number multiplication to multiplication of two binomials)
Apply to Real-World Problems: Level G
• Apply mathematics to solve non-routine, real-world problems
• Apply mathematics in contexts outside of mathematics
Non-Specific Cognitive Demand: Level Z
Abstract (if available)
Abstract
Despite A Nation at Risk being almost 40 years old with the claims that American schools were failing and had declining educational standards (National Commission on Excellence in Education, 1983) the spotlight on failing schools and low educational standards has not gone away. This study aims to uncover if mathematics content and cognitive demand are consistent from high school to college. About a third of first- and second-year college undergraduates reported taking a remedial course in the 2011-2012 year (Skomsvold, 2014). Mathematics was the remedial course most enrolled in. A content analysis was conducted using the Surveys of Enacted Curriculum, to determine the level of consistency between the Common Core State Standards for high school and an introductory college-level (non-STEM) mathematics textbook. The analysis was to determine which topics and cognitive demand in the book or standards had a high degree of alignment or low degree of alignment (misalignment). The results showed an overall lower level of alignment than the Polikoff (2015) study. The top 20 over emphasized topics made up about half of the book’s content while the standards were 8%. Regarding cognitive demand, 18 of 20 of the top 20 over emphasized topics focused on procedures that were consistent with Polikoff (2015). The top 20 under emphasized topics were in 2% of the textbook and 23% of the standards. Regarding cognitive demand, 14 of 20 of the top 20 under emphasized topics were focused on procedures. The textbook showed poor alignment to the standards in content and cognitive demand. The study suggests that the textbook was not evaluating new content, it was reteaching many high school concepts. While teachers rely heavily on textbooks to inform curriculum (Chingos & Whitehurst, 2012), further research could include studying the alignment of instruction, other curriculum materials, and assessments to include student outcomes and textbook comparisons.
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Reyes Oda, Julie Ku'uleialoha
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Core Title
A study of alignment between the Common Core State Standards in math and an introductory college textbook
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Rossier School of Education
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Doctor of Education
Degree Program
Education (Leadership)
Degree Conferral Date
2022-05
Publication Date
02/02/2022
Defense Date
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), Picus, Lawrence O. (
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