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An examination of teacher-centered Explicit Direct Instruction and student-centered Cognitively Guided Instruction in the context of Common Core State Standards Mathematics…

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An examination of teacher-centered Explicit Direct Instruction and student-centered Cognitively Guided Instruction in the context of Common Core State Standards Mathematics…

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Content i

AN EXAMINATION OF TEACHER-CENTERED EXPLICIT DIRECT INSTRUCTION AND
STUDENT-CENTERED COGNITIVELY GUIDED INSTRUCTION IN THE CONTEXT OF
COMMON CORE STATE STANDARDS MATHEMATICS COLLEGE READINESS: A
DOCUMENT ANALYSIS

by

Phoebe Moore

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 2021

Copyright 2021 Phoebe Moore

ii
Dedication
I would like to dedicate the achievement of this milestone to my family. My endeavor to
complete a project such as this was possible because of the love, encouragement, support, and
many sacrifices you have given me. I love you all.

iii
Acknowledgements
I would like to express my deepest gratitude to the dissertation committee, Dr. Sandra
Kaplan, Dr. Raymond Gallagher, and Dr. Silvia Ybarra. The detailed feedback at every step in
my dissertation process has been invaluable and provided a volume of additional learning for me.
I truly appreciate the time the committee has spent working with me, and thank you so much for
pushing me to do my best work. Undeniably, the committee's timely encouragement and support
turned what seemed impossible into reality.

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Table of Contents
Dedication ....................................................................................................................................... ii
Acknowledgements ........................................................................................................................ iii
List of Tables ................................................................................................................................. vii
List of Figures ................................................................................................................................ ix
List of Abbreviations ...................................................................................................................... xi
Abstract ......................................................................................................................................... xii
Chapter One: Overview of the Study .............................................................................................. 1
Background of the Problem ....................................................................................................... 2
High School Graduation and College Enrollment Data ...................................................... 2
Pre-College Mathematics Proficiency Achievement Data .................................................. 3
Various Reports ................................................................................................................... 3
Post-Secondary Remediation .............................................................................................. 5
Statement of the Problem .......................................................................................................... 6
Teachers' Lack of Understanding of the SMPs ................................................................... 7
Teachers’ Lack of Understanding on How to Integrate CCSSM-SMPs ........................... 11
Dichotomous Pedagogical Beliefs .................................................................................... 13
Purpose of the Study ............................................................................................................... 17
Research Questions ................................................................................................................. 18
Statement of Methodology ...................................................................................................... 18
Theoretical/Conceptual Framework ........................................................................................ 19
Significance of the Study ........................................................................................................ 19
Definition of Terms ................................................................................................................. 20
Organization of the Study ....................................................................................................... 21
Chapter Two: Review of Literature .............................................................................................. 23
Literature Background on the History of Mathematics ........................................................... 24
Late 1800s - 1930: The Era of Mathematic Debate .......................................................... 25
1940s - 1980s: Crisis, Reform, & Reaction in Mathematics ............................................. 27
The 1990s - No Child Left Behind .................................................................................... 30
No Child Left Behind ........................................................................................................ 31
Mathematics History in Summary ..................................................................................... 31
Literature Background on the Study Variables ....................................................................... 32
Brief Overview of the Common Core State Standards Mathematics ................................ 32
Standards for Mathematical Practice (SMPs) College Readiness Definition ................... 33
Two Selected Pedagogies: EDI and CGI .......................................................................... 44
Theoretical Framework ........................................................................................................... 63
Impacting Teachers’ Actions Through Knowledge, Beliefs, and Goals ........................... 64
Interconnectedness of Knowledge, Beliefs, and Goals ..................................................... 68
Operationalizing Teachers’ Actions Through Structured Knowledge .............................. 72
Summary ................................................................................................................................. 73
Chapter Three: Methodology ........................................................................................................ 74
Overview ................................................................................................................................. 74
Restatement of the Problem .................................................................................................... 75
Restatement of the Purpose ..................................................................................................... 75
Research Question ................................................................................................................... 76
v
Research Methodology ............................................................................................................ 76
Sample Selection ..................................................................................................................... 78
Data Collection and Analysis .................................................................................................. 80
Building a Content Analysis Framework using Mayring’s Four-step Model ................... 80
Summary ................................................................................................................................. 90
Chapter Four: Results and Findings .............................................................................................. 91
Statement of the Problem ........................................................................................................ 91
Research Question ................................................................................................................... 94
Findings: Steps 1 & 2 .............................................................................................................. 94
Steps 1 & 2: Selection of Materials and Descriptive Analysis ......................................... 94
Findings (Steps 1 & 2): Analytic Variable 1 (AV 1) ........................................................ 95
Findings (Steps 1 & 2): Analytic Variables 2 and 3 (AV 2 and AV 3) ............................ 95
Linking Variables (Selected through Research): College Readiness Definitions ............. 96
Findings (Steps 1 & 2): Linking Variable 1 (LV 1) .......................................................... 96
Findings: Step 3 ..................................................................................................................... 103
Categorizing and Connecting Strategy ............................................................................ 103
Findings (Step 3): Analytic Category 1 – Problem Solving ............................................ 105
Findings (Step 3): Analytic Category 2 – Reasoning ...................................................... 108
Findings (Step 3): Analytic Category 3 – Argumentation .............................................. 110
Findings (Step 3): Analytic Category 4 – Modeling ....................................................... 112
Findings (Step 3): Analytic Category 5 – Tools ............................................................. 114
Findings (Step 3): Analytic Category 6 – Precision ........................................................ 116
Findings (Step 3): Analytic Category 7 – Structure ........................................................ 118
Findings (Step 3): Analytic Category 8 – Regularity ...................................................... 120
Findings: Step 4 ..................................................................................................................... 122
Findings (Step 4): Analytic Category 1 – Problem Solving ............................................ 123
Findings (Step 4): Analytic Category 2 – Reasoning ...................................................... 126
Findings (Step 4): Analytic Category 3 – Argumentation .............................................. 129
Findings (Step 4): Analytic Category 4 – Modeling ....................................................... 131
Findings (Step 4): Analytic Category 5 – Tools ............................................................. 133
Findings (Step 4): Analytic Category 6 – Precision ........................................................ 135
Findings (Step 4): Analytic Category 7 – Structure ........................................................ 137
Findings (Step 4): Analytic Category 8 – Regularity ...................................................... 139
Chapter Five: Discussion ............................................................................................................ 141
Statement of the Problem ...................................................................................................... 141
Summary of the Study ........................................................................................................... 142
Sample Selection ................................................................................................................... 143
Data Analysis ........................................................................................................................ 144
Validity and Reliability ......................................................................................................... 145
Research Question ................................................................................................................. 146
Discussion of Findings .......................................................................................................... 146
Key Findings: SMP 1 ...................................................................................................... 146
Key Findings: SMP 2 ...................................................................................................... 147
Key Findings: SMP 3 ...................................................................................................... 148
Key Findings: SMP 4 ...................................................................................................... 149
Key Findings: SMP 5 ...................................................................................................... 150
vi
Key Findings: SMP 6 ...................................................................................................... 150
Key Findings: SMP 7 ...................................................................................................... 151
Key Findings: SMP 8 ...................................................................................................... 152
Implications ........................................................................................................................... 153
Theoretical Implications .................................................................................................. 153
Practical Implications ...................................................................................................... 155
Future Implications and Recommendations .................................................................... 157
Limitations ............................................................................................................................ 159
Conclusion ............................................................................................................................. 160
References ................................................................................................................................... 162

vii
List of Tables
Table 1. RAND Corporation Study: Teachers’ Understanding of CCSSM 9
Table 2. EdInsights Study: Teachers’ Concern and Confusion around CCSSM-SMPs 12
Table 3. Differences between SMP and CCSS-M 34
Table 4. The Common Core State Standards for Mathematical Practice 35
Table 5. Mathematical Reasoning and Communication Rubric 39
Table 6. EDI Lesson Design 47
Table 7. EDI Lesson Delivery Strategies 49
Table 8. Levels of CGI Instruction 53
Table 9. CGI: Framework Components 54
Table 10. CGI: Classroom Practices 55
Table 11. CGI: Getting Started 56
Table 12. CGI: Components of CGI Mathematics Instruction 57
Table 13. Single Digit – Types of Problems 58
Table 14. Direct Modeling – Addition and Subtraction 58
Table 15. Counting 59
Table 16. Direct Modeling – Multiplication and Division 60
Table 17. Multidigit Problem-Solving Strategies 61
Table 18. Invented Algorithms 63
Table 19. Mayring’s Four-Step Model 78
Table 20. Revised Bloom’s Taxonomy: The Knowledge Dimension 97
Table 21. Revised Bloom’s Taxonomy: The Cognitive Process Dimension 98
Table 22. Revised Bloom’s Taxonomy: Knowledge Dimension Structure 101
viii
Table 23. RBT – Cognitive Process Dimension 102
Table 24. Analytic Categories 104

ix
List of Figures
Figure 1. College Readiness Gap 5
Figure 2. Implications of college mathematics remedial courses. 6
Figure 3. Rand Corp. Study: Percentage of Teachers Engaging Students in CCSSM/SMPs 11
Figure 4. Education Timeline: Constructivism and Progressivism in Mathematics 32
Figure 5. Schoenfeld’s Five Phase Model of Problem-Solving 36
Figure 6. Open-Ended Questions to Promote Problem Solving 37
Figure 7. Using Tools Appropriately 41
Figure 8. Tips for Attending to Precision 42
Figure 9. Reasons Student are Not Successful in the Classroom 45
Figure 10. Instructional Approach Guidelines 46
Figure 11. Teacher Behaviors that Characterize Well-Structured Lessons 46
Figure 12. EDI Circle 50
Figure 13. Knowledge, Beliefs, and Practices 52
Figure 14. Strategies for Solving Multidigit Problems 62
Figure 15. Model of Schoenfeld’s Theory of Teaching-in-Context 65
Figure 16. Findings: Analytic Steps 1 & 2 84
Figure 17. Findings: Analytic Steps 3 & 4a 86
Figure 18. Findings: Analytic Steps 4b-4d 88
Figure 19. Document Analysis Flow Chart 90
Figure 20. Analytic and Linking Variables 94
Figure 21. Alignment of CCSSM and Two Dimensions of RBT. 100
Figure 22. Definition Alignment – Analytic Category 1: Problem Solving 105
x
Figure 23. Definition Alignment – Analytic Category 2: Reasoning 108
Figure 24. Definition Alignment – Analytic Category 3: Argumentation 110
Figure 25. Definition Alignment – Analytic Category 4: Modeling 112
Figure 26. Definition Alignment – Analytic Category 5: Tools 114
Figure 27. Definition Alignment – Analytic Category 6: Precision 116
Figure 28. Definition Alignment – Analytic Category 7: Structure 118
Figure 29. Definition Alignment – Analytic Category 8: Regularity 120
Figure 30. EDI and CGI Comparison –Analytic Category 1: Problem Solving 124
Figure 31. EDI and CGI Comparison –Analytic Category 2: Reasoning 126
Figure 32. EDI and CGI Comparison Analytic Category 3: Argumentation 129
Figure 33. EDI and CGI Comparison Analytic Category 4: Modeling 131
Figure 34. EDI and CGI Comparison Analytic Category 5: Tools 133
Figure 35. EDI and CGI Comparison Analytic Category 6: Precision 135
Figure 36. EDI and CGI Comparison Analytic Category 7: Structure 137
Figure 37. EDI and CGI Comparison Analytic Category 8: Regularity 139

xi
List of Abbreviations
CCR College and Career Readiness
CCSS Common Core State Standards
CCSSI Common Core State Standards Initiative
CCSSM Common Core State Standards Mathematics
CCSSO Council of Chief State School Officers
CGI Cognitively Guided Instruction
EDI Explicit Direct Instruction
NGA National Governor’s Association
PD Professional Development
RBT Revised Bloom’s Taxonomy
SMPs Standards for Mathematical Practice

xii

Abstract
An Examination of Teacher-Centered Explicit Direct Instruction and Student-Centered
Cognitively Guided Instruction in the Context of Common Core State Standards Mathematics
College Readiness: A Document Analysis
Phoebe Moore

Despite many reform efforts in K-12 education, the declining trend among high school
students’ achievement in Mathematics and the lack of student success in post-secondary
mathematics courses indicate a disconnect and misalignment in the mathematics instructions
students receive in K-12 institutions. The mathematics instruction students receive in K-12
institutions fails to develop students' critical knowledge and skills to succeed in postsecondary
mathematics courses (Callan et al., 2006; Cohen, 2008; Woods et al., 2018). This qualitative
document analysis seeks to examine two pedagogical approaches: teacher-centered, Explicit
Direct Instruction (EDI), and student-centered Cognitively Guided Instruction (CGI), to identify
whether the two pedagogies support the college readiness skills represented in the Standards for
Mathematical Practice (SMPs). The research question will reveal how the two pedagogies
support the SMPs’ college readiness in mathematics in the context of Common Core State
Standards Mathematics. Theoretically, this study is viewed through the lens of Schoenfeld’s
(1998) Theory of Teaching-in-Context. Data will be collected and analyzed within a systematic
content analysis of the documents related to college readiness in mathematics using Mayring’s
(2000) four-step model. The model will provide a structured process to determine how the two
selected pedagogies, EDI and CGI support the college readiness in SMPs. The structured process
xiii
provided a way to analyze the study documents, including preselected documents related to the
SMPs, EDI, and CGI. The research identified additional documents critical to the document
analysis, such as Revised Bloom’s Taxonomy and various documents related to the definition of
college readiness in mathematics. The study’s findings revealed that the two selected pedagogies,
EDI and CGI, support the SMP’s expectations for college readiness in mathematics in the
context of CCSSM.

TEACHER UNDERSTANDING OF TEXT COMPLEXITY

1
Chapter One: Overview of the Study
College readiness in mathematics has become a growing concern for all levels of
stakeholders in education, including education policymakers, practitioners, and researchers in the
K-12 and higher education landscape (Callan et al., 2006; Jackson & Kurlaender, 2013; Tierney
& Sablan, 2014). The Common Core State Standards-Mathematics (CCSSM) were created to
address this concern and “to ensure that all students graduate from high school with the skills and
knowledge necessary to succeed in college, career, and life…” (CCSSI, 2010, para. 2). The
CCSSM comprises both content standards and the Standards for Mathematical Practice (SMPs).
The CCSSM content standards address what mathematics knowledge and skills teachers need to
teach and students need to know and master at each grade level. The SMPs outline the cognitive
process competencies of how students should engage in critical thinking, problem-solving, and
communication when learning mathematics as they progress through their K-12 education
(California Department of Education, 2014). The two standards “support instruction with an
equal focus on developing students’ ability to engage in the practice standards and on developing
conceptual understanding of and procedural fluency in the content standards” (Yakes & Sprague,
2015, p. 2). The CCSSM content standards, the knowledge component, have shifted mathematics
instruction away from rote memorization to instruction that focuses on a conceptual
understanding of mathematics. The SMPs, the cognitive process component, require the
implementation of high-level cognitive process verbs, such as apply, analyze, evaluate,
synthesize, and create new knowledge and understandings with the knowledge they have
acquired (CCSSI, 2010; Conley et al., 2010). However, after one decade of CCSSM
implementation, a great deal of time and resources have been spent addressing the content
standards through curricular development and professional development. Conversely the SMPs

2
have not been addressed with the same intensity making the shift in teachers’ instruction so that
there is also a focus on the SMPs has not happened at the pace expected (Delvin, 2014; Seago &
Carroll, 2018; Walder, 2020).
This study examined the issues related to integrating college readiness cognitive process
skills required by the SMPs into teacher- and student-centered pedagogical approaches. The
abbreviation SMPs will be used in this study to represent college readiness in Mathematics, as
required in the Standards of Mathematical Practice within the context of the Common Core State
Standards in Mathematics.
Background of the Problem
The concerning issue regarding K-12 mathematics instruction leading the discourse
among educators and researchers is that student achievement in mathematics remains
consistently low, and students graduating from K-12 institutions are not prepared for
postsecondary education (Butrymowicz, 2020). These findings are evidenced by a declining
trend in the mathematics achievement data of college-bound seniors (American College Testing
[ACT], 2018; Butrymowicz, 2020; National Assessment of Educational Progress [NAEP], 2020;
College Board Scholastic Aptitude Test [SAT], 2018).
High School Graduation and College Enrollment Data
The persistence of the past reform efforts toward college readiness in mathematics has
resulted in increasing the number of students who graduate from high school (The Condition of
Education, 2019). According to The Condition of Education 2019, the most recent national data
on high school students graduating on time increased from 73.4% in 2005 to 85% in 2017-2018.
Conversely, the number of high school graduates enrolling in degree-granting postsecondary
institutions did not follow the same pattern (NAEP, 2010-2018). Specifically, undergraduate

3
enrollment in the nation’s colleges and universities decreased by 8% between 2010 and 2018
(from 18.1 million to 16.6 million students), leaving us with the overall college enrollment rate
for 18-to 24-year-old at 41% (NAEP, 2010-2018).
Pre-College Mathematics Proficiency Achievement Data
Based on the performance report by NAEP, 12
th
-grade students achieving at or above the
NAEP proficiency level in mathematics remained little changed from 2005 at 23% to 2015 at
25% (NAEP, 2015). Also, mathematics proficiency scores from the Scholastic Aptitude Test
(SAT) and American College Testing (ACT) among college-bound seniors have been stagnant at
the below college readiness benchmark for decades (National ACT, 2019; College Board SAT,
2018). Likewise, the ACT (2019) college readiness mathematics benchmarks for high school
graduates from 2015 to 2019 have dropped from 42% to 39% (National ACT, 2019). The
Washington Post wrote, “ACT scores for the high school Class of 2019 show that rates of college
readiness in…math have sunk to record lows, testing officials reported” (Anderson, 2019, para
1). Overall, based on the ACT 2018 report, the level of college readiness in Mathematics of U.S.
College-Bound Seniors have steadily declined since 2014 and leaving us with only 26% of ACT-
tested 2018 graduates likely to have the foundational work readiness skills needed for more than
nine out of 10 jobs recently profiled in the ACT JobPro database (National ACT, 2018).
Various Reports
Various reports, documents, and initiatives have shown for decades that high school
students are graduating without the knowledge, skills, and cognitive thinking ability to tackle
college-level mathematics. The various landmark reports represented these, documents, and
initiatives through An Agenda for Action, A Nation at Risk, Curriculum and Evaluation
Standards for School Mathematics, Principles and Standards for School Mathematics,

4
the Common Core State Standards for Mathematics, and Principles to Actions (National
Commission on Excellence in Education, 1983; National Council of Teachers of Mathematics
[NCTM], 2000; 2014; National Governors Association and Council of Chief State School
Officers, 2010). In the Catalyzing Change in High School Mathematics report (2018), the trend
in high school mathematics achievement has been flat. Although many researchers and studies
have worked to solve the college readiness gap, the readiness for college among today’s high
school students remains an issue. Based on a special report by The National Center for Public
Policy and Higher Education and the Southern Regional Education Board (2010),
while access to college remains a major challenge, states have been much more
successful in getting students into college than in providing them with the knowledge and
skills needed to complete certificates or degrees. Increasingly, it appears that states or
postsecondary institutions may be enrolling students under false pretenses. Even those
students who have done everything they were told to do to prepare for college find, often
after they arrive, that their new institution has deemed them unprepared. Their high
school diploma, college preparatory curriculum, and high school exit examination scores
did not ensure college readiness. (p.1)
See Figure 1 for the college readiness gap in three types of postsecondary institutions: highly
selective four-year institutions, somewhat selective four-year institutions, and nonselective or
open-access two-year colleges (SREB, 2010). The readiness gap is insignificant in the most
selective universities due to their screening of most underprepared students through its
admissions criteria and process. Conversely, the gap is huge in the other two sectors of higher
education. Furthermore, it is in the two sectors which serve between 80 to 90 percent of
undergraduates in public institutions.

5
Figure 1
College Readiness Gap
Public Postsecondary
Enrollments
Highly selective institutions require high school diploma + college-prep
curriculum + high grade-point average + high scores + extras
Readiness
Gap
10%
Less selective institutions require high school
diploma + college-prep curriculum + usually a
combination of grade-point average and/or test
scores (but lower than most selective institutions)
Readiness Gap
30%
Nonselective (open access)
institutions require a high
school diploma
Readiness Gap
60%
Note. Adapted from Beyond the Rhetoric: Improving College Readiness Through Coherent State Policy [Report, NCPPHE &
SREB], 2010, Online Research Access Service (https://edinsightscenter.org). p.2. Copyright 2010 by the author.
Post-Secondary Remediation
To address students' lack of preparedness when they exit the K-12 system, postsecondary
institutions offer non-college credit remedial mathematics (Alliance for Excellent Education,
2011; Ganem, 2009; Grubb, 2013). While remedial math courses support some students, it does
not help all. Close to 30% of students testing into remedial courses do not enroll in
postsecondary institutions. Less than half of the students who enroll in remedial courses persist
in completing the sequence (Bailey et al., 2010; Grubb, 2013). Furthermore, the graduation rate
for the students who begin with remedial courses is lower for both students completing
community college in three years and undergraduate degrees in six years (Complete College
America, 2012). Accordingly, students who enroll in remedial mathematics courses are more
likely to drop out of higher education institutions than those who do not take remedial courses
(Mireles et al., 2011; Rothman, 2012). See Figure 2.

6
Figure 2
Implications of college mathematics remedial courses
Implications of College Mathematics Remedial Courses
Of the Students Who Test into College
Remedial Mathematics Courses:
30% do not enroll in postsecondary institutions.
Of the Students Who Enroll in College
Remedial Mathematics Courses:
Less than half of the students persist in completing the sequence.
Graduation Rate of the Students Who Begin
College with Remedial Mathematics Courses:
Students who begin with remedial courses is lower for both students
completing community college in three years and undergraduate
degrees in six years.
Drop Out Rate of the Students Who Begin
College with Remedial Mathematics Courses:
Students are more likely to drop out of higher education institutions
than those who do not take remedial courses.

Statement of the Problem
Despite K-12 reform efforts, the declining trend among high school students'
achievement in mathematics is concerning. Additionally, the persistent drop in student
proficiency data implies that there is a disconnect and misalignment between K-12 instruction in
mathematics and college readiness in mathematics. The mathematics instruction students receive
in K-12 institutions fails to develop students' critical knowledge and skills to succeed in
postsecondary mathematics courses (Callan et al., 2006; Cohen, 2008; Woods et al., 2018). A
common assumption is that high school students earning a diploma are ready for college-level
coursework; however, the data suggest otherwise (Grubb, 2013). Addressing the misalignment
found in the essential skills and knowledge needed in college and the skills taught in high
schools is critical as numerous investigative studies have highlighted how the quality of
mathematics preparedness for high school students impact success at college-level mathematics
(Adelman, 1999; Boaler, 1997; Callan et al., 2006; Cohen, 2008; Grubb, 2013; Grubb &
Worthen, 1999).
Ganem (2009) states, mathematics instruction is in the "midst of a paradox" as the
number of students unprepared for college mathematics is on the rise, and the cause is due to the

7
"disconnect between the high school and college math curricula" (Ganem, 2009, para. 3). Ganem
(2009) states, "math involves knowledge and understanding of symbolic representations for
abstract concepts, and it is extremely difficult to short cut development" (para., 16). Ganem
(2009) suggests the disconnect between high school and college math is rooted in three areas:
1. Teachers confuse difficulty with rigor by attempting difficult problems without the
proper foundation, which impedes the rigor of independent thinking and problem solving
(para. 10);
2. Teachers mistake the conceptual understanding of the mathematics procedure with
memorization and computational prowess, which is the opposite of college mathematics
that requires an understanding (para. 15); and
3. Teachers focus on mathematical concepts that are not developmentally appropriate.
This study addresses the misalignment and confusion that prevents teachers from
integrating the SMPs into their pedagogical practice. Three contributing factors to the
misalignment and confusion were identified: (a) teachers' lack the knowledge and understanding
of the components and alignment of the SMPs, (b) teachers' lack the knowledge and
understanding on how to integrate the SMPs into the pedagogical process, and (c) persistent
dichotomous pedagogical beliefs regarding effective mathematics instruction (Freedberg, 2016;
Mueller & Gozali-Lee, 2013).
Teachers' Lack of Understanding of the SMPs
There is a misalignment across K-12 and college institutions about how to engage
students in particular college readiness mathematics thinking and selecting mathematics topics
that are essential for student success in postsecondary coursework (Melguizo & Ngo, 2020). Two
factors play a role in why this is happening. First, teachers' lack of understanding of the SMPs
has impacted moving students toward college readiness cognitive process skills (Opfer et al.,
2018). Secondly, teachers lack clarity in defining college readiness cognitive skills vital to
students' success in taking college-level mathematics courses, and what they assume in practice

8
do not match what is expected (Lewis et al., 2016; Rebora, 2013; Rothman, 2012; Schmidt &
Burroughs, 2012). An observational study conducted in thirteen community colleges in
California confirmed the misalignment in K-12 mathematics instruction. The evidence was based
on the fact that the college mathematics instruction was dominated by remedial level
mathematics content. To add to this, the instructors could not move the students to college-level
mathematics content (Grubb, 2013; Grubb & Worthen, 1999). The lack of student preparation for
college mathematics demonstrates a lack of teacher preparedness to teach college-ready
mathematics. In another study conducted with over 12,000 teachers, a survey confirmed a lack of
teacher preparation in mathematics. Less than 50% of the elementary teachers, 60% of the
middle school teachers, and 70% of the high school mathematics teachers did not believe they
were well prepared to teach the math topics in CCSSM (Schmidt & Burroughs, 2012). The study
also indicated that mathematics teachers were not clear regarding the expectations of CCSSM
grade-level content standards, SMPs, and college readiness aligned mathematics instruction
(Rebora, 2013; Rothman, 2012; Schmidt & Burroughs, 2012).
While this study's focus is on the SMPs, it is essential to note that there is a larger issue in
mathematics instruction, that is, teachers are challenged with both the CCSSM content standards
and the SMPs. The study will bring the matter to the forefront; however, there will not be an in-
depth analysis of the content standards. Despite a great deal of time and resources spent
addressing the CCSSM content standards through curricular and professional development,
teachers remain unclear about the grade-level content standards. A study conducted by the
RAND Corporation in 2018 provides insight regarding how teachers are challenged with both
the CCSSM content standards and the SMPs.

9
The RAND study on the CCSSM Content Standards
In a study conducted by the RAND Corporation (2018), teachers across the United States
participated in a survey to collect data on teachers' understanding of the CCSSM standards and
the standards-aligned practices they use in mathematics instruction. The survey evoked responses
focusing on various topics related to CCSSM and standards-aligned practices including the
percentage of teachers who can identify the math standards they teach at their specific grade
level (Table 1), and to what extent are teachers engaging their students in standards-aligned
Mathematical Practices (Figure 3). The rationale behind the focus of the survey was rooted in the
fact that "if teachers are to teach the standards at the depth that is intended, they need to
understand which standards are to be taught at their grade level and which are not" (Opfer et al.,
2018, p. 11). Knowledge of which standards to teach is not enough, as teachers also need a deep
understanding of the standards "to engage their students in [rigorous] practices aligned with
those standards" (Opfer et al., 2018, p. 3).
Table 1
RAND Corporation Study: Teachers’ Understanding of CCSSM
Grade
Percentage of Teachers who Identified
Common Core Aligned Topics
Percentage of Teachers who respond with,
“I Don’t Know” the Grade Level Topics
Kindergarten 65.1 3.0
First 11.9 4.0
Second 47.8 3.0
Third 10.2 3.0
Fourth 46.4 9.0
Fifth 15.1 10.0
Sixth 11.6 9.0
Seventh 20.7 10.0
Eighth 4.1 14.0
Algebra 2.4 12.0
Geometry 17.8 8.0
Note. Source: Opfer et al., 2018, p.3.
Table 1 shows the percentage of mathematics teachers identifying CCSSM standards
versus distractor standards for the grade levels they teach. In the survey, teachers were given two
grade-level standards, and two distractor standards-distractor standards are not grade-level

10
standards. In the lower grade levels, kindergarten, second, and fourth grades, the teachers
demonstrated a higher knowledge of their grade-level standards by identifying the two grade-
level standards and none of the distractor standards. In contrast, in the higher grade levels,
seventh, eighth, Algebra, and Geometry, the teachers demonstrated a much lower knowledge of
their grade-level standards by choosing distractor standards instead of their grade-level
standards. One possible reason the upper-grade teachers were not able to distinguish grade-level
standards from distractor standards is they “conflate the need for understanding of prior
standards with the specific standards to be taught at their grade level or in their specific
mathematics subject” (Opfer et al., 2018, p.11).
The Rand study on the SMPs.
Figure 3 shows 30% to 53% of the teachers surveyed reported engaging students in six
CCSSM/SMPs. However, less than 20% of the teachers reported engaging students in two of the
CCSSM/SMPs— SMP 7 - looking for and making use of structure, and SMP 3 - constructing
viable arguments (Opfer et al., 2018).

11
Figure 3
Rand Corporation Study: Percentage of Teachers Engaging Students in CCSSM/SMPs

Note. Rand Corporation Study: Percentage of Teachers Engaging Students in CCSSM/SMPs. Adapted from “Aligned Curricula
and Implementation of Common Core State Mathematics Standards,” by Opfer et al., 2018, p.17. Copyright 2018 by the author.
Teachers’ Lack of Understanding on How to Integrate CCSSM-SMPs
Another issue is the confusion that exists within K-12 institutions about how to teach the
SMPs effectively. Although the CCSSM content standards and SMPs outline what teachers need
to understand about college readiness in mathematics conceptually, it remains unclear how
teachers operationalize the conceptual understanding of college readiness components
(Letwinsky & Cavender, 2018; Saracusa & Willingham, 2010). Moreover, the CCSSM and
SMPs were developed to work in tandem, as highlighted by the Conference Board of
12%
21%
41%
38%
42%
37%
42%
53%
14%
20%
30%
36%
40%
49%
46%
50%
SMP 3 - Construct viable arguments and critique
the reasoning of others
SMP 7 - Look for and make use of structure (e.g.,
patterns in numbers, shapes, or algorithms)
SMP 4 - Apply mathematics to solve problems in
real-world contexts
SMP 5 - Choose and use appropriate tools (e.g.,
pencil and paper, concrete models, a ruler,…
SMp 8 - Use repeated practice to improve their
computational skills
SMP 6 - Explain and justify their work
SMP 1 - Make sense of problems and persevere
in solving them
SMP 2 - Use mathematical language and symbols
appropriately when communicating about…
Percentage of Teachers Engaging Students in Standards-Aligned
Mathematics Practices to a Great Extent
Elementary teachers Secondary teachers
SMP 1 – Use mathematical language and
symbols appropriately when communicating
about mathematics

SMP2 – Make sense of problems and persevere
in solving them

SMP 6 Explain and justify their work

SMP 8 – Use repeated practice to improve their
computational skills

SMP 5 – Choose and use appropriate tools (e.g.,
pencil and paper, concrete models, a ruler,
software) when solving a problem

SMP 4 – Apply mathematics to solve problems
in real world contexts

SMP 7- Look for and make use of structure (e.g.,
patterns in numbers, shapes, or algorithms)

SMP 3 – Construct viable arguments and
critique the reasoning of others

12
Mathematical Sciences (CBMS) (Beckman, 2012). According to Beckman (2012), “the features
of mathematical practice described in [the CCSSM] standards are not intended as separate from
mathematical content…” (p. 24). Bartel et al. (2017) underscore the connection between CCSSM
and SMPs based on research from past reform efforts, emphasizing that standards must have
explicit or companion teaching practices or the standards will fail to achieve students’ desired
goals. In research conducted by the Education Insights Center (EdInsights) (2016), many
teachers noted that the emphasis of skills in the SMPs such as critical thinking, collaboration,
communication was a more substantive change than the shift in content based on the standards
only (Lewis et al., 2016). The EdInsights (2016) study highlighted numerous concerns teachers
have about their knowledge and skills in navigating mathematics content and practices that are
college and career readiness (CCR) aligned. Table 2 underscores these concerns:
Table 2
EdInsights Study: Teachers’ Concern and Confusion around CCSSM-SMPs
Category of Concern or Confusion Educators’ Concerns or Confusion
CCR Aligned Pedagogy: Content Teachers are unclear regarding their role in aligning curricular and
instructional expectations between high school and college.
CCR Aligned Pedagogy: Content Teachers are overwhelmed and confused about who should do the
work in reaching out to colleges to learn about entrance and
placement expectations, and to collaborate regarding curricular and
instructional needs of students transitioning from High school to
college.
CCR Aligned Pedagogy: Content Teachers are unclear about how to adjust curricula and instruction to
support the full range of postsecondary options.
CCR Aligned Pedagogy: Content and SMPs Teachers believe the college and career readiness expectations are
“abstract” in the Common Core, so it is difficult to determine whether
they are hitting the mark in reaching a goal that is ill-defined.
CCR Aligned Pedagogy: SMPs Teachers believe there has not been adequate training in the kinds of
instructional strategies that will support the development of critical
thinking and other non-academic skills.
CCR Aligned Pedagogy: SMPs Teachers are confused as to whether their instructional strategies
around critical thinking, collaboration, and the development of non-
academic skills are effective
CCR Aligned Pedagogy: SMPs Teachers find it difficult to identify examples of instructional
strategies that foster high-quality CCR skills.
Alignment between the CCSSM-SMPs and
College Readiness.
Teachers are uncertain whether—and, if so, how—the changes relate
to the connection and alignment between CCSS and CCR. The teachers
are concerned that not all of the changes in math and ELA would be
aligned with entrance and credit-bearing course expectations in

13
systems of higher education. While teachers have received
professional development during the first five years of
implementation of CCS, the vast majority of the professional
development opportunities were exclusively based on the changes
made to math and ELA standards.
Note. Adapted from “Supporting High School Teachers’ College and Career Readiness Efforts: Bridging California’s Vision with
Local Implementation Needs,” by Lewis et al., 2016. Copyright 2016 by the author.
Dichotomous Pedagogical Beliefs
Another factor that adds to the lack of progress in moving students forward as CCSSM-
SMPs intended is the confusion within K-12 mathematics instruction brought on by a
disagreement among educators regarding the appropriate mathematics pedagogy to prepare
students for post-secondary mathematics courses (Kirst & Venezia, 1998). For decades, studies
have been conducted to find the most effective pedagogical approach to teaching and learning
mathematics. However, it is still unclear which instructional practices teachers should employ
(Clements & Sarama, 2012; Lemov, 2010; Montague & Jitendra 2012; National Association for
the Education of Young Children, 2013; Rohrer & Pashler 2010; Rosenshine, 2012).
Historically there have been two major philosophical thoughts on how students learn best
and how they should be taught (Adelson, 2004; Chall, 2000; Hmelo-Silver et al., 2007; Kuhn,
2007; Schimidt et al., 2007; Simon et al., 2004; Spillane & Zeuli, 1999). One philosophical
thought is the teacher-centered pedagogical approach. Grounded in Essentialism, the teacher-
centered approach to teaching “adheres to a belief that a core set of essential skills must be
taught to all students, a universal pool of knowledge needed by all” (Educational Psychology,
n.d., Section 3, para., 5). According to Ornstein & Levine (2003), Essentialist classrooms are
teacher-centered in instructional delivery that focuses on lecture and teacher modeling. In
contrast is the student-centered pedagogical approach. Grounded in Constructivism, the student-
centered approach focuses on active student participation. Ganly (2007) posits that the student-

14
centered constructivist pedagogical approach allows students to explore their own ideas, share
concepts with one another, and participate in hands on activities.
Studies analyzing teacher- and student-centered pedagogical approaches have been
conducted through a comparative dichotomous and oppositional lens to determine which method
effectively increases student achievement (National Mathematics Advisory Panel, 2008). Both
positions in this dichotomy are supported by empirical studies that claim their way of educating
children is the best approach (Andelson, 2004; Schmidt et al., 2007). Practitioners are led to
believe that one approach is better than the other depending on which side asserts their approach
with greater force and persistence.
Teachers are caught in the middle of a tug-of-war between the lecture, discussion, skill
modeled, and teacher-facilitated direct instruction of the teacher-centered methodology versus
the collaborative, self-guided, and learner-facilitated student-centered methodology. The teacher-
and student-centered methodologies are often the focus of district initiatives to reform
instruction. On one side of the tug-of-war regarding pedagogical methodologies, some districts
seek a progressive approach to pedagogical practices. These districts conform to a type of
educational PC—or pedagogical correctness—that views content-focused experts using direct
instruction as an outmoded drill and kill approach to which students should never be subjected
(Ullman, 2018). On the other side of the tug-of-war, many districts emphasize the more
structured traditional approach of direct instruction that recognizes the teacher’s central role as
the subject matter expert who ensures all students master the content (Cothran, 2018; Ullman,
2018). The ongoing pedagogical tug-of-war incorrectly emphasizes the fidelitous implementation
of a particular instructional model rather than emphasizing the instructional shifts required of
teachers in the SMPs (Dole et al., 2015; Kaufman & Stein, 2010).

15
With nearly one decade into the implementation of the CCSS, the confusion around how
to use a pedagogical process to implement the SMPs has contributed to students’ lack of
preparation in SMPs (Chazen et al., 2016; Venezia & Jaeger, 2013). The lack of students’
preparation in the standards was not the intent of CCSSM, highlighting two critical points to be
made regarding the creation and development of the standards. First, the CCSSM were designed
to provide clear expectations regarding students’ knowledge and skills needed to succeed in
college, career, and life after graduating from high school. Secondly, the SMPs provide a
framework for strengthening the teaching and learning of mathematics by outlining the eight
areas that address how students should engage in critical thinking, problem-solving,
communication, and application of mathematics. The CCSS (2010) states,
The Standards for Mathematical Practice describe ways in which developing student
practitioners of the discipline of mathematics increasingly ought to engage with the
subject matter as they grow in mathematical maturity and expertise throughout the
elementary, middle and high school years. (p. 8)
For these two critical points to flourish in the mathematics classroom, teachers in the K-12
setting need first to understand the definition and alignment of the SMPs. Whether using a
teacher-centered or student-centered pedagogical approach, teachers also need to be adequately
prepared to deliver instruction that prepares students to be college-ready in mathematics
(Kaufman & Stein, 2010).
In a 2016 Policy Brief, Lewis et al. (2016) agree that teachers do not believe the SMPs
are clear. The authors report that teachers believe,
since college and career readiness expectations are abstract in the Common Core, they
face difficulties in figuring out whether they have hit the mark. It is difficult for them to
know if they are reaching their goal when the goal is ill-defined. This challenge is
heightened because teachers must develop their instructional strategies to help students
think critically, collaborate more effectively, and develop other non-academic skills, and
they want to know how effective those strategies are. Teachers could not cite any tools
they could use to help them understand if they are succeeding in these areas. (pp. 4-5)

16
To reiterate, teachers in the K-12 setting need a clear understanding of the kind of
instruction necessary for students to be college-ready in mathematics; instruction that develops
critical thinkers and problem-solvers (Seago & Carroll, 2018). Haber (2020) asserts that the
development of students who critically think is accomplished through
deliberate practice that specifically focuses on the development of critical-thinking
skills…explicit instruction on critical-thinking principles and techniques, deliberate
practice opportunities that put those techniques to work, encouraging transfer between
domains, and inspiring students to practice thinking critically on their own—all represent
high-leverage critical-thinking practices applicable to any domain. (section 3, para. 3;
section 4, para. 1)
The studies aforementioned have underscored the fact that teachers do not have clarity
regarding the connection between pedagogical implementation and the college readiness
expectations of the SMPs (Akkus, 2016; Lewis, et al., 2016; Rebora, 2013; Rothman, 2012;
Schmidt & Burroughs, 2012). These studies highlight the need for a structured process for
teachers to clarify and implement effective college readiness in mathematics instruction. The
critical problem in implementing the college readiness aspect of SMPs is the teachers’ belief that
the SMPs are abstract. (Lewis et al., 2016). The problem of abstractness surrounding the SMPs is
critical to this study because the SMPs are the catalyst that determines whether the instruction in
the CCSSM will lead students to reach college-level thinking in mathematics. Delvin (2014)
asserts that SMPs epitomize the aggregate of mathematical knowledge, skills, abilities, habits,
and attitudes, as the practices support the acquisition and application of mathematics content
knowledge. According to the Conference Board of Mathematical Sciences (2012), there must be
intentional opportunities for teachers to learn, fully understand, and implement the expectations
of the SMPs.
The urgency to resolve students' inadequate preparation for college readiness in
mathematics, evidenced in students' low mathematics achievement data, is critical. A resolution

17
to these problems requires a call to action that targets the barriers teachers face in providing
effective mathematics instruction and to move toward affecting the intended results of the SMPs
so that every high school graduate is college, career, and life ready (CCSSI, 2010).
Purpose of the Study
The purpose of the study was to conduct a qualitative document analysis using the
college readiness factors of the SMPs to compare if the teacher-centered, EDI and student-
centered CGI pedagogical approaches support the SMPs college readiness in mathematics in the
context of CCSSM. First, it was essential to understand the definition of college readiness in
mathematics, as defined by post-secondary mathematicians and researchers in education. Next,
the study verified the alignment between the college readiness definition identified by post-
secondary mathematicians and researchers in education and the SMPs. Additionally, the study
sought out to identify and integrate the use of a comprehensive cognitive processing framework
(Radmehr & Drake, 2019). The intent in using a comprehensive cognitive processing framework
was to support the creation of a structured process to understand better what teachers’ and
students’ actions are when engaging in critical thinking, problem-solving, and communicating
mathematics (Radmehr & Drake, 2019). Finally, the study conducted a comparative analysis to
identify if the two selected pedagogies, a teacher-centered EDI and a student-centered CGI,
supported the identified teacher and students’ actions of SMPs. The decision to choose EDI and
CGI was based on name recognition and to select one pedagogical approach that represented a
teacher-centered approach, EDI, and another that represented a student-centered instructional
approach, CGI. To clarify, the aim of this study was not to critique or judge the two selected
pedagogies. The intent was to find a possible systematic and structured process to verify the
alignment of college readiness expectations of the SMPs and use the structured process to

18
identify whether the two selected pedagogies consisted of teacher- and student actions that
support the college readiness skills represented in the SMPs. Therefore, this study aims to add to
the practitioners’ knowledge and understanding of the SMPs and offer insight into aligning the
mathematics instructional process that will engage teachers and students in the intended college
readiness designed by the SMPs.
Research Questions
To achieve the purpose of this study, the following research question was used:
In what ways do the teacher-centered, EDI and student-centered CGI pedagogical approaches
support the SMPs college readiness in mathematics in the context of CCSSM?
Statement of Methodology
This study will use a qualitative Comparative Document Analysis grounded in Content
Analysis Methodology. This study seeks to look at the essential characteristics or nature of the
variables and not the quantity. The researcher is interested in extrapolating the specific teachers’
and students’ actions of the SMPs’ college readiness juxtaposed to the two selected pedagogies,
EDI and CGI. In this comparative document analysis, the study intends to address the nature of
the problem and uncover an understanding of the alignment between the college readiness
characteristics of the SMPs and pedagogical aspects.
The comparison may appear simple; however, the comparison is difficult to achieve
without the variables’ similarity and contiguity (Dey, 2016; Maxwell & Chmiel, 2014). The
study will deploy categorizing and connecting strategies to integrate a comprehensive cognitive
processing model as linking variables to create the comparability of categories (Dey, 2016;
Maxwell & Chmiel, 2014; Radmehr & Drake, 2019). In other words, the linking variables will
allow the researcher to make comparisons of similar attributes and interactive pairing capacities

19
(Dey, 2016; Maxwell & Chmiel, 2014). Additionally, based on the cognitive thinking model, the
researcher will use the following logical processes: 1.) abduction - the discovery of meaningful
rules within the structure of the variables, 2.) deduction - making predictions to link the
connecting constituent parts to add to the existing meaning within the structures, and 3.)
induction - testing the newly rediscovered model to the selected samples from the purposive or a
critical case sampling (Douven, 2017; Reichertz, 2014).
Theoretical/Conceptual Framework
The process of examining whether EDI, a teacher-centered pedagogy, and CGI, a
student-centered pedagogy, supported the SMPs college readiness was viewed through the lens
of Schoenfeld’s Theory of Teaching-in-Context (1998), which is grounded in cognitive studies
(Clark & Peterson, 1986; Leinhardt & Greeno, 1986; Schoenfeld, 1998). Schoenfeld’s Theory of
Teaching-in-Context (1998) provided a focus on the importance of what teachers bring to
teaching and learning and how it ultimately results in the college readiness of students in
mathematics. Through this framework, a focus was placed on the practical application of the
findings to produce a work that can impact mathematics instruction and not merely provide a
philosophical summary of the current reality in mathematics instruction.
Significance of the Study
The impact of focusing on teachers’ knowledge of the SMPs and their ability to
understand and implement the classroom standards could yield an increase in students’
mathematics achievement toward college readiness. Therefore, the development of teachers’
knowledge and understanding of college readiness in mathematics prescribed by the SMPs is
imperative. As Schoenfeld (1998) described, teachers’ decisions in the classroom that impact the
quality of teaching and learning depend on teachers’ knowledge, beliefs, and goals about content,

20
pedagogy, and college readiness in mathematics. Developing a structured process for teachers to
understand and operationalize the implementation of the SMPs will address the lack of students’
college readiness in mathematics.
Definition of Terms
Revised Bloom’s Taxonomy
A two dimensional carefully developed definitions for knowledge and cognitive process
domains. Knowledge domain consisting of: (a) Factual knowledge, (b) Conceptual knowledge,
(c) Procedural knowledge, and (d) Metacognitive knowledge. Cognitive process domain consists
of: (a) Remember, (b) Understand, (c) Apply, (d) Analyze, (e) Evaluate, and (f) Create. Like
Bloom’s Original Taxonomy, Bloom’s Revised Taxonomy is widely known and translated into
22 languages (Anderson & Krathwohl, 2001). Bloom’s Taxonomy has been used for the “basis
of determining for a particular course or curriculum the specific meaning of broad educational
goals, such as those found in the currently prevalent national, state, and local standards”
(Anderson & Krathwohl, 2001, p.1).
Cognitively Guided Instruction (CGI)
CGI is a student-centered pedagogy, and “a research-based model of children’s thinking
that teachers can use to interpret, transform, and reframe their informal or spontaneous
knowledge about student’s mathematical thinking” (Carpenter et al., 1996, p.1).
Explicit Direct Instruction (EDI)
EDI is a teacher-centered pedagogy, and “a strategic collection of instructional practices
combined together to help teachers design and deliver well-crafted lessons that explicitly teach
content, especially grade-level content, to all students” (Hollingsworth & Ybarra, 2018, p. 16).
Pedagogy

21
Pedagogy is defined broadly “to refer to the full set of instructional techniques and
strategies that enable learning to take place…” (Siraj, 2010, p.151)
Pedagogical Space
“The spaces, norms, and pedagogical scaffolds that emerge around shared classroom
practices form important constituents of ‘pedagogical space’” (Brown, 2005, p. 2). “Context,
therefore, may be seen as a process of ‘contextualizing’ where agents make sense of their activity
through negotiating actions and by privileging certain ways of knowing (e.g., pedagogical
relationships) and doing (e.g., pedagogical practices) over others” (Brown, 2005, p. 2)
Student-centered instruction
“Student-centered instruction refers to an instructional philosophy that aims to position
students at the center of inquiry and problem solving” (Talbert, et al., 2018, p.328).
Teacher-centered instruction
“Teacher-centered pedagogy is often described as being based upon a model of an active
teacher and a passive student” (Mascolo, 2009, p.4).
Organization of the Study
The remainder of this study is organized into five chapters, a bibliography, and
appendices in the following manner. Chapter 2 presents a review of the related literature dealing
with the historical aspect of college readiness in mathematics to the current SMPs, defining the
study variables, and the presentation of the theoretical framework through which the researcher
reviewed the study: (a) evolving trends and significant influences that occurred over the history
of mathematics, (b) a brief overview of CCSSM, (c) SMPs, (d) the two selected pedagogies, EDI
and CGI, and (e) Schoenfeld’s Theory of Teaching-in-Context. Chapter 3 describes the design of
the research and the methodology of the study. The study includes a description of the data

22
collection and analysis process and the procedures followed. Chapter 4 presents the analysis of
the data and findings. Chapter 5 contains the discussion of the findings, implications,
recommendations, summary, and conclusions of the study. The study concludes with a
bibliography and appendices.

23
Chapter Two: Review of Literature
The CCSSM were created with college and career readiness as a central focus to support
students prepared to succeed in college-level mathematics (CCSSI, 2010). For the past forty
years, standards and recommendations toward preparing students to be college-ready in
mathematics in the U.S. have evolved to highlight mathematical practices, i.e., SMPs, which
describe mathematical thinking competencies that should be developed for students at all levels
(Max & Welder, 2020). The developers of the SMPs accentuate that “the SMPs are not a
checklist; they are the basis of mathematics instruction and learning” (NGA, Center for Best
Practices, and CCSSO, 2010). There is a clear need to move beyond the competitive discourse
that prevents education from making progress in mathematics instruction. Instead of allowing the
instructional dichotomous thinking to treat teacher-centered and student-centered as oppositional
and separate, the paradigm shift may be to approach the two distinct instructional approaches as
one (Yang & Lin, 2016). Marzano validates the sense of urgency to change and progress in our
mathematics instruction (Marzano & Toth, 2014). Marzano states, “for years, the majority of
U.S. high schools are simply not matriculating students who are ready for college or career”
(Marzano & Toth, 2014, p. 7). We need practitioners to clearly understand how K-12 education
can align mathematics instruction with the SMPs college readiness expectations. In this effort,
this study examines the ways in which the two selected pedagogies, EDI and CGI, align with the
expectations of the SMPs’ college readiness.
To answer the research question, this chapter presented the literature related to the
historical trend in mathematics content and instruction. The literature review regarding
mathematics history began with the history of mathematics, paying close attention to the
mathematics pedagogy and content of each era leading to the current CCSSM. Next, the chapter

24
examined the literature related to the predetermined study variables selected for this study, which
included:
1. Standards for Mathematical Practice;
2. Explicit Direct Instruction, a teacher-centered pedagogy; and
3. Cognitively Guided Instruction, a student-centered pedagogy.
Finally, the chapter ended with a review of the theoretical framework, which serves as the lens
through which the study will be examined.
Literature Background on the History of Mathematics
The vision for mathematics content and pedagogy, the goals of mathematics learning, and
the knowledge required for mathematics learning changed throughout time dating back to 1726
when Harvard hired its first professor of mathematics (Waggener, 1996). As Jones and Coxford
(1970) mention, throughout history in mathematics, two persistent questions continue to be (a)
What should the nature of mathematics be—facts, skills, and procedures or concepts and
understanding? and (b) How should students learn mathematics—teacher-directed with a focus
in memorization, or student-centered through reasoning and discovery?
For this study, the history of mathematics from the late 1800s will be examined, focusing
on mathematics content, pedagogy, and the significant influences in mathematics at the time: (a)
The study will look at the history through the lens of mathematics content, and the focus will be
on the specific mathematics courses offered at the time and in what schools offered grade levels
the courses, (b) Mathematics pedagogy will be examined, looking at the teacher/student
interactions to determine the teacher’s role in direct instruction or inquiry-based learning, and the
role of the student as a master of assigned tasks and depository of new information or an explorer
or collaborator in new learning, and (c) The study will examine math history through the
significant influences present throughout each period. It is essential to the study to view history

25
through specific lenses to provide a clear understanding of the current teaching and learning
components related to the specific components of college readiness in mathematics, such as
critical thinking, problem-solving, and reasoning. Canty-Jones (2020) points to the American
author, David McCullough’s quote, “History is who we are and why we are the way we are”
(p.1), a quote most appropriate, as mathematics is examined through the past to help us
understand the present and where we are in meeting the demands of college readiness in the
CCSSM.
Late 1800s - 1930: The Era of Mathematic Debate
During this period in mathematics, there was a persistent philosophical tug-of-war
between the proponents of mathematics based on procedures, skills, rules, facts, and
memorization where teachers do most of the mathematics. In contrast, it was the conceptual
understanding and sense-making approach to mathematics where students are engaged in
mathematics (Jones & Coxford, 1970). The teacher-centered characterization of mathematics in
the late 1800s demonstrated an approach where there was little focus on the learner and more on
what and how teachers should teach content (Schoenfeld, 2016). During this time and leading up
to the 1900s, there was an emphasis on mathematics that would lead to college attendance. As a
result, the mathematics curriculum became an area of concern that ultimately led to the reshaping
of junior high and elementary mathematics education (Schoenfeld, 2016). This reshaping of
secondary education led to the rise of algebra and geometry classes in secondary schools, and by
1908, almost all secondary schools in the U.S. provided one year of algebra and geometry
(Willoughby, 1967).
At the turn of the century, the College Entrance Examination Board (CEEB) was
founded, and the purpose of this board was to standardize college entrance requirements. While

26
the CEEB was establishing criteria for college entrance, the CEEB had no influence on
curriculum and instruction in secondary schools (Waggener, 1996). However, the CEEB college
requirements influenced the emergence of standardized intelligence and proficiency tests for
secondary students (Schoenfeld, 2016; Waggener, 1996).
This period leading up to the 1923 “Reorganization of Mathematics in Secondary
Education” report was a time of reform resulting from the criticism that mathematics instruction
was dominated by drill and the use manipulative (Center for the Study of Mathematics
Curriculum [CSMC], 2005, para. 2). The 1923 report was written in response to calls for reform,
as students were failing algebra and geometry courses, and the number of students taking algebra
and geometry was on the decline (CSMC, 2005). By 1922, only 40 percent of American students
took algebra, and 23 percent took geometry (CSMC, 2005; Whitney, 2016). As a result, the
report recommended that some of the algebra topics be introduced in junior high school, with
only the one mastering the content that would move on to take algebra and geometry in high
school (Bidwell & Clason, 1970). Additionally, the report responded to a leading influential
educator and professor at Teachers College, William Kilpatrick, who had influenced the writing
of a report in 1920, “The Problem of Mathematics in Secondary Education.” This report
espoused the views of Kilpatrick, who believed “algebra and geometry were a waste of time for
most students” (Whitney, 2016; para. 13). By 1934, the number of students taking high school
math had decreased to 30 percent of American students taking algebra, and 17 percent took
geometry.
The 1930s, as Schoenfeld (2016) describes, was a period focused on how students think
about math became an interest during this time. According to Schoenfeld (2016), this era in
mathematics brought about a greater focus on the learner, as it became clear that “what you

27
choose to think about, represent, and characterize what goes on in the minds of learners, are
consequential” (p. 500). Brownell and Chazal (1935) argued that arithmetic instruction should
develop students’ ability to think about mathematics and not merely to perform operations with
automaticity. Knight (1930) was at the forefront of the change in mathematics instruction, as he
believed mathematics instruction did not provide conceptual comprehension of the math content
to the learners.
1940s - 1980s: Crisis, Reform, & Reaction in Mathematics
The 1940s marked a new era and was known as the progressive era in mathematics, and it
was the beginning of the U.S. government’s interest in mathematics education (Waggener, 2016).
The interest was that the WWII recruits did not have basic mathematics skills, such as
computation and problem-solving skills (Fey & Graeber, 2003). William Brownell became
prominent when he published a 1944 article called The Progressive Nature of Learning
Mathematics (Brownell, 2006). Brownell and Chazal (1935) asserted the importance of focusing
on the process of learning. He promoted the teaching of content more in-depth and believed that
drill and practice only interfered with meaningful learning. A group formed the educational
program called the Life Adjustment Movement in the 1940s, and the group thought schools were
“too devoted to an academic curriculum. Educational leaders at the time presumed that 60% of
the students did not have the intellectual ability for college or skilled occupations” (Klein, 2003,
p.4). The Life Adjustment Movement promoted an agenda that high school mathematics should
focus only on preparing students for everyday living. The movement pushed students to learn
mathematics for “consumer buying, insurance, taxation, and home budgeting, not on algebra,
geometry, and trigonometry” (Klein, 2003, p.4). Increasing criticism of public schools by the end

28
of the 1940s damaged the Life Adjustment Movement’s substantial support. The damage was too
significant, and the movement began to lose its following by the end of 1949 (Klein, 2003).
By 1950, the American schools were under attack from business, the military, and the
public for graduating students who could not meet the rigor of college at that time (Klein, 2003).
The progressive agenda in education had to retreat to make way for New Math that lasted
through the 1960s (Klein, 2003). The New Math was the cause for significant division between
psychologists and mathematicians, as it sought to displace progressivism’s disdain for procedural
mastery and replace it with a “coherent logical explanations for the mathematical procedures
taught in the schools” (Klein, 2003, p.6; Tampio, 2017).
In 1957, the U.S.S.R. launched Sputnik, the first space satellite, which caused significant
humiliation in the United States, rooted in pride and national security concerns (Waggener,
2016). The humiliation called attention to low-quality math and science instruction in public
schools (Klein, 2003). The 1958 Defense Act was passed that poured financial resources into
mathematics education programs that resulted in the development of textbooks focusing on
mathematical structure, the real number system, and deductive proofs (Waggener, 1996). The
New Math Movement did not have the significant impact desired, as demonstrated in a 1963
survey conducted by the College Board (Waggener, 1996). The College Board discovered that
30% of the school’s mathematics classes did not teach several topics central to New Mathematics
programs (Waggener, 1996).
In the 1960s, New Mathematics focused heavily on the conceptual learning of problems
and solutions, exercises using reasoning, and a continuation of calculation arithmetic (Baker et
al., 2010). The New Mathematics approach ushered in plenty of studies in mathematics, such as
the one conducted by D.A. Johnson on what he observed in the area of conceptual learning of

29
mathematics. He stated, “learning takes place somewhat in proportion to the involvement of the
learner in learning activities” and “the role of the teacher as a guide…” (Johnson, 1957, p. 402).
The early 1970s marked the end of New Mathematics, and the decade was known as the
Back to Basic era. Back to Basics abandoned the precise, structuralist language of sets, logic, and
algebraic structures of the New Math in favor of an emphasis on computation and algebraic
manipulation (Herrera & Owens, 2001). Computational mastery was now more important than
conceptual understanding (Tampio, 2017), which helps us understand why there was not a great
deal of mathematics instruction in the classrooms. Teachers were free to teach what they wanted,
and mathematics instruction worked through the math problems assigned. During this era,
students had to pass minimum competency tests in basic skills. The scores on these standardized
tests were steadily declining and bottomed out in the early 1980s (Kenney & Silver, 1997; Klein,
2003).
The 1980s were known as the period of standards-based reform. Mathematics in the 1980s
was influenced by the release of An Agenda for Action in 1980 by the National Council of
Teachers of Mathematics, another attempt at reforming education. Agenda in Action pushed for a
focus on problem-solving because “requiring complete mastery of skills before allowing
participation in challenging problem solving is counterproductive” (Fey & Graeber, 2003; Klein,
2003, p. 9). According to Klein (2003), the Agenda for Action resulted from studies indicating
that between 1975-1980, remedial mathematics courses in colleges increased by 72 percent and
were one-quarter of the math courses taught in the colleges. The deficit in mathematics
achievement raised the concerns of those in government and education. Achievement in
Mathematics awakened these concerns greatly due to the publication of 1983, A Nation at Risk,
that reported on average only 31% of high school students complete algebra and only 6 percent

30
complete calculus (Klein, 2003; Tampio, 2017). In response to the heightened awareness and
concern about mathematics in the U.S., the National Council of Teachers of Mathematics
(NCTM) created the Curriculum and Evaluation Standards for School Mathematics. These
standards sought to change the role of the teacher from the “transmitter of knowledge to a new,
therefore uncomfortable, position as a facilitator--one who engages the class in mathematics
investigation, orchestrates classroom discourse, and creates a learning environment that is
mathematically empowering” (Falba & Weiss, 1991, p. 90). The NCTM standards were criticized
for overemphasizing the application of mathematics over basic computational skills. Another
criticism is that students were encouraged to use calculators. One group, called Mathematically
Correct, were in opposition to the standards stating, “the [textbooks] don’t even give explicit
definitions or procedures...students [have to] discover all of the mathematics for
themselves...many of these programs don’t even teach standard algorithms for the operation of
arithmetic…” (Herrera & Owens, 2001, p.90).
The 1990s - No Child Left Behind
The NCTM standards continued into the 1990s and paved the way for state standards and
curricular frameworks. By the mid-1990s, forty-one states had implemented state standards or
curricular frameworks consistent with the NCTM standards (McLeod, 2003). However, by the
end of the 1990s, the NCTM standards witnessed extensive criticisms. Parents criticized the
standards because they expected the teachers to fulfill the traditional role of a transmitter of
knowledge; however, the reformers responsible for the NCTM standards wanted the students to
think about math (McLeod, 2003). The criticisms were all too familiar, as the standards did not
emphasize procedural skills, direct instruction, practice, and memorization. According to Klein,
in the NCTM standards, (a) fundamental algebra and arithmetic skills were not developed, (b)

31
elementary schools encouraged students to invent their math and discouraged the use of standard
algorithms for addition, subtraction, multiplication, and division, and (c) encouraged the excess
use of calculators at the elementary level. The criticism of the NCTM standards and the
disagreement among parents, mathematicians, and professional educators ended the 20th century
with education in flux. Students were entering middle school, unable to progress in algebra. The
lack of student preparedness in math caused great concern among university mathematicians who
argued that reformers, who advocate “pedagogical practices based on opinions rather than
research,” had taken over mathematics instruction (Klein, 2003, p. 21).
No Child Left Behind
The No Child Left Behind Act (NCLB) of 2001 solidified what became known in the
2000s as the era of outcome-based education that ushered in an emphasis on standardizing pre-
college education through standards-based education and high stakes testing. During this era,
mathematics curricula changed to offer a “balanced diet of skills, concepts, and problem-
solving” (Schoenfeld, 2016). The balanced approach, according to Larson and Kanold (2016),
influenced mathematics instruction such that the content covered on the standardized
assessments is the content that was taught in the classroom. Overall, both instruction and student
achievement did not meet the expectations of NCLB, and the blame by many pointed to the
“surprisingly poor levels of coherence of state standards and assessments, the key instruments
intended to drive teachers’ instruction” (Polikoff et al. 2011).
Mathematics History in Summary
The history of mathematics demonstrates a teetering between a constructivist and a
progressivist approach to education. See Figure 4 for the Constructivism and Progressivism
Timeline.

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Figure 4
Education Timeline: Constructivism and Progressivism in Mathematics
Constructivist and Progressivist Approach to Education
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Constructivism Progressivism

Evolving trends and significant influences occurred over the history of mathematics. In
mathematics curricula, algebra and geometry were a focus, and then it was not, and then it was
again. Arithmetic was important until it was not, and then it was again. Back to basics was the
call of many until it gave way to discovery learning, and then back to basics again. Until the
1960s, 85 percent of math instruction was focused on arithmetic, and by 2000, it was only 64
percent of math instruction (Hartnett, 2016). The percentage has dropped between 1960 and
2000, as mathematics reformers have worked diligently to get students to reason mathematically
and conceptually understand mathematical ideas. However, some believe this has led to students
being “engaged in activities and messing around,” according to a College of Education Dean at
Michigan State University (Hartnett, 2016, para., 15; Polikoff et al., 2011).
Literature Background on the Study Variables
Brief Overview of the Common Core State Standards Mathematics
The release of the CCSS-M in 2010 is the most recent development in K-12 mathematics
(CCSSI, 2010). Purportedly, CCSSM reflects a newly achieved consensus about what should be
taught in our nation’s classrooms. Munter et al. (2015) affirmed, “In mathematics, the CCSSM
represent an unprecedented agreement--across previously divided parties--regarding which ideas

33
should be included in the K-12 mathematics curriculum” (p.3). As stated in Chapter 1, the
CCSSM encompasses two types of standards: Mathematical Content Standards and Standards for
Mathematical Practices (CCSSI, 2010). The mathematical content standards are different at each
grade level; however, the Standards for Mathematical Practice (SMPs) are the same at each
grade level. The CCSSM are viewed as “a balanced combination of procedure and
understanding” (CCSSI 2010, p. 8), with the content standards addressing what math to teach at
different grade levels to prepare students for college readiness content. On the other hand, the
SMPs delineate how students should “engage with the subject matter as they grow in
mathematical maturity and expertise throughout the elementary, middle, and high school years”
to prepare students for college readiness mathematical thinking (CCSSI 2010, p. 8).
Standards for Mathematical Practice (SMPs) College Readiness Definition
The SMPs are critical to the success of mathematics learning in the CCSSM, and the
emphasis on incorporating mathematical practices into math instruction has been clear, yet there
is evidence that mathematical practices are not a part of mathematical pedagogy (CCSSI, 2010;
NCTM, 2000; NCTM, 2006; Venkat, 2015). Russell (2012) states, “If the Standards for
Mathematical Practice are taken seriously, we must focus on them in the same way we focus on
any other standards—with targeted, intentional, planned instruction” (p. 52). As part of the
CCSSM, SMPs are process standards that are separate from the content standards; however, they
are an essential part of the content standards. The SMPs describe mathematical ways of thinking
as well as the way in which students should do mathematics (CCSSI, 2010; Mateas, 2016).
Mateas (2016), said, “...the Content Standards determine the topic and the SMPs describe the
mathematical thinking (p. 96). The SMPs connect students to the content standards and provide

34
cognitive process skills that allow additional understanding of the content standards (CCSS,
2011). Table 3 describes the differences between the SMP and the CCSSM.
Table 3
Differences between SMP and CCSS-M
Note. Source: Mateas, 2016, p. 94

The SMPs were developed using the NCTM’s process standards as well as the
mathematical proficiencies found in the National Research Council’s (NRC) report called
Adding it Up (Kilpatrick et al., 2001). There are five strands in Adding it Up that were
instrumental in developing the processes and proficiencies that formed the eight SMPs: 1) make
sense of problems and persevere in solving them, 2) reason abstractly and quantitatively, 3)
construct viable arguments and critique the reasoning of others, 4) model with mathematics, 5)
Use appropriate tools strategically, 6) Attend to precision, 7) Look for and make use of structure,
and 8) Look for and express regularity in repeated reasoning (CCSS, 2011). O’Connell and
SanGiovanni (2013) also agree that “all three documents embrace practices that are
interconnected and that build understanding, reasoning, and application (p.7). The description on
the interconnectedness of the SMPs, NCTM Math Process Standards, and NRC Strands of
Mathematical Proficiency is seen in Table 4.

Standards for
Mathematical Practices
Standards for
Mathematical Content

What are they?
Descriptions of ways of thinking that
mathematically proficient students use
Descriptions of what students should understand
and be able to do mathematically.

How are they
organized?
The same set of eight standards to be used at
all grade levels, from kindergarten through
grade 12. They develop over time, broadening
in meaning as students encounter new
content and becoming more sophisticated as
students develop cognitively.
A different set of standards at each grade level;
grouped by cluster within a grade level and by
domain across grade levels. (High school
standards are not organized by grade level but
by conceptual category related to mathematical
domains.)

35
Table 4
The Common Core State Standards for Mathematical Practice
SMPs NCTM Math Process Standards
NRC Strands of Mathematical
Proficiency
SMP 1
Make sense of the problems and
persevere in solving them.
Problem Solving Strategic Competence
SMP 2
Reason abstractly and quantitatively.
Representation
Communication
Reasoning and Proof
Problem Solving
Adaptive Reasoning
SMP 3
Construct viable arguments and
critique the reasoning of others.
Reasoning and Proof
Communication
Representation
Conceptual Understanding
Adaptive Reasoning
SMP 4
Model with mathematics.
Representation
Communication
Strategic Competence
Conceptual Understanding
SMP 5
Use appropriate tools strategically.
Problem Solving
Reasoning and Proof
Conceptual Understanding
Procedural Fluency
SMP 6
Attend to precision.
Communication
Representation
Procedural Fluency
Conceptual Understanding
SMP 7
Look for and make use of structure.
Reasoning and Proof
Problem Solving
Representation
Adaptive Reasoning
Productive Disposition
SMP 8
Look for and express regularity in
repeated reasoning.
Reasoning and Proof
Representation
Communication
Adaptive Reasoning
Conceptual Understanding
Productive Disposition
Note. The Common Core State Standards for Mathematical Practice were influenced by both the NCTM Standards and the
Strands of Mathematical Proficiency. Source: Adapted from O’Connell and SanGiovanni (2013).
SMP 1: Make sense of Problems and Persevere in Problem-solving
Teachers who are developing students’ capacity to “make sense of problems and
persevere in solving them” develop ways of framing mathematical challenges that are clear and
explicit, and then check-in repeatedly with students to help them clarify their thinking and their
process (Inside Mathematics, 2020, para., 1). The NCTM (2000), defines problem-solving as
“engaging in a task for which the solution method is not known,” and “...the teacher plays a
critical role in supporting students in problem-solving.” “It is [the teacher’s] responsibility to
select, adapt, design, and implement appropriate mathematical tasks for the students in front of
them” (p. 2). This fact is a monumental task for teachers, and as Mateas (2016) points out, many

36
teachers feel the challenge of teaching math content and teaching the “thinking and habits of the
mind required to do mathematics” (p. 95).
Problem-solving in mathematics gained prominence in mathematics discourse when
George Pólya (1945) wrote How to Solve It. He introduced four phases of problem-solving,
which were strategies or “patterns of productive thinking that often help one better understand
and/or make progress toward the solution of a problem” (Schoenfeld, 2016, p. 507). Schoenfeld
(1985) developed Pólya’s work by operationalizing a five-phase model of problem-solving,
which are (a) Analysis, (b) Design, (c) Exploration, (d) Implementation, and (e) Verification.
According to Biggerstaff (1995), problem-solving strategies are difficult for teachers to
implement, as problem-solving strategies cannot replace the problem solver's knowledge base
and experiences. “If a student does not know the relevant mathematics facts, formulas, and
relationships…”, the problem-solving strategies cannot be effectively applied (p.13). Refer to
Figure 5 for Schoenfeld’s Five Phases of Problem-solving.
Figure 5
Schoenfeld’ s Five Phase Model of Problem-Solving

Schoenfeld’s Five Phase Model of Problem-Solving

1. Analysis
a. Understanding the statement
b. Simplifying the problem
c. Reformulating the problem

3. Exploration
a. Essentially equivalent problems
b. Slightly modified problems
c. Broadly modified problems
2. Design
a. Structuring the argument
b. Hierarchical decomposition

4. Implementation
a. Step-by-step execution
b. Local verification

Note. Source: Schoenfeld, 2016, p. 507.

5. Verification
a. Specific
b. General test

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Additionally, as seen in Figure 6, O’Connell and SanGiovanni (2013) provide insight into
how open-ended questions can be used to promote problem solving.
Figure 6. Open-Ended Questions to Promote Problem Solving
Open-Ended Questions to Promote Problem Solving
These questions guide our students thinking as they solve math problems.
• Before
o What is the question?
o What data will help you find the solution?
o How will you get started solving this problem?
o Does this problem remind you of any others you have solved?
o What did you do to solve that problem? Will it work here?
• During
o Are you blocked? Should you try a new approach?
o Does your answer make sense? Why or why not? If it doesn’t make sense, what could you do?
• After
o How did you solve the problem?
o Why did you solve the problem that way?
o What was easy/hard about solving this problem?
o Where did you get stuck? How did you get unstuck?
o Were you confused at any point? How did you simplify the task or clarify the problem?
o Can you describe another way to solve the problem? Which way might be more efficient?
o Is there another answer? Explain.
Note. Source: O’Connell & SanGiovanni, 2013, p.25
SMP 2: Abstract/Quantitative Reasoning
Teachers who are developing student's capacity to "reason abstractly and quantitatively"
help their learners understand the relationships between problem scenarios and mathematical
representation, as well as how the symbols represent strategies for a solution (Inside
Mathematics, 2020, para., 1). The CCSSI (2010) states that quantitative reasoning entails habits
of (1) creating representations of a problem, (2) considering the units involved, and (3) attending
to the meaning of quantities (p. 6). According to NCTM (2000), abstract and quantitative
reasoning is a 3-step process: (1) decontextualizing problems by representing a problem context
using mathematical symbols, (2) manipulating symbols, such as performing calculations or
solving an algebraic equation, and (3) contextualizing problems by periodically connecting the
mathematical symbols back to the problem context (p. 24).

38
Mason (1989) describes abstract/quantitative reasoning as a follow-up to a generalizing
activity. Students move from manipulating objects (physical, pictorial, symbolic, mental) to
“expressing properties or features of those objects in ways that can form the basis for further
manipulation” (p. 3). Galasso (2016) called it contextualizing--taking abstract mathematical
representation and putting it into context, or decontextualizing--taking context and representing it
abstractly. The application of this mathematical practice would be in teaching students how to
reason quantitatively to see the effects of operations on numbers, as opposed to practicing
operational procedures through calculations (Schifter, 2011).
SMP 3: Constructing Viable Arguments and Critique the Reasoning of Others
Teachers who are developing students’ capacity to "construct viable arguments and
critique the reasoning of others" require their students to engage in active mathematical discourse
(Inside Mathematics, 2020, para., 1). Developing students' ability to construct viable arguments
is grounded in active mathematical discourse where students explain and discuss their thinking
processes aloud (Rumsey & Langrall, 2016). Kilpatrick et al., (2001) contends that the ability to
construct a viable argument is having the capacity to build networks of logical statement or
argumentation chains with awareness of the warrants that underlie them. In a more practical
sense, Rumsey and Langrall, (2016) view mathematical argumentation as a process of social
discourse for discovering new mathematical ideas. Students provide evidence and reasoning to
convince others that their claim is valid (para., 2). The aim of having students construct viable
arguments is to strengthen students’ skills of argumentation as a mathematical practice while
deepening their understanding of mathematical content (Rumsey, 2012; Rumsey & Langrall,
2016). See Table 5 for O’Connell and SanGiovanni’s (2013) example of a rubric organized into

39
mathematical reasoning and communication that can guide the expected student outcomes for
argumentation.
Table 5
Mathematical Reasoning and Communication Rubric
Expected Student Outcomes
The student is expected to create an argument to clearly justify an answer or clearly defend a mathematical decision. A clear
justification includes the use of specific math vocabulary or symbolic notation and a logical organization so the argument can
be understood. It also includes specific data, examples or counterexamples, diagrams, definitions, and/or explanations of
the student’s reasoning that support the answer or decision. Some complex arguments may require multiple pieces of
evidence.

RUBRIC FOR A BRIEF JUSTIFICATION
CATEGORY 4 3 2 1 0
Mathematical
Reasoning
The argument does
not contain flaws.

There is specific and
appropriate data or
reasoning to support
the student’s
decision/position.
The argument
contains at least one
of the following:
specific example
and/or
counterexample, a
related definition, a
labeled diagram, or
other appropriate
data to back up the
main point.

The student may
include more than
one way to justify
his/her solution
The argument may
contain minor
flaws.

There is adequate
data or reasoning
to support the
student’s
decision/position.
The argument
contains at least
one of the
following: an
example, a
definition, a
diagram, or other
appropriate data
to back up the
main point.
The argument
may contain
flaws.

There is
minimal data
or reasoning to
support the
student’s
decision/
position.
There is no
mathematical
reasoning or data
to support the
student’s
position/decision.
Bank; No
response.
CATEGORY 4 3 2 1 0
Communication The writing is
organized.

The argument is
presented in a clear,
logical sequence.

Precise math
language and
appropriate
symbolic notation
are evident.
The writing is
generally clear and
organized. It
includes
appropriate math
language and
symbolic notation.
The writing is
somewhat
organized but
may lack
clarity.
The writing may be
difficult to
understand and
may not contain
appropriate math
language or
symbolic
notations.
Bank; No
response.

Note. Source: O’Connell & SanGiovanni, 2013, p.59.

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SMP 4: Model with Mathematics
Teachers who are developing students’ capacity to "model with mathematics" move
explicitly between real-world scenarios and mathematical representations of those scenarios
(Inside Mathematics, 2020, para., 1). Galasso (2016) asserts that modeling with mathematics has
common foundational elements that include (a) students making assumptions about different
constraints present in a task, (b) students being asked to find specific information or students
could be given the information once they have identified they need it, and (c) students making
multiple assumptions and then being expected to find the values using various tools and
resources, providing justification for the assumptions they make and values they chose (p. 3).
Modeling with math is also about noticing patterns, finding structure, and attending to
“juxtaposing examples in ways that draw attention to variations, invariances, and connections
between examples” (Venkat, 2015, p.3).
SMP 5: Use Appropriate Tools Strategically
Teachers who are developing students' capacity to "use appropriate tools strategically"
make clear to students why the use of manipulatives, rulers, compasses, protractors, and other
tools will aid their problem-solving processes (Inside Mathematics, 2020, para., 1). According to
Van De Walle (2004), “strategically using appropriate skills refers to the use of tools as an
object, picture, drawing, or figure that represents a mathematical concept used by students to
make meaning of the mathematics, or on which the relationship for that mathematics, helping
them to construct mathematical meanings and explain their ideas” (section, 1). According to
Sharma (2016), “Teachers must recognize that tools do not produce understanding, problem
solving and solutions. These come when teachers ask questions and make connections between
the tool and the concept and when students do the same. The teacher should bring to students’

41
attention the strength and limitations of the tool and its usage” (p. 2). See Figure 7 for O’Connell
and SanGiovanni’s (2013) elaboration on students using tools appropriately.
Figure 7
Using Tools Appropriately
USE TOOLS APPROPRIATELY
Do Students know how to use number lines?
• Where do they begin on the number line?
• What direction should they move?
• How do they know the quantity for each “jump”?
Do students know how to use rulers?
• Where do they place the ruler on the object to be measured?
• Do the marking begin at the edge of the ruler, or is there an indentation before the measurement markings begin?
• Do they understand that the length is the difference between 2 points on the ruler, not just the distance from 1 to
another point on the ruler?
• Do they understand the fractional markings in order to be able to read the measurement?
Do students know how to use protractors?
• Do they know how to accurately place the protractor on one of the rays?
• Do they understand the concepts of acute and obtuse, thus determining which of the two measurements is the
appropriate one for the angle they are measuring?
• Do they understand that the protractor measures the rotation of one segment or ray from another ray?
Do students know how to use graphing calculators?
• Do they understand how to input the data?
• Can they manipulate the data that is graphed?
• Are they able to follow button procedures to complete tasks?
• Can they make sense of the representations and apply them to different situations?
• Do they understand concepts of linear equations, slope, and functions and the resulting graphs?
Note. Source: O’Connell & SanGiovanni, 2013, p.81.
SMP 6: Attend to Precision
Teachers who are developing students’ capacity to “attend to precision” focus on clarity
and accuracy of process and outcome in problem-solving (Inside Mathematics, 2020, para., 1).
Attending to precision is developing students’ capacity to attend to precision, focus on clarity
and process and outcome in problem-solving (CCSSI, 2010). Sharma (2016) states “the key
word in this standard is the verb attend. The primary focus is attention to precision of
communications in mathematics, in thinking, in speech, in written symbols, in the usage of
reasoning, in applying it in problem-solving, and in specifying the nature and units of quantities
in numerical answers and in graphs and diagrams” (para., 1). Sharma (2016) emphasizes the

42
teachers’ role in helping students attend to precision by (a) being attentive to precision in the way
they teach and to insist on its presence in students’ work, (b) demonstrating, and expecting
precision are all aspects of students’ interactions relating to mathematics, and (c) focusing on
clarity and accuracy of process and outcomes of mathematics learning and in problem-solving
(para., 4). See Figure 8 for an example of attending to precision when communicating about
math.
Figure 8
Tips for Attending to Precision
Tips for Attending to Precision in Math Communication
While knowing the words of mathematics contributes to our students’ abilities to precisely communicate their
ideas, being able to formulate clear explanations requires more than a grasp of math vocabulary. Our students
are asked to explain how they solved problems, justify solutions, and describe math concepts. Consider the
following tips to help students more effectively talk and write about their math ideas:
• When students are asked to explain a process (e.g., how they solved a problem or performed a
complex calculation), steps and order are key elements. Writing their ideas in a numbered list
simplifies the task for students as it helps them organize and clearly communicate the steps of the
process.
• When asked to justify an answer, expect students to provide the solution and data or reasoning
that defends that solution. Have students think about a because statement, “The solution is ____
because…” Thinking about a because statement reminds them that proof for their solution is an
integral part of the teacher’s expectation.
• When describing math concepts (e.g., congruent, symmetry, multiple, factorial, exponent, etc.),
the brief responses we tend to get from students lack the detail necessary to adequately describe
the concept. Remind students to sue words, picture, numbers, and examples to provide additional
elaboration.
Note. Source: O’Connell & SanGiovanni, 2013, p.103.
SMP 7: Look for and make use of structure
Teachers who are developing students’ capacity to look for and make use of structure, help
learners identify and evaluate efficient strategies for solutions (Inside Mathematics, 2020).
Looking for and making use of structure go beyond the information that is given in the problem
and shifts the perspective to discerning relationships between pieces of information either
explicitly or implicitly or predicting the structure in the information. The steps include: (a) look
for relationships between the given information in the problem and what is hidden, (b) look for a

43
pattern or structure in the problem or concept under discussion, (c) step back for an overview or
shift, (d) see something as a whole or as a combination of parts, and (e) using familiar/known
structures to see something in a different way (Sharma, 2016). O’Connell and SanGiovanni
(2013) adds proficient students can:
1. See the flexibility of numbers in that numbers can be composed (put together) and
decomposed (broken apart);
2. Understand properties, e.g., commutative, associative, distributive, and identity; and
3. Recognize patterns and functions, such as patterns in the hundred chart or multiplication
chart and ratio tables.
SMP 8: Look for and express regularity in repeated reasoning
Teachers who are developing students’ capacity to look for and express regularity in repeated
reasoning attend to and listen closely to their students’ noticing’s, as well as create the conditions
for students to look for and express regularity in repeated reasoning (Inside Mathematics, 2020,
para., 1). As teachers develop lessons and instruction around the mathematical practice, they also
need to develop prerequisite skills in addition to introducing interesting and relevant content.
However, prerequisite skills should be integrated rather than isolated exercises. These
prerequisite skills include: (a) the ability to follow sequential directions, (b) classification, (c)
organization, (d) quantitative and spatial reasoning, (e) visualization, and (f) deductive and
inductive reasoning (Sharma, 2016). Mathematically proficient students can perform in
answering the following (O’Connell & SanGiovanni, 2013):
● What did you notice about the signs of the sums and products?
● Did what you saw make sense?
● Would the product of a positive and negative integers always be negative?
● Could you explain it?

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Two Selected Pedagogies: EDI and CGI
Explicit Direct Instruction: EDI
Explicit Direct Instruction (EDI) is a teacher-centered lesson design framework merging the
terms of explicit instruction and direct instruction developed by Hollingsworth and Ybarra
(2009). “EDI is a strategic collection of researched-based instructional practices pioneered by
pedagogues and cognitive researchers, such as Hattie, Rosenshine, Marzano, Sousa, Hunter,
Goldenberg…” (Dataworks Educational Research, n.d., para. 3).
How EDI Research Began. The authors of EDI started their company, DataWORKS
Educational Research, in 1997 “with a single purpose of using real data to improve student
learning, especially for underperforming students (Hollingsworth & Ybarra, 2018, p.2). In the
inception of DataWORKS, the authors analyzed student achievement data for over 600 schools
per year, supporting schools and school districts to assess and disaggregate data accurately to
determine student achievement. However, there was a turning point for DataWORKS that shifted
the focus away from assessment to teaching. The shift was a result of a single event when one of
the principals pointed out to DataWORKS emphatically stating, “Don’t show me the test scores.
Show me how to raise the test scores” (Hollingsworth & Ybarra, 2018, p.3). With the newfound
focus, DataWORKS began moving away from focusing solely on analyzing and interpreting
student achievement data to a focus on impacting student performance. Hollingsworth and
Ybarra (2009, 2018) broadly expanded the data to include measurements of classroom teaching
practices. Hollingsworth and Ybarra (2018) pursued classroom data collection effort to include:
(a) What students were being taught, and (b) How students were being taught. The authors
coined the phrase, “It’s better inputs that produce better outputs” and believed that the secret to

45
true school reform is “Every time teaching improves…even a little bit…students learn more, and
that’s how test scores go up” (Hollingsworth and Ybarra, 2018, p. 3).
EDI’s Philosophy and Purpose. EDI is rooted in a teacher-centered, direct instructional
model. EDI is grounded in observing over 35,000 teachers and surveying more than 100,000
educational stakeholders. The observations and surveys identified a trend of students not
succeeding in the classrooms. According to Hollingsworth and Ybarra (2018), the common
reasons students are not successful are directly tied to teacher decisions. Hollingworth and
Ybarra highlights the acronym GIFT, Great Initial First Teaching, to address the common
reasons why students are not successful. The change needs to happen at the teachers’ Great
Initial First Teaching level. See Figure 9 for common reasons students are not successful in the
classrooms.
Figure 9
Reasons Student are Not Successful in the Classroom
Common Reasons Students are not Successful in the Classrooms
• Independent practice does not match learning objective
• Lesson components of teaching is bypassed, and students are working independently on their own
• Checking for understanding, and providing corrections when necessary is not practiced
• Various ways of students understanding the concept is not practiced.
Note. Source: EDI Workshop PowerPoint, 2019.
Accordingly, Hollingsworth and Ybarra (2018) concluded that “Students learn more and
learn faster when the teacher delivers a well-designed, well-taught lesson, using the most
effective strategies to explicitly teach the whole class how to do it” (p.11). Based on the data
collected, the two authors determined to develop a framework that teachers can follow to design
and deliver effective lessons to students, and as a result established the Instructional Approach
Guidelines to help guide the process. See Figure 10 for DataWORKS’ five guidelines.

46
Figure 10
Instructional Approach Guidelines
Instructional Approach Guidelines
1. The instructional approach is effective [students learn] and efficient [students learn quickly].
2. The instructional approach is based on research, and the strategies can be used over and over again.
3. The lesson planning process is clear and well defined.
4. The lesson planning proves is independent of grade level, content, and age.
5. The instructional approach produces a high percentage of successful students.
Note. Source: Hollingsworth & Ybarra, 2018, p.10.
EDI is a version of direct instruction, an approach that encompasses components to
address the learning needs of the low-performing students. Hollingsworth and Ybarra (2018)
state “Explicit Direct Instruction is a strategic collection of instructional practices combined
together to design and deliver well-crafted lessons that explicitly teach content, especially grade-
level content, to all students” (p.16). EDI adheres to the research of Rosenshine and Stevens
(1986), the Handbook of Research on Teaching. See Figure 11 for the components of teacher
behaviors that characterize well-structured lessons which was a synthesis of all empirical studies
conducted by the researchers, Rosenshine and Stevens (1986).
Figure 11
Teacher Behaviors that Characterize Well-Structured Lessons
Synthesis of All Studies: Teacher Behaviors that Characterize Well-Structured Lessons
● Start lessons by reviewing prerequisite learning
● Provide a short statement of goals.
● Present new material in small steps, with student practice after each step.
● Give clear and detailed instructions and explanations.
● Provide a high level of active practice for all students.
● Ask a large number of questions, check for understanding, and obtain responses from all students.
● Guide students during initial practice.
● Provide systematic feedback and corrections.
● Provide explicit instruction and practice for seatwork exercises and, where necessary monitor students during
seatwork.
Note. Source: Hollingsworth and Ybarra, 2018, p.12.

47
The EDI framework has eight core lesson components: learning objective, activate prior
knowledge, concept development, skill development, guided practice, relevance, lesson closure,
and independent practice (Hollingsworth & Ybarra, 2009). See Table 6 for EDI Lesson Design.
Table 6
EDI Lesson Design
EDI Lesson Design
Lesson Component Lesson Component Description
Learning Objective (LO) Focused statement describing the skill and concept of the lesson.
• Written Learning Objective with Concept and Skill
• Footnoted vocabulary definition
• CFU questions
• Sentence Frame
Activate Prior Knowledge (APK) Quick review of skill or experience with a statement about how it is relevant to the
learning objective.
• Text for students to read
• Definitions, if necessary
• Footnoted vocabulary definition
• Universal Experience Or Sub-skill Review
• Matched problems for teacher and students
• CFU questions
• Sentence frame
• Connection of APK to new learning
Concept Development (CD) Concept definitions and big ideas supported by labeled examples. Precise academic
language established. Questions to Check for Understanding.
• Written Concept definitions
• Labeled Examples
• Non-Examples, if applicable
• CFU questions
• Academic text for student to read
• Graphic organizer, if applicable
• Footnoted vocabulary definition
• Sentence Frames
• Additional resources (word bank, tables, illustrations, etc.)
Skill Development/Guided
Practice (SK/GP)
Concept-based steps using academic and content vocabulary. One problem for the
teacher to Model (Skill Development) followed by a matching problem for students to
work step-by-step (Guided Practice).
• Written Steps
• Rule of Two (Matching problems – one for teacher model, and student guided
practice)
• CFU questions
• Academic text for student to read
• Graphic organizer, if applicable
• Footnoted vocabulary definition
• Sentence Frames
• Additional resources (word bank, tables, illustrations, etc.)
Relevance (REL) Personal, academic, and real-life reasons why the lesson is important to learn.

48
• Written Relevance reasons
• Examples to illustrate the reasons
• CFU questions
• Footnoted vocabulary definition
• Academic text for students to read
• Sentence frames
Closure (CL) A skill-based problem, concept-based problem, and writing summary to provide
evidence of learning.
• Concept Closure Questions
• Skill Closures Questions
• Summary Closure (What did you learn today?)
• Academic text for student to read
• Assessment – Type Closure
• Footnoted vocabulary definition
• Sentence frames
• Additional resources (word bank, tables, illustrations, etc.)
Independent Practice (IP) Students practice what they have just been taught. Teacher works with students who
are not successful. (Classroom)
• Problem that match the lesson
• All problem types form the lesson are included
• Footnoted vocabulary definition
• Additional resources (word bank, tables, illustrations, etc.)
Homework (HW) Students practice what they were taught with no support. (Outside of the classroom)
Periodic Review (PR) Distributed Practice (day 1, 2, 7, 30)
Enhanced Selected Response, Extended Constructed Response
• Problem that match the lesson
• Three Periodic Reviews after the initial day 1 practice
• Footnoted vocabulary definition
• Additional resources (word bank, tables, illustrations, etc.)
Assessments Use results to improve or modify instruction.
Student Support Anticipate student needs.
Note. Source: Hollingsworth & Ybarra, 2018, p.12; EDI Workshop PowerPoint, 2019.
In addition to the lesson components, EDI incorporates the use of lesson delivery
strategies throughout the lesson. These strategies are not related to any specific lesson
component aforementioned above. Rather, these strategies are integrated to facilitate teaching
and learning. See Table 7 for EDI Lesson Delivery Strategies.

49
Table 7
EDI Lesson Delivery Strategies
EDI Lesson Delivery Strategies
Strategy Strategy Description
Student Engagement Creating Academic Student Engagement
• Pronounce with Me
• Track with Me
• Read with Me
• Gesture with Me
• Pair-Share
• Attention Signal
• Whiteboards
• Complete Sentences
Checking Understanding Rule of Two – pairing of two problems: one for the teachers and the other for the
student
Teaching Strategies • Explain
• Model thinking
• Physical demonstration
Effective Feedback • Provide cues and prompts.
• I’ll come back to you.
• De-escalate to multiple choice.
• Explain your thinking.
• Read the answer.
• Two consecutive wrong answers, reteach.
• Pair-share again.
Vocabulary Development Content and Academic Vocabulary Development
English Learner Support Content Access Strategies to make
English easier to understand
Language Strategies to promote English
language acquisition
• Comprehensible Delivery
• Context Clues
• Accessible Text
• Vocabulary Development: Content
Vocabulary, Academic Vocabulary,
Support Vocabulary
• Language objective: Listen, Speak,
Read, Write

Differentiation and Scaffolding Adjust sub-skills and time for diverse learners.
Cognitive Strategies Help students remember and retrieve information.
Student Support Identify students for grade-level or sub-skill support
Note. Source: Hollingsworth and Ybarra, 2018, p.12; EDI Workshop PowerPoint, 2019.
To put it all together, the EDI Circle summarizes how the EDI lesson design components
and EDI lesson delivery strategies work in tandem to support students’ learning. The lesson
delivery components are divided into two sections: (a) Teacher delivery of the lesson preparing
students to learn the lesson skills and concepts, and (b) Teacher delivery of the lesson presenting

50
the skills and concepts students need to learn. Simultaneously, the above-mentioned EDI lesson
delivery strategies are used throughout the two phases of the lesson delivery components. See
Figure 12.
Figure 12
EDI Circle

Note. Source: Hollingsworth & Ybarra, 2018.
Cognitively Guided Instruction
How CGI Research Began. Cognitively Guided Instruction (CGI) is based on the
Vygotsky’s (1978) idea that children begin mathematical learning before they begin formal
education. Carpenter et al., (2000) uses Vygotsky’s idea as the bases for Cognitively Guided
Mathematics Instruction. Carpenter et al., (2015) provide the following definition of Cognitively
Guided Instruction: CGI is based on an integrated program of research focused on: (a) the
development of students’ mathematical thinking, (b) instruction that influences student’s
mathematical thinking development, (c) teachers’ knowledge and beliefs that influence their

51
instructional practices, and (d) the way that teachers’ knowledge, beliefs, and practices are
influenced by their understanding of students’ mathematical thinking.
CGI began over 30 years ago and was initially intended to be a project lasting three years
to educate teachers solely in a professional development setting to convey how students learn
and think about mathematics. However, the initial intent shifted when the authors’ realized that
unless the knowledge is implemented, the potential impact of the knowledge is not actualized.
Additionally, the authors understood that the knowledge teachers bring to the table regarding the
practical implementation, that is, “the practicalities of teaching and about children” was needed
to implement the CGI framework (Foster, 2000). At this time, the authors shifted their approach
and began focusing on implementing their research in the classroom. The authors were amazed
at the incredible impact on children’s learning that their work had started (Foster, 2000).
CGI’s Philosophy and Purpose. CGI is a student-centered methodology, and according
to Carpenter et al., (1989), it is grounded in three tenets: (a) instruction must be based on what
each student knows, (b) instruction should take into consideration how children’s mathematical
ideas develop naturally, and (c) children must be mentally active as they learn mathematics (p.
203). To meet the expectations of the three tenets, teachers are challenged to make changes that
involve three main factors: beliefs, knowledge, and practice (Fennema & Nelson, 1997, p. 255).
Figure 13 demonstrates how beliefs, knowledge, and practice influence students’ learning.

52
Figure 13
Knowledge, Beliefs, and Practices

Teachers’
Knowledge

Teachers’
Decisions

Classroom
instruction

Students’
Cognition

Students’
Learning

Teachers’
Beliefs

Students’
Behavior

Note. Source: Fennema, et al., 1989, p. 204.
CGI Framework. CGI is a student-centered approach to teaching math that focuses on
problem-solving and mathematical thinking, as teachers facilitate student learning by asking
questions and using problem-solving strategies during instruction (Heinemann, n.d., section 1;
Carpenter et al., 1999). CGI begins with teachers connecting new learning with students' prior
knowledge and building on their natural number sense and intuitive approaches to problem-
solving (Carpenter et al., 1999). CGI utilizes scenarios of problem-solving to address the skills
all students need for addition, subtraction, multiplication, and division. These skills are taught in
context as opposed to being taught in isolation (Carpenter et al., 1999). A goal of CGI is to help
students with the application of the “intuitive, analytic modeling skills exhibited by young
children to analyze problem situations” to help students “avoid some of their most glaring
problem-solving errors” (Carpenter et al., 1999, p. 55). CGI is not a math program or curriculum,
but rather, a way of listening to students, asking questions, and engaging students in thinking for
the purpose of uncovering and expanding students' mathematical understanding (Heinemann,
n.d., section 1).

53
CGI is built upon four levels of instruction developed by the founders of the CGI
methodology. The four levels of instruction are designed to track and categorize the progress in
which teachers are making during the implementation of a new mathematics concept. See Table
8 for the Levels of CGI Instruction.
Table 8
Levels of CGI Instruction
CGI – Four Levels of Instruction
Level Level Description
Level
1
• Teaching revolves around an adopted textbook
• Teacher believe students must be shown step-by-step what to do
• Provide few, if any, opportunities for student to problem solve or share their thinking
Level
2
• Teacher relies less on the adopted textbook
• Teachers begin to allow and even solicit discussions around other ways to solve problems
• Provides limited opportunities for students to problem solve or share their thinking
• Elicits or attends to students’ thinking or uses what they share in a limited way.
Level
3
• Teacher provides opportunities for students to problem solve and share their thinking.
• Teacher begins to elicit and attend to what students share, but do not used what is shared to make
instructional decisions.
Level
4-A
• Teacher provides opportunities for students to solve a variety of problems.
• Teacher elicits their thinking and provides time for sharing their thinking.
• Instructional decisions are usually driven by general knowledge about his or her students’ thinking, but
not by individual students’ thinking
Level
4-B
• Teacher provides opportunities for students to be involved in a variety of problem-solving activities.
• Teacher elicits students’ thinking and attends to students’ sharing their thinking and adapts instruction
according to what is shared.
• Instructional is driven by teachers’ knowledge about individual students in the classroom.
Note. Source: Carpenter et al., 1999; 2003; 2015.
The goal of CGI is for teachers to reach Level 4-B. At this level, the pinnacle of CGI, that
teachers are focused on the individual needs of each student. Carpenter et al., (1989) points out
that level 4-B denotes that the teacher continuously grows in their understanding of their
students. At this level, teachers can easily switch out of whole-class instruction to instruction that
is tailored to a particular student’s needs and then back again to whole-class instruction
(Fennema & Nelson 1997).

54
The authors of CGI posit “young children’s conceptions of addition, subtraction,
multiplication, and division are quite different from adults” (Carpenter et al., 2015, p. 1). With
CGI’s focus on helping elementary students learn mathematics using their intuitive way of
understanding mathematics, the founders of CGI studied and analyzed elementary school
students worldwide, resulting in multiple large-scale data collection. Through such extensive
data, patterns on how young students intuitively understand mathematics emerged. The CGI
authors developed a framework describing the overarching principles underlying young
children’s thinking in mathematics with the collected data. CGI framework describes young
students’ thinking and learning in mathematics through a unifying structure of two components
that are necessary to create a range of ways to organize lessons and support for student: (1)
Strategies – for solving problems, and (2) Types of Problems. See Table 9 for the CGI
framework structural components.
Table 9
CGI: Framework Components
CGI Framework Components
Types of Problems
Strategies – for
solving problems
Single Digit Numbers Multidigit Numbers
• Direct Modeling
• Counting Strategies
• Flexible Choice of
Strategies
• Derived Facts/Number
Facts
• Relational Thinking

Addition and Subtraction
● Join (unknowns: Result, Change,
Start)
• Separate (unknowns: Result,
Change, Start)
• Part-Part-Whole (unknowns:
Whole, Part)
• Compare (unknown: Difference,
Compared Set)
Multiplication and Division
• Multiplication
• Measurement Division
• Partitive Division
Multidigit Problems
• Direct Modeling with Ones
• The development of Base-ten Number
Concepts
• Direct Modeling with Tens
• Invented Algorithms: Incrementing,
Combining the Same Units, and
Compensating

Note. Source: Carpenter et al., 1999; 2003; 2015.

55
The implementation of the CGI framework is dependent on teachers developing
classroom practices that includes: (a) Posing Problems, (b) Eliciting Student Thinking, (c)
Principles for Eliciting Thinking, and (d) Levels of Engagement. See Table 10.
Table 10
CGI: Classroom Practices
CGI – Developing Classroom Practices
Teacher Action Teacher Action Description
Posing Problems
• Unpacking the Problem
• Students Share Ideas About a Familiar Context
• Making Sense of Problems
Eliciting Student Thinking

• Eliciting Student Thinking After Students Have Solved the Problem
• Asking a Series of Follow-up Questions
• Eliciting Student Thinking When Student Solutions are Incomplete or Incorrect
• Eliciting Student Thinking from Student Written Work
• Eliciting Student Thinking to Extend Mathematical Ideas
Principles for Eliciting Student
Thinking
• Consistently ask student to share their thinking.
• Find ways for each student to explain his thinking to you or other students.
• Follow up with specific questions drawing from what the student shared or did.
• Support students to work all the way through the details of their strategies.
• Ask about correct, incorrect, and incomplete strategies.
• Watch for students to tell or show you that they are ready to be supported to
adapt their strategy or try a new one.
• Listen, observe. Try to not impose your ideas on students.
Levels of Engagement
• Comparing an idea to another student’s idea
• Attending to the details of another student’s ideas
• Building on or adding to another student’s ideas
Note. Source: Carpenter et al., 1999; 2003; 2015.
A typical CGI classroom begins with teachers posing a problem to see what students
would do. CGI authors leave the implementation open to teachers to move forward with their
own ideas about getting started (Carpenter et al., 2015). However, for teachers who rely on some
direction about getting started, CGI provides the following steps seen in Table 11.

56
Table 11
CGI: Getting Started
How to Get Started with CGI
Teacher Action
Teacher Action Description
Teachers Choose a Problem
Teachers can choose a problem from this book or write one of their own.
Keep in mind, students will need to be able to understand the context and objects in
the problem.
Teachers Choose the Students
Teachers choose the students they will work with. Many teachers start by posing
problems to the whole class. Others prefer to start with a small group of students.
Teachers Pose a Problem
Teachers provide manipulatives: blocks, counters.
Before posing the problem: teachers tell the students that they want students to solve
the problem in a way that makes sense to the students.
Teachers read the problem aloud.
Teachers unpack the problem at the beginning of the lesson to support sense making
around the context, and it is not to help students come up with a strategy or an answer.
Teachers may or may not have the students produce a written representation of how
they solved the problem.
Teachers may provide a written copy of the problem to each student or have the
problem written on their board if it is helpful. Teachers may need to read the problem
several times.
If student do not know how to get started solving the problem:
• Check to ensure that students understand the story.
• Give students some time to think. Students can talk with another student and come
back.
Teachers can think about how to make the problem easier for the student(s) – perhaps
using smaller numbers.
Teachers Observe Students
Solving a Problem
Teachers observe their students solving the problem. Many teachers find it useful to
just observe without saying anything as their students solve the problem during the
first few problem-solving sessions.
What Teachers do After Students
Solve a Problem
After students are done solving the problem, or teachers can try the following:
• Ask students to share how they solved the problem with a partner.
• Ask a few students to present to the class how they solved the problem.
• Collect the students’ written work (if there was any) and think about what questions
you have for these students. Return to the students the next day and ask them these
questions.
Collect students’ written work (if there was any) and choose two or three students to
present their strategies to the class the following day.
Note. Source: Carpenter et al., 1999; 2003; 2015.
Strategies for Solving Problems. Within the framework of CGI, there are four strategies
for solving problems and are built upon a hierarchy of complexity moving from concrete to
abstract way of thinking: (1) Direct Modeling Strategies, (2) Counting Strategies, (3) Flexible
Choice of Strategies, and (4) Derived Facts/Number Facts Strategies. Students begin solving

57
problems by working with manipulatives such as cubes, their fingers, and written representation
to model the action and relationship stated in the problem. Using concrete objects and drawing is
the first level in the hierarchy of complexity referred to as Direct Modeling (Carpenter et al.,
1999). Over time, these physical modeling strategies lead to more efficient counting strategies,
flexible choice of strategies, and derived facts and number facts, a progression of an increased
degree of abstractness in the forms of solving problems. Once students experience different
problem types and strategies to solve various problems, they learn to develop and use invented
algorithms and relational thinking strategies to solve more complex problems. With the frequent
use of the more abstract levels of solving problems with recalling numbers facts and derived
algorithms, students become very efficient when dealing with very large numbers (Carpenter et
al., 1999). See Table 12 for the four components of problem-solving strategies in CGI.
Table 12
CGI: Components of CGI Mathematics Instruction
CGI – Four Strategies in Solving Problems
Strategies Teacher Actions Student Actions
Direct Modeling
• Act out the problem
• Follow the sequence of steps for completing the
problem
• Use a paper-and-pencil presentation
• Use manipulatives
• Use composition or decomposition of numbers
• Pose problems
• Listen to students’ thinking
• Ask questions
• Make decisions based on
students’ understanding
• Differentiate problems
• Help make connections
• Help represent
mathematical thinking
• Facilitate
• Solve problems
• Speak (explain)
• Listen (ask questions)
• Challenge themselves to
think deeply
• Think flexibly
• Learn from one another
• Make connections
• Represent mathematical
thinking
• Comparing an idea to
another student’s idea
• Attending to the details
of another student’s
ideas
• Building on or adding to
another student’s ideas
Counting Strategies
• Use the following strategies: counting-all, counting -
on, & counting back
Flexible Choice of Strategies
• Flexibly choose from the possible addition &
subtraction, and multiplication & division strategies
to solve the problem
Derived Facts/Number Facts
• Recall basic facts for addition, subtraction,
multiplication, and division
• Use invented or alternative algorithms
• Use derived facts, Base-ten number concept,
doubles or near doubles

58
Note. Source: Adapted from Sencibaugh et al., 2016; Carpenter et al., 1989.
Types of Problems. The CGI framework addresses two specific groups of numbers: (a)
single-digit, and (b) multidigit problems. Within the two categories of numbers the four
operations – addition, subtraction, multiplication, and division are carried out. See Figure___
Within the single-digit numbers, the types of problems are dependent on the correlating
unknowns tied to the operation groups of addition and subtraction or multiplication and division
are organized into the following. See Table 13.
Table 13
Single Digit – Types of Problems
Single Digit Numbers – Types of Problems
Addition and Subtraction
Problem Types Unknowns
Join Result, Change, Start
Separate Result, Change, Start
Part-Part-Whole Whole, Part
Compare Difference, Compared

Multiplication and Division
Problem Types Unknowns
Multiplication Total
Partitive Division Amount per group
Measurement Division Number of groups

Note. Source: Carpenter et al., 1999; 2003; 2015.
Single-digit Numbers – Direct Modeling Strategies for Solving Problems. As shown in
Tables 14, 15, and 16, depending on the unknowns of the single digit operations, students’ use of
strategies varies.
Table 14
Direct Modeling – Addition and Subtraction
Direct Modeling – Addition and Subtraction in Single Digit Problems
Problem Type Direct Modeling Strategy Description
Join (Result Unknown)
Ellen had 3 tomatoes. She picked 9 more
tomatoes. How many tomatoes does Ellen have
now?
Joining All
Student constructs a set of 3 objects and a set of 9 objects, then finds
the answer by counting all the objects in the two sets.
Join (Change Unknown)
Chuck has 3 dollars. How many more dollars
does he need to buy a stuffed animal that costs
12 dollars?
Joining To
Student constructs a set of 3 objects, then adds objects to this set until
there is a total of 12 objects. Student finds the answer by counting the
number of objects added.
Join (Start Unknown) Trial and Error

59
Deborah had some books. She went to the
library and got 9 more books. Now she has 12
books altogether. How many books did she have
to start with?
Student constructs a set of objects, then adds a set of 9 objects to the set
and counts the objects in the resulting set. If the set final count is 12,
then the number of objects in the initial set is the answer. If it is not 12,
student tries a different initial set.
Separate (Result Unknown)
There were 12 seals playing. 9 seals swam away.
How many seals were still playing?
Separating From
Student constructs a set of 12 objects, then removes 9 objects. Student
finds the answer by counting the remaining objects.
Separate (Change Unknown)
There were 12 people on the bus. Some people
got off. Now there are 3 people on the bus. How
many people got off the bus?
Separating To
Student counts out a set of 12 objects, then removes objects from the
set until the number of objects remaining is equal to 3. Student finds the
answer by counting the removed objects.
Note. Source: Carpenter et al., 2015, p.22.
Table 15
Counting
Counting – Addition and Subtraction in Single Digit Problems
Problem Type Counting Strategy Description
Join (Result Unknown)
Ellen had 3 tomatoes. She picked 9 more tomatoes. How
many tomatoes does she have now?
Counting On from First
Student begins counting with 3 and continues on 9 more
counts. The answer is the last number in the counting
sequence.
Join (Result Unknown)
Ellen had 3 tomatoes. She picked 9 more tomatoes. How
many tomatoes does she have now?
Counting On from Larger
Student begins counting with 9 and continues on 3 more
counts. The answer is the last number in the counting
sequence.
Join (Change Unknown)
Chuck has 9 dollars. How many more dollars does he need to
save to buy a stuffed animal that costs 12 dollars?
Counting On To
Student counts forward starting from 9 and continues until
reaching 12. The answer is the number of counting words int
eh sequence.
Separate (Result Unknown)
There were 12 seals playing. 3 seals swam away. How many
seals were still playing?
Counting Down
Student counts backward starting from 12. The sequence
continues for 3 more counts. The last number in the
counting sequence is the answer.
Separate (Change Unknown)
There were 12 people on the bus. Some people got off. Now
there are 3 people on the bus. How many people got off the
bus?
Counting Down To
Student counts backward from 12 and continues until
reaching 3. The answer is the number of counting words in
the sequence.
Note. Source: Carpenter et al., 2015, p.27.

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Table 16
Direct Modeling – Multiplication and Division
Direct Modeling – Multiplication and Division in Single Digit Problems
Problem Type Direct Modeling Strategy Description
Multiplication
Bart has 4 boxes of pencils. There are 6 pencils in each
box. How many pencils does Bart have altogether?
Grouping
Make 4 groups with 6 counters in each group. Count
all the counters to find the answer.
Measurement Division
Art has 24 pencils. They are packed 6 pencils to a box.
How many boxes of pencils does he have?
Measurement
Put 24 counters into groups with 6 counters in each
group. Count the groups to find the answer.
Partitive Division
Bart has 6 boxes of pencils with the same number of
pencils in each box. Altogether he has 24 pencils. How
many pencils are in each box?
Partitive
Divide 24 counters into 6 groups with the same
number of counters in each group. Count the counters
in one group to find the answer.
Note. Source: Carpenter et al., 2015, p.55.
Students gradually replace direct modeling with counting strategy when solving
problems, as seen in Figure 14 and Figure 16. However, the counting method to solve
multiplication and division problems is rather difficult for the students. As a result, students
generally do not use the counting strategy as a method in the early stages of solving
multiplication and division problems (Carpenter et al., 2015). Students also learn to solve
problems with strategies that do not follow the structure of the problem. They develop an
understanding of part-whole relationships, which allow them to think about the problem
holistically in terms of the parts of the problem (Carpenter et al., 2015). Understanding part-
whole relationships enables students to be more flexible in their strategies and solve problems
that are challenging to model with direct modeling or counting strategies. As students’ progress
through understanding part-whole concepts, they can achieve thinking at the level of derived
facts and number facts, as well as relational thinking.
Multidigit Numbers. CGI authors point out, in the past, students having base-ten number
concepts is a prerequisite to students adding, subtracting, multiplying, and dividing multidigit

61
numbers (Carpenter et al., 1999). However, this has proven to be invalid. Carpenter et al., (1999)
argue that “as long as students can count, they can solve problems involving two-digit numbers
even when they have limited notions of grouping by ten” (p.63). The authors of CGI support the
students’ use of direct modeling and counting with ones to solve problems with two- and three-
digit numbers, as the process provides a context for the learners to develop an understanding of
base-ten number concepts. Therefore, as the base-ten concept increases, students can use this
knowledge to solve multidigit problems. See Table 17.
Table 17
Multidigit Problem-Solving Strategies
Problem
Strategies
Counting by Ones Counting by Tens Direct Place Value
Multiplication
John has 6 pages of stickers.
There are 10 stickers on
each page. He also has 4
more stickers. How many
stickers does he have in all?
Makes 6 groups of counters
with 10 counters in each
group. Adds 4 additional
counters and counts the set
by ones.
Counts, “10, 20, 30, 40, 50,
60, 61, 62, 63, 64,” keeping
track on fingers.
Says, “64. 6 tens is 60 and 4
more is 64.”
Measurement Division
Mary has 64 stickers. She
pastes them in her sticker
book so that there are 10
stickers on each page. How
many pages can she fill?
Counts out 64 counters and
puts them into groups with
10 in each group. Counts the
number of groups.
Counts, “10, 20, 30, 40, 50,
60,” putting up a finger with
each count. Counts fingers
with each count. Counts
fingers to get answer of 6.
Says, “6. There are 6 tens in
60.”
Note. Source: Carpenter et al., 2015, p.89.
As students progress in understanding the base-ten concept, it transfers to using direct
modeling with tens. It then eventually leads to incrementing invented algorithms, combining the
same units invented algorithms, compensating invented algorithms, and written representation of
invented algorithms. Additionally, young learners, either individually or in a whole class setting,
invent their algorithms. It is said that not all invented algorithms are correct, “but the benefit of
children inventing their own algorithms is they have the basis for understanding and correcting

62
their errors” (Carpenter et al., 2015, p.110). See Figure 14 for the list of strategies used for
solving multidigit problems.
Figure 14
Strategies for Solving Multidigit Problems
Strategies for Solving Multidigit Problems
Direct Modeling with Ones
The development of Base-ten Number Concepts
Direct Modeling with Tens
Incrementing Invented Algorithms
Combining the Same Units Invented Algorithms
Compensating Invented Algorithms
Written Representation of Invented Algorithms
Note. Source: Carpenter et al., 1999; 2003; 2015.
Students solving multidigit addition and subtraction problems involves the problem types
and the use of invented algorithm strategies. The problems are organized into: (a) Join – Result
Unknow, (b) Separate – Result Unknown, and (c) Join – Change Unknown. The strategies used
to solve the three types of addition and subtraction multidigit problems are: (a) Incrementing –
students add or subtract in increments, e.g. in increments of tens or fives, and the ones, (b)
Combining the Same Units – students add or subtract by combining the same units, e.g., tens
together, and ones together, and (c) Compensating – students add or subtract by compensating
number to make tens or fives, and then add or subtract the value that was compensated
(Carpenter et al., 1999; 2003; 2015). See Table 18.

63
Table 18
Invented Algorithms
Invented Algorithms– Addition and Subtraction Multidigit Problems
Problem Incrementing Combining the Same Units Compensating
Join (Result Unknown)
Paul had 28 strawberries in
his basket. He picked 35
more strawberries. How
many strawberries did he
have then?
“20 and 30 is 50, and 8 more
is 58. 2 more is 60, and 3
more than that is 63.”
“20 and 30 is 50. 8 plus 5 is
like 8 plus 2 and 3 more, so
it’s 13. 50 and 13 is 63.”
“if I change 28 to 30, I have
to take 2 from 35. 30 plus 33
is 63.”
Separate (Result Unknown)
Paul had 83 strawberries in
his basket. He gave 38
strawberries to his friend.
How many strawberries did
Paul have left?
“83 take away 30 is 53 and
take away 3 is 50. Then take
away 5 more. That’s 45.”
“80 take away 30 is 50. 3
take away 8 makes 5 more to
take away. 50 take away 5 is
45.”
“83 take away 38 is the same
as 85 take away 40. That’s
45.”
Join (Change Unknown)
Paul has 47 strawberries in
his basket. How many more
strawberries does he have to
pick to have 75 altogether?
“47 and 3 is 50 and 20 more
is 70. So that’s 23, but I need
5 more, so it’s 28.”

“47 [pause], 57, 67. That’s
20. 67 and 3 is 70, and 5
more is 75. So, 8 and then
20, 28.”
Combining the Same Units is
not commonly used for Join
(Change Unknown).
“If it were 45, it would be 30.
But it’s 47, so it’s 2 less. 28.”
Note. Source: Carpenter et al., 2015, p.108.
Theoretical Framework
Although the CCSS-M resolved the question of what should be taught in mathematics at
each grade level, it left open how the math should be taught (Munter et al., 2015). The general
concern is that CCSSM is another version of a list of items to cover rather than a framework on
which cohesive strong teaching and learning are made explicit to the practitioners (Russell,
2012). The CCSSI (2010) states “These Standards do not dictate curriculum or teaching
methods”, nor do they offer a teaching sequence or connect across various topics (CCSSI 2010,
p. 5).
How teachers engage students in mathematics is critical, as the focus is on students’
actions related to the SMPs. Equally critical to the study are the teachers’ actions in mathematics
instruction that ultimately activate students’ learning and result in increased student achievement

64
in mathematics. Additionally, the SMPs suggest how students should engage in the content and
cognitive process of mathematics. Teachers’ understanding of how to guide the cognitive process
is paramount in maximizing the students’ ability to mathematically practice what is expected.
Students’ actions in the learning process are maximized when “effective teaching engages
students in meaningful learning through individual and collaborative experiences that promote
their ability to make sense of mathematical ideas and reason mathematically” (NCTM 2014, p.
7).
To counter the lack of teacher action in preparing students to be college ready in
mathematics, teachers need to understand how to operationalize the implementation of the SMPs
into a pedagogical process. The aim of this study is to address the abstractness of
operationalizing the SMPs through a structured process. To that end, a systematic structure to
interpret the findings and the impact of this research is provided in the following theoretical and
conceptual frameworks: (a) Schoenfeld’s Theory of Teaching-in-Context theoretical framework
addresses the interconnectedness of teachers’ knowledge, beliefs, and goals, which affect the
teachers’ actions in a pedagogical space, and (b) A conceptual framework such as Revised
Bloom’s Taxonomy provides a structured approach to operationalize the implementation of the
SMPs by focusing on specific teachers’ and students’ actions involved in particular cognitive
process categories and levels within a pedagogical space.
Impacting Teachers’ Actions Through Knowledge, Beliefs, and Goals
Schoenfeld (1998) asserts in his Theory of Teaching-in-Context that teachers are the
activators of their knowledge, beliefs, and goals that impact their moment-by-moment decisions
regarding the subject matter to be taught through the use of a pedagogical approach in the
classroom. According to Schoenfeld, these moment-by-moment decisions made by teachers

65
ultimately impact students' learning outcomes. The particular activation level of the teachers’
knowledge, beliefs, and goals is an indication of how important what it is to be taught is at the
time. In the Theory of Teaching-in-Context, there are two core assumptions underlying the
model: (a) the activation levels of knowledge, beliefs, and goals at any moment will be assigned
so that the highest priority knowledge, beliefs, and goals are consistent and mutually supportive;
and (b) actions taken by the teacher will be selected in a way to be consistent with the teacher's
current knowledge, beliefs, and goals (Schoenfeld, 1998, p. 3). Figure 15 provides a pictorial
representation of the components of Schoenfeld’s Theory of Teaching-in-Context.
Figure 15
Model of Schoenfeld’ s Theory of Teaching-in-Context

Teachers’
Knowledge

Teachers’
Decision-Making
(Actions)
and
Mechanism (Resources
and Tools)

Teachers’
Beliefs

Teachers’
Goals
Note. Source: Adapted from Toward a Theory of Teaching-in-Context, by A. H. Schoenfeld, 1998.
Represented in the three outer boxes in Figure 15 are the components of the model:
knowledge, beliefs, and goals. The center box represents the teachers’ decision making through
their actions and the use of resources and tools such as a pedagogical process. The model
demonstrates the interconnectedness of teachers' knowledge, beliefs, and goals, which impacts

66
their instructional practices in preparing students with the subject matter, i.e., preparing students
to be college ready in mathematics.
Knowledge
According to Schoenfeld (2019), the knowledge component of the model is defined as
teachers having a profound and deep understanding of subject matter that translates into a
profound pedagogical knowledge. Schoenfeld (1998) highlights three types of knowledge: (a)
subject matter knowledge, (b) general pedagogical knowledge, and (c) pedagogical content
knowledge.
Subject Matter Knowledge. Subject matter knowledge is the teachers’ knowledge that
transcends the facts, terms, and concepts of a particular discipline. Borko and Putnam (1996)
support Schoenfeld’s assertion by stating that the knowledge of subject matter is the “knowledge
of organizing ideas, connections among ideas, ways of thinking and arguing, and knowledge
growth within the discipline is an important factor in how they will teach the subject” (p. 676).
Extending Schoenfeld’s assertion about teachers’ subject matter knowledge, based on the
expectations of CCSSM, both content standards and standards for mathematical practices need to
be addressed in the manner described by Borko and Putnam (1996). In this study, the focus is on
teachers’ knowledge of standards for mathematical practices.
General Pedagogical Knowledge. General pedagogical knowledge is teachers’
knowledge is more than the basic instructional routines. Borko and Putnam (1996) echo
Schoenfeld’s view by stating that general pedagogical knowledge is “about teaching, learning,
and learners that transcends a particular subject matter domain. It includes…classroom
management, instructional strategies for conducting lessons and creating learning environments,
and more fundamental knowledge and beliefs about learners, how they learn, and how that

67
learning can be fostered by teaching” (p.675). The words of Schoenfeld (2019) and Borko and
Putnam (1996) capture the importance of general pedagogical knowledge the teachers should
possess and apply toward preparing the students with a particular subject matter, i.e., the
components of the CCSSM.
Pedagogical Content Knowledge. Pedagogical content knowledge, which encompasses
the teachers’ knowledge about: (a) the purpose for teaching a subject matter…the nature of the
subject and what is important for the students to learn, (b) knowledge of students' understandings
and potential misunderstandings of a subject area ... [including] preconceptions, misconceptions,
and alternative conceptions about topics such as division of fractions, negative numbers, (c)
knowledge of curriculum and curricular materials, and (d) knowledge of strategies and
representations for teaching particular topics (Borko & Putnam, pp. 676-677).
Schoenfeld (1998) emphasizes that teacher actions in the classroom are fundamentally
shaped by the knowledge they bring to a given situation, and the intellectual resources are the
teachers’ knowledge base. According to Schoenfeld (1998), knowledge, beliefs, and goals are
interdependent, as “the beliefs serve as a backdrop for what the teacher is trying to accomplish -
the goals. The teacher has accessed various kinds of knowledge in the service of those goals and
has a set of expectations about the ways things are likely to unfold” (Schoenfeld, 1998, p. 16).
Beliefs
According to Schoenfeld (1998), the beliefs component of the model is defined as the
beliefs a teacher has relevant to a particular segment of instruction. Schoenfeld (1998) asserts
that beliefs can be interpreted as “mental constructs that represent the codification of people’s
experiences and understandings” (p.19), and he continues by stating how these beliefs shape how

68
teacher perceive various circumstances, “what they consider to be possible or appropriate in
those circumstances, the goals they might establish in those circumstances” (p. 19).
Goals
According to Schoenfeld (1998), the goals component of the model is defined as the
“relevant goals regarding content, community, and individual students held by a teacher…These
range from very long-term goals (overarching pedagogical, content, and social goals for the
whole of instruction) to very short-term goals (e.g., very specific content goals for specific
subsegments of instruction)” (p.32).
Interconnectedness of Knowledge, Beliefs, and Goals
Schoenfeld (1998) emphasizes the interconnectedness of the components of knowledge,
beliefs, and goals in his Theory of Teaching-in-Context, as it is the interconnectedness of the
components that shapes instruction. Schoenfeld (1998) establishes the interconnectedness of the
beliefs that affect teachers’ actions in the classroom, and he emphasizes the four beliefs “must be
examined in a comprehensive model of teaching” (p. 25). The beliefs are: (a) beliefs about the
nature of subject matter (in general and with regard to the specific topics being taught), (b)
beliefs about the nature of the learning process (both cognitive and affective), (c) beliefs about
the nature of the teaching process and the roles of various kinds of instruction, and (d) beliefs
about particular students and classes of students (Schoenfeld, 1998). For this study, greater
emphasis will be placed on beliefs (a) and (c); however, a brief description of beliefs (b) and (d)
will be provided to offer comprehensive understanding of the interconnectedness of the
components of the Theory of Teaching-in-Context.

69
Nature of Subject Matter in Mathematics
According to Schoenfeld (1998), the first belief affecting teachers’ actions in instruction
is their belief about the nature of subject matter in mathematics. The belief about the nature of
subject matter is directly connected to the teachers’ knowledge base in a particular subject or
content, which is foundational in supporting the teachers’ beliefs in establishing the importance
of teaching the subject or content. In this study, the subject matter is the SMPs. Teachers need to
first develop a strong knowledge base in the SMPs. The development of a strong teacher
knowledge base in the SMPs will activate the teachers’ beliefs regarding the importance of
teaching the standards. With a strong knowledge base, along with a strong belief that the SMPs
should be taught, teachers will begin to establish goals regarding how to teach the SMPs.
Regarding the interconnectedness of the components of Schoenfeld’s model, Cobb (1986) points
out that “beliefs are allocated the link between goals and the actions arising as a consequence of
them: The general goals established, and the activity carried out in an attempt to achieve those
goals can therefore be viewed as expressions of beliefs. In other words, beliefs can be thought of
as assumptions about the nature of reality that underlie goal-oriented activity” (p.406).
Schoenfeld (2003, 2006) states that a teachers’ beliefs influence the priority and pursuance of
goals when planning for instruction, so any shift in a teachers’ goals indicates the beliefs he or
she may hold.
Nature of the Learning Process in Mathematics
According to Schoenfeld (1998), the second belief affecting teachers’ actions in
mathematics instruction is the teachers’ beliefs about the affective and cognitive processes in
mathematics teaching and learning. In a presentation at Berkeley University, Schoenfeld (2011)
emphasized that students who are learning in any academic discipline are not merely “learning

70
fact and procedures, but developing particular habits of mind”, which enables them to use their
knowledge “productively” (para. 2). Schoenfeld also emphasizes the importance of teaching
students habits of good metacognitive control in their problem solving and work processes.
Developing students’ habits of mind is key to the affective domain. McLeod (1985; 1992)
asserts that the affective domain significantly contributes to students’ performance in
mathematics and should be considered in the planning and delivery of mathematics instruction.
Teachers’ consideration of the affective domain is critical because it is often necessary to change
students’ beliefs about mathematics. Hart (1989) states, "a positive or negative attitude toward
mathematics could be inferred from one's emotional reaction to mathematics, one's behavior in
approaching or avoiding mathematics, and one's beliefs about what mathematics is and how it
may be used" (p. 39). Students who struggle with mathematics often perceive mathematical tasks
as a threat to their self-esteem (Cemen, 1987), or the students feel a strong sense of anxiety about
mathematics that interferes with their ability to complete mathematical problems (Suinn, 1969).
Nature of the Teaching Process in Mathematics
According to Schoenfeld (1998), the third belief affecting teachers' actions in
mathematics instruction is the teachers' beliefs about the nature of the teaching process and the
roles of various kinds of instruction. Shulman (1987) states that teaching “begins with a teacher's
understanding of what is to be learned and how it is to be taught…[and] proceeds through a
series of activities during which the students are provided specific instruction and opportunities
for learning. Teaching ends with new comprehension by both the teacher and the student.
Although this is certainly a core conception of teaching, it is also an incomplete conception.
Teaching must properly be understood to be more than the enhancement of understanding” (p.7)

71
The specific instructional components of the SMPs are critical to this study, and at the
core of what needs to happen as a result of instruction is the students’ development of critical
thinking and problem-solving skills. Schoenfeld (2011) defines “problem solving as confronting
a situation that does not have a ready answer” (Section 3). Schoenfeld (2011) describes four
categories of knowledge that determine a teachers’ success in the area of problem-solving in
mathematics. These four areas include: (a) The teachers’ knowledge base: The knowledge base
referenced here is about the teachers’ ability to look beyond differentiating instruction. The
teachers needs to be able to evaluate the interpretive filters students have developed about
mathematics, to recognize how the students know and understand the mathematics content, and
to be able to intervene when students’ understanding falters (Schoenfeld, 2011, section 4); (b)
The teachers’ problem solving strategies: Teachers need to know and understand how to teach
problem-solving strategies in order for students to learn problem-solving strategies (Schoenfeld,
2011, section 5); (c) The teachers’ control and monitoring of self-regulation (i.e. metacognition):
Teachers need to know and understand how to teach metacognitive control in order for students
to learn metacognitive control of their problem-solving processes (Schoenfeld, 2011, section 6);
and (d) The teachers’ beliefs and practices: Teachers need to know and understand how to
implement enriching classroom practices, in a disciplinary context, in order for students to learn
more productive beliefs and behaviors (Schoenfeld, 2011, section 7).
Beliefs About Students
According to Schoenfeld (1998), the third belief affecting teachers’ actions in
mathematics instruction is the teachers’ beliefs about particular students and classes of students.
Shulman (1987) stated that “knowledge of learners and their characteristics” is a main
component of teachers’ knowledge” (p. 18). For this study, there will not be an in-depth look at

72
Schoenfeld’s belief about students beyond how students may be viewed within a teacher-
centered or student-centered classroom. Schoenfeld (1987) describes how student learning may
be viewed within a teacher-centered class, where students are “passively receiving knowledge, as
receptacles into which mathematical knowledge is poured, as blank slates entering the
classroom”. On the contrary, a student-centered classroom student learning where students are
often engaged in collaborative work involves sense-making by building on students’ prior
knowledge (Walters et al., 2014).
Operationalizing Teachers’ Actions Through Structured Knowledge
There is not much research that focuses on teacher actions. Hunter (2008) states,
“effective teaching is not part of the equation when it comes to school improvement.” The focus
is on “curriculum alignment, selection of textbooks, analysis of student achievement data,
selection of intervention programs, technology, and everything but improving the daily
interactions of teachers and students in the classroom” (p.134). Marzano (2007) posits that
“while factors influence student learning, the greatest contributor to student achievement is
classroom instruction,” making the point that what a teacher does in the classroom is important
(p.16). The CCSSM initiative provides autonomy and freedom for state and local school districts
to determine the how in accomplishing the expectation set forth in the CCSS Initiative (CCSSI,
2010). With autonomy and freedom there is a heightened responsibility to achieve the student
outcomes, in light of the high expectations, high-stakes accountability, and the school
dashboards (Marzano, 2007). Currently, the autonomy and freedom to work out the how has not
supported the level of student achievement needed to meet the college readiness expectations of
CCSS-M.

73
A need at this time is a structure to accurately understand students’ actions – what
students are doing in the learning process, as well as teachers’ actions – what teachers are doing
to activate students’ actions (Bloom, 2001). As simple as it may sound, the paradigm shift is in
understanding that there are two actors and actions that are always present in the pedagogical
space, the teacher and the student. The confusion brought on by the opposing views in the
teacher/student-centered dichotomy is that one actor, either the teacher or the student, become
invisible in the pedagogical space. Once we understand that there are always two actors in a
pedagogical space, and actions from both actors are essential, this creates a basis to think about
each actor’s actions that become one. Each actor has an important role. The teacher’s role is to
activate the student’s thinking, and the student’s role is to be activated and think at the various
cognitive processing levels to learn the content. Once this is made clear, then we can begin to
effectively use the pedagogical space to teach and learn regardless of whether it is a teacher-
centered or a student-centered approach.
Summary
In summary, this chapter provided an overview of the history of mathematics with a
focus on curriculum content, pedagogical approaches, and examining the influence of
progressivism and constructivism in each era. The chapter also highlighted each of the key
variables of the study, including: (a) CCSSM—a limited overview on the standards
encompassing both the Mathematical Content Standards and the Standards for Mathematical
Practice, (b) Standards for Mathematical Practices college readiness definition, and (c) EDI and
CGI—the two pedagogies, one student-centered and one teacher-centered were highlighted.
Additionally, the chapter provided an extensive elaboration on Schoenfeld's theoretical
framework on Teaching-in-Context through which the researcher will examine the study.

74
Chapter Three: Methodology
In the pedagogical space, there are two main actors: teachers and students. Teaching and
learning are complementary, interconnected, and interdependent actions of teachers and students
occupying the same pedagogical space where learning takes place (Yang & Lin, 2016).
Regardless of the teacher-centered or student-centered style of pedagogy, the expectation set by
the CCSSM is that teaching and learning result in students who are ready for college in
mathematics before they leave K-12 education (Clements et al., 2013). Therefore, the actions of
teachers and students’ that exemplify teaching and learning in a pedagogical space need to match
the outcomes of students’ preparedness for college readiness intended by CCSS-M expectations.
Marzano and Toth (2014) assert “a major focus of the new standards is the emphasis on higher-
order thinking skills and the ability to solve complex problems” (Marzano & Toth, 2014).
Anderson and Krathwohl (2001) also affirm the importance of engaging students using higher
order thinking to problem solve. However, Anderson and Krathwohl (2001), also highlight that
in order for the students to engage in the higher-ordered thinking in the cognitive process
dimension, students will need the cognitive process skills such as Remembering, Understanding,
Applying, Analyzing, Evaluating, and Creating. What is critical for students’ preparedness for
college readiness in mathematics is that cognitive process engagement intentionally provides
opportunities for the students to exercise the actions necessary for the particular cognitive
process to become the habit of critically thinking and problem-solving.
Overview
This chapter explains the qualitative document analysis methodology used to conduct this
study, which focused on comparing the two selected pedagogies, EDI, and CGI, to the college
readiness definition of CCSS-M. A review of the problem statement, purpose statement and

75
research question is included in this chapter. The research design, sampling, methods of data
collection, and analysis are also described. The final section of this chapter concludes with a
summary of the study.
Restatement of the Problem
The direction has not been clear as to how to teach math that aligns with the CCSS-M
expectations of college readiness (Callan et al., 2006; Opfer et al., 2018; Rothman, 2012; Woods
et al., 2018), and research literature documents the misalignment of focus and the mathematics
skills needed at the high school level (Callan et al., 2006; Cohen, 2008; Rebora, 2013; Schmidt
& Burroughs, 2012). The lack of clarity on how to teach mathematics in K-12 has impacted
students by graduating high school students who are underprepared for college-level
mathematics (Adelman, 1999; Boaler, 1997; Callan et al., 2006; Cohen, 2008; Opfer et al., 2018;
Woods et al., 2018). Finding clarity in understanding the definition of CCSS-M college readiness
and aligning pedagogy with the expectations of college readiness in the CCSS-M with precision
can bring the change that is needed to address the lack of college readiness problem.
Restatement of the Purpose
The purpose of this study was to use the college readiness components in the CCSSM,
specifically the eight attributes of mathematically proficient students and expertise described in
the SMPs and conducted a qualitative comparative document analysis to determine the ways in
which the two selected pedagogies, EDI and CGI, support the college readiness in CCSSM-
SMPs. This study intended to analyze the pedagogical alignment of EDI and CGI by breaking
down the constituent parts of the variables in the following frameworks and definitions:
(a)SMPs; (b)College Readiness in Mathematics Definition According to Post-Secondary
Mathematicians and Educational Researchers; (c) a comprehensive cognitive process framework;

76
and (d) the two pedagogies EDI, and CGI to determine how the pieces are related to one another
and ultimately to the overall structure of the SMPs college readiness. Based on the perspective
from which the problem of students' lack of college readiness in mathematics was viewed, this
study expected to clarify the college readiness in mathematics definition and present a direct path
of understanding the opportunities of SMPs college readiness alignment the pedagogies offered.
It is expected that these findings may provide the practitioners in the field of education with the
knowledge base that will enable them to deliver a college readiness aligned pedagogy in
mathematics as prescribed by the SMPs to the students. Consequently, the practitioners will be in
a position to address the problem of students graduating high school underprepared for college-
level mathematics.
Research Question
The researcher sought to answer the following research question:
Research Question: In what ways do the teacher-centered, EDI and student-centered CGI
pedagogical approaches support the SMPs college readiness in mathematics in the context of
CCSSM?
Research Methodology
The research study enlisted a qualitative Comparative Document Analysis grounded in
Content Analysis Methodology. The content analysis has been used by many researchers for the
purpose of analyzing text’s characteristics, content, and contextual meaning (Lindkvist, 1981;
McTavish & Pirro, 1990; Tesch, 1990). The content analysis is broad in nature and the
methodology encompasses a two-level process. The first level in the analysis process is Manifest
Content Analysis, which utilizes a quantitative process of counting the document’s frequency of
specific words to develop the categories (Maxwell & Chmiel, 2014). For this study, the Manifest

77
Content Analysis process of developing the categories was accomplished through the structure
of the SMPs document. The categories of the SMP numbers that already existed in the
organization of the SMPs served as the analytic categories. The second level in the analysis
process is Latent Content Analysis, a qualitative process that refers to the “process of
interpretation of content” (Hsieh & Shannon, 2005, p. 1284). The study’s goal was to analyze the
content to look for underlying meanings of the words or content within the selected documents
and then connect the college readiness in mathematics expectations of teachers’ and students’
actions. (Babbie, 1992; Hsieh & Shannon, 2005; Mayring, 2000; Seuring & Gold, 2012).
To accomplish the two-level Content Analysis process, the researcher selected Mayring’s
(2008) four steps to frame the Content Analysis Methodology as the foundation to support the
Document Analysis. The four steps are: (a) material collection, (b) descriptive analysis, (c)
categorizing and connecting strategies, and (d) material evaluation. An example of Content
Analysis using Mayring’s (2000) four-step model is exemplified in the Seuring and Gold (2012)
literature review study. Similar to the Seuring and Gold study, this study applied analytic
categories already present within the analyzed documents, which in this study, are the Standards
for Mathematical Practice (SMPs). See Table 19 for Mayring’s Four-step Content Analysis
Model.

78
Table 19
Mayring’ s Four-Step Model
Note. Source: Adapted from Mayring, 2008.
This study’s primary research objective was focused on comparing existing theories and
frameworks based on preconceived categories derived from prior relevant theories, research, and
literature (Cavanagh, 1997; Kondracki et al., 2002). Similar to a qualitative document
comparative analysis is a grounded theory research approach. However, grounded theory is not
appropriate for this study since it does not use intensive open-ended and interactive processes
that simultaneously rely on data collection, coding, and memo writing that leads to theory
development (Strauss et al., 1994).
Sample Selection
The sampling method used in this qualitative study was purposive sampling, which
enabled the researcher to focus on specific documents of interest that would best serve in
answering the research question. Purposive sampling is a sampling method extensively used in
Steps Description
Steps 1 & 2:
Material Selection, and
Descriptive Analysis
Analytic Variables (AV):
Pre-selection of variables made by the researcher.
Linking Variables (LV):
The researcher selected Linking Variables (LV) to connect the similarity and contiguity
aspects of the Analytic Variable (AV).
Step 3:
Categorizing and Connecting
Verifying that the SMPs college readiness in mathematics definitions align with post-
secondary mathematicians’ and educational researchers’ definitions.
Step 4:
Material Evaluation
Cognitive process framework utilized to analyze the SMPs’ cognitive process verbs:
connecting the cognitive process verbs in a selected framework with the cognitive
process verbs within the SMPs framework.
The identification of teachers’ and students’ actions tied to a cognitive process
framework.
Additional identification of teachers’ and students’ actions of the SMPs from another
supporting document.
Final comparison of the analytic variables: two selected pedagogies – EDI and CGI with
the SMPs college readiness.

79
qualitative research to identify and select information-rich documents for the most effective use
of limited resources (Patton, 2002). Purposive sampling relies on the judgment of the researcher
when it comes to selecting the units that are to be studied and are selected according to the needs
of the study; therefore, the alternate name is deliberate sampling (Patton, 1990, 2002). Although
the nature of the sampling does not enable statistical inferences with the broader population, this
type of sampling method was needed to suit the needs of this study (Lund Research Ltd, 2012;
Maxwell, 1996; Yin, 2003). This study was aimed at examining the college readiness alignment,
specifically between the selected two pedagogies in the context of SMPs college readiness. There
are several subtypes of purposive sampling method. The subtype of purposive sampling used in
this study was critical case sampling technique. Critical case sampling is dependent upon the
recognition by the researcher of critical dimensions that make for a critical case. This is
especially important when the study is limited in sample resources (Patton, 2002). Critical case
sampling was selected because it “can be decisive in explaining the phenomenon of interest”
(Lund Research Ltd, 2012, Section 10), which was particularly useful in this qualitative research
study given the researcher’s aim of finding a connection between the analyzed documents.
Although there is a question of generalizing the results to a broader scale, as Patton (1990) states,
“If it can happen there, it can happen anywhere (p.170)”, so it can be argued that critical cases
can help in making logical generalizations (Lund Research Ltd., 2012). The researcher sought to
develop a process that would yield and resemble a standardized method's capacity with which
other pedagogies can be evaluated for their alignment to the SMPs college readiness--that is,
making a teacher-centered pedagogy, EDI, and a student-centered pedagogy, CGI a critical case.
Thereby, the subsequent uses of this study’s process with other pedagogies would then offer a
generalization.

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Data Collection and Analysis
The data collection and data analysis were conducted within a systematic content analysis
of the documents related to college readiness in mathematics by using Mayring’s (2000) four-
step model. The content analysis followed a clear and purposeful process structure (Kassarjian,
1977). Mayring’s four-step model was well suited for this study, as the model has been
successfully used in previous Content Analysis studies (Seuring and Gold (2012).
Building a Content Analysis Framework using Mayring’s Four-step Model
Mayring’s (2000) four-step model consists of the following: (a) Step 1: Selection of
Materials, (b) Step 2: Descriptive Analysis, (c) Step 3: Categorizing Strategy to generate and
structure categories and Connecting Strategy to identify and describe the key relationships that
tie the data together, and (d) Step 4: Material Evaluation.
The key components to the four-step content analysis were the two types of strategies
called categorizing and connecting strategies, which are “respectively based on the identification
of similarity relations and contiguity relations (Maxwell & Chmiel, 2014, p. 22). To compare the
two pedagogies, EDI and CGI, the categorizing and connecting strategies of analysis targeted the
use of linking data, a comprehensive cognitive process framework, which was a critical element
of the overall qualitative data analysis (Maxwell & Chmiel, 2014).
Step 1 & 2: Selection of Materials and Descriptive Analysis
The comparative document analysis began by applying the first two steps in Mayring’s
(2000) Four-Step model to make the sample selections and provide a descriptive analysis of the
documents that were analyzed in the study. The process of selecting samples was targeted and
specific to the research topic and purpose of the study, which is to determine in what ways do the
two selected pedagogies, EDI and CGI, support the college readiness in SMPs. The researcher

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conducted a search and analysis of the existing literature from various search engines such as
ProQuest, JSTOR, Google Scholar, ERIC (EBSCO), ERIC (OCLC), as well as original
documents, such as books written by the original authors of (a) EDI: The Power of the Well-
Crafted, Well-Taught Lesson, 1st and 2nd editions (Hollingsworth & Ybarra, 2009, 2018), and
(b) CGI: Children’s Mathematics Cognitively Guided Instruction (Carpenter et al., 1999),
Thinking Mathematically Integrating Arithmetic & Algebra in Elementary School (Carpenter et
al., 2003), Children’s Mathematics Cognitively Guided Instruction (Carpenter et al., 2015), and
(c) A Taxonomy for Learning, Teaching, and Assessing: A revision of Bloom’s taxonomy of
educational objectives (Anderson & Krathwohl, 2001).
The extant documents related to the existing principles and frameworks of the study
variables were selected and analyzed using critical case sampling technique (Lund Research Ltd,
2012; Maxwell, 1996; Yin, 2003). The study variables were organized into two groups: (a)
Analytic Variables, and (b) Linking Variables. The study began with the preselected analytic
variables followed by the selection of linking variables listed below.
Analytic Variables (AVs) - Preselected Documents. The analytic variables are
variables the researcher pre-selected prior to conducting the analysis to create a critical case for
the comparative document analysis. In this comparative document analysis, all variables are
represented in the form of documents. Therefore, the analytic variables in the study are the
preselected documents.
● AV 1: Standards for Mathematical Practice – analytic variable representing college
readiness definition in mathematics. The expanded description on the SMPs is in Chapter
2. The researcher focused on making sense of explicit and implicit meanings centered
around the components of college readiness as defined by SMPs and was open to

82
discovering the connecting points between the major frameworks that were selected for
this study. The nature of a qualitative document analysis approach calls for an integrated
method, procedure, and technique for locating, identifying, retrieving, and analyzing
documents for the relevance, significance, and meaning (Altheide et al., 2008).
● AV 2: Explicit Direct Instruction, a teacher-centered pedagogy – analytic variable to
determine in what ways the pedagogy supports the SMPs. The resource selected as
preselected documents were in the form of books: Explicit Direct Instruction (EDI) – The
Power of the Well-Crafted Well-Taught Lesson, First Edition (Hollingsworth & Ybarra,
2009); Explicit Direct Instruction (EDI) – The Power of the Well-Crafted Well-Taught
Lesson, Second Edition (Hollingsworth & Ybarra, 2018). The expanded description on
EDI is in Chapter 2.
● AV 3: Cognitively Guided Instruction, a student-centered pedagogy – analytic variable
to determine in what ways the pedagogy supports the SMPs. The resource selected as
preselected documents were in the form of books: Children’s Mathematics Cognitively
Guided Instruction (Carpenter et al., 1999), Thinking Mathematically Integrating
Arithmetic & Algebra in Elementary School (Carpenter et al., 2003), and Children’s
Mathematics Cognitively Guided Instruction (Carpenter et al., 2015). The expanded
description on CGI is in Chapter 2.
In addition to having name recognition, the two pedagogies were selected because they
offer distinct characteristics representing teacher-centered and student-centered pedagogies.
(Patton, 2002). This study aimed to examine the pedagogical alignment to college readiness,
specifically between the selected two pedagogies in the context of SMPs college readiness as a
critical case.

83
Linking Variables (LV) – Selected Documents through Research. Maxwell and
Chmiel (2014) stated that “linking data [variable] involves recognizing substantive...and formal
connections between things'' (p. 24). A formal connection focuses on how things are similar or
different, and a substantive connection focuses on “how things interact within the same space
and time” (Dey, 2016, p. 152). In this qualitative study, the use of linking strategies played a
critical part in connecting the analytic variables’ aspects based on their similarities or interactive
pairing capacities (Dey, 2016).
The linking variables are variables the researcher selected through the research. The
linking variables were selected based on the needs of the study. All of the linking variables in
this document analysis are represented in the form of documents. The researcher selected
Linking Variables (LV) to connect the similarity and contiguity aspects of the Analytic Variable
(AV).
● Various documents were selected that were related to college readiness in mathematics
definitions by the post-secondary mathematicians and educational researchers;
● A comprehensive cognitive process framework to assist in the analysis of the cognitive
process verbs in the SMPs; and
● Supporting documents to assist in the analysis of teachers’ and students’ actions of the
SMPs.
A graphic organizer was used for the purpose of selecting the document samples and
providing a descriptive analysis of the sample documents. The graphic organizer is used to begin
the first two steps in Mayring’s (2000) Four-Step model. See Figure 16 for the graphic organizer
to be used in Steps 1 and 2.

84
Figure 16
Findings: Analytic Steps 1 & 2
Findings: Analytic Steps 1 & 2
Step 1
Selection of
Materials
List of Analytic
Variables and
supporting
documents.
Pre-selected:
1. SMPs
2. EDI
3. CGI
List of Linking Variables and supporting documents.
Selections made during the initial research and
analysis process of the Analytic Variables:
1. Various documents were selected that were
related to college readiness in mathematics
definitions by the post-secondary
mathematicians and educational
researchers.
2. A comprehensive cognitive process
framework was selected to analyze the
SMPs’ cognitive process verbs.
3. Another supporting document was selected
to assist in understanding the SMPs’
teachers’ and students’ actions tied to each
SMP.
Step 2
Descriptive Analysis
Descriptions of the sample documents selected for the analysis.

Step 3: Categorizing and Connecting Strategy
After the sample documents of analytic variables and linking variables were selected, the
researcher applied Step 3 in Mayring’s (2000) Four-Step model. In this process, the initial
comparative document analysis began with comparing and verifying that the SMPs college
readiness in mathematics definitions aligned with:
a. College readiness in mathematics definitions by the post-secondary
mathematicians; and
b. College readiness in mathematics definitions by the educational researchers.
A graphic organizer was used for the purpose of collecting the data and for the data
analysis. The researcher analyzed various documents to accomplish the initial comparative
analysis of verifying the alignment between the SMP’s definition of college readiness in
mathematics with the definition of college readiness mathematics as defined by post-secondary
mathematicians and educational researchers. In the top section of the two-column graphic
organizer, the alignment between the college-ready mathematics definitions will state “aligned”

85
or “not aligned.” Both the right and left columns will display the definitions of college-ready
mathematics cognitive skills students need to have to be successful at the college level
mathematics. The graphic organizer’s left column will contain the analytic category SMP
definition of college readiness in mathematics. The right column will contain the post-secondary
mathematicians’ and Researchers’ definitions of college readiness in mathematics. To verify the
alignment of the definitions within the K-12 college readiness definition as represented in SMPs,
the researcher abstracted the college-ready cognitive process verbs within the post-secondary
mathematicians’ and educational researchers’ definitions of college-ready mathematics. Then,
the researcher checked to verify that the abstracted college readiness cognitive process verbs
were found within the SMPs’ college readiness definition. Finally, the cognitive process verbs
from the selected comprehensive cognitive process framework that connect with the verification
process of college readiness between the SMP and post-secondary mathematicians and
educational researchers were placed at the bottom row of the organizer. See Figure 17 for the
graphic organizer used to verify the SMPs’ college readiness in mathematics definitions.

86
Figure 17
Findings: Analytic Steps 3 & 4a
Findings: Analytic Step 3 of Category 1
College Readiness in Mathematics Definition Alignment
There is alignment of the college readiness correlating cognitive process verbs and
phrases within the definitions of the Post-Secondary Mathematicians, Educational
Researchers, and SMP 1.
Steps 3a & 3b
Verifying that the
SMPs college
readiness in
mathematics
definitions align
with post-secondary
mathematicians’
and educational
researchers’
definitions.

SMP 1: Problem Solving
(CCSSI 2010, p.6)
Post-Secondary Mathematicians
and
Educational Researchers
Step 3a:
SMP—College readiness definition was
placed in the left column. Cognitive
verbs and phrases were highlighted.
Step 3a:
Post-Secondary Mathematicians’ and
Educational Researchers’ definitions
were placed in the right column.
Cognitive verbs and phrases were
highlighted.

Step 3b:
Analytic Findings—Key cognitive verbs
and phrases were placed in the left
column.

Step 3b:
Analytic Findings—Key cognitive verbs
and phrases were placed in the right
column.

Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the Revised Bloom’s Taxonomy

Analysis of cognitive process verbs and/or meaning to
verify alignment between the college readiness
mathematics definitions

87

Step 3c
Cognitive process
framework utilized to
analyze the SMPs’
cognitive process verbs:
connecting the cognitive
process verbs in a
selected framework with
the cognitive process
verbs within the SMPs
framework.
Step 3c
Correlating Cognitive Verbs from the Revised Bloom’s Taxonomy was placed in this
area.

Step 4 Material Evaluation
Following the verification of the alignment between the SMPs definition of college-ready
mathematics with post-secondary mathematicians and educational researchers’ definitions, the
researcher applied Step 4 in Mayring’s (2000) Four-Step model. In this process, the researcher
utilized a selected cognitive process framework to connect the SMPs’ cognitive process verbs of
college readiness in mathematics with the cognitive process verbs and meaning of the cognitive
process framework. Next, the correlating teachers’ and students’ actions tied to the identified
cognitive process verbs were extrapolated from the comprehensive cognitive process framework,
as see in Seen 4b. Additionally teachers’ and students’ actions were extrapolated from the
supporting document, as seen in Step 4c. Finally, in step 4d, the identified teachers’ and
students’ actions were juxtaposed with source documents of the two selected pedagogies to
specifically identify the teachers’ and students’ actions correlating with each of the SMPs. Based
on the teachers’ and students’ actions found or not found will determine whether the pedagogies
support or not support the college readiness expectations of the SMPs. The extrapolation of
teachers’ and students’ actions helped understand what college readiness cognitive process verbs
should look like in a pedagogical space. See Figure 18 the Findings Table for the collection of
data in Step 4 Material Evaluation process.

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Figure 18
Findings: Analytic Steps 4b-4d

Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP
and RBT Connection
Analytic Category 1: Problem Solving
Make sense of problems and persevere in solving them. (CCSSI 2010, p.6)

Step 4b
The identification of
teachers’ and
students’ actions tied
to a cognitive process
framework.

Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in
Connection with SMP 1 (Anderson & Krathwohl, 2001)

3.1 Executing – Teachers provide structured exercises.
Students:
• Carry out known skills, procedural knowledge, and algorithms
to perform the task.

Step 4c
Additional
identification of
teachers’ and
students’ actions of
the SMPs from
another supporting
document.

Step 4c: Additional Teachers’/Students’ Actions in Connection with SMP 1 (O’Connell
& SanGiovanni, 2013)

Students Actions - Determine and articulate what the problem is asking.
Teachers Actions:
• Ask students to restate the problem in their own words.
• Have students turn to a partner to state the problem.

Step 4d
Final comparison of
the analytic variables:
two selected
pedagogies – EDI and
CGI with the SMPs
college readiness
teachers’ and
students’ actions.

Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical
approaches support the SMPs’ college readiness in mathematics in the context of
CCSSM?

0 Actions Identified =
Does not support
1 or More Actions Identified =
Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 1.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Concept Development
• The students are provided with different
approaches to solve a problem.
Skill Development/Guided Practice
• The students are asked process questions
in which they need to determine how they
know which procedures to use in solving
the problems.

Activate Prior Knowledge
Concept Development X
Skill Development/Guided
Practice
X
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews

89

CGI – Examples of teachers’ and students’ actions supporting the SMP 1. Carpenter et
al., (2015)

CGI Framework “X” if Present Description
Teachers Choose a
Problem

Teachers Pose a Problem
• Teachers pose a problem that can easily be
directly modeled.
• Teachers use Unpacking Problems strategy to
engage students in making sense of the context
and action or situation of the problem to ensure
that all students understand the problem.

Teachers Choose the
Students

Teachers Pose a Problem X
Teachers Observe
Students Solving a
Problem
X
What Teachers do After
Students Solve a Problem
X

Additionally, the content analysis Flow Chart, Figure 19 describes the document analysis
process as a visual to support the understanding of the data collection and analysis process. The
cognitive verbs of the college readiness definition of the SMPs initiated the document analysis
flow with the existing SMPs category numbers creating the analytic categories. Then, ends with
the final analysis phase in step 4, the derived teachers’ and students’ actions of SMPs are used as
a reference to determine whether these actions occur in the two selected pedagogical spaces of
EDI and CGI. Additionally, based on the cognitive thinking model, the researcher used the
logical processes of deductive, inductive, and abductive reasoning to (a) discover meaningful
rules within the structure of the variables, (b) make predictions to link the connecting constituent
parts to add to the existing meaning within the structures, and (c) test the newly rediscovered
model to the selected samples from the purposive or a critical case sampling (Douven, 2017;
Reichertz, 2014).

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Figure 19
Document Analysis Flow Chart
Summary
In summary, this chapter provided an overview of the methodology for a qualitative
document comparative analysis approach of investigation to determine the ways in which the
two selected pedagogies, EDI and CGI, supported the SMPs. This chapter reviewed the problem
statement, purpose statement, research question, selecting sampling, building a content analysis
framework using Mayring’s four-step model, and data collection and analysis.

Document Analysis Flow Chart

Document Analysis Starting Point
Preselected Variable:
Cognitive Verbs from the Verified SMPs’ College Readiness Definition
(CCSSIM, 2010)

Categorizing Strategies:
Linking Variable 1 (LV 1):
A comprehensive document
guiding the interpretation of
SMPs

Linking Variable:
A Formal Connection – used similarity relations to connect the
cognitive verbs of the SMPs and a comprehensive cognitive process
framework

Connecting Strategies:
Extrapolation Teachers’ and
students’ actions tied to the
SMPs college readiness
definition
A Substantive Connection – used contiguity relations to connect
Teachers’ and students’ actions tied to the cognitive verbs in a
comprehensive cognitive process framework that correlates with
the SMPs cognitive verbs

Preselected Variables – Comparative Analysis:
In what ways do the teacher-centered, EDI and student-
centered, CGI pedagogical approaches support the SMPs’
college readiness in mathematics in the context of
CCSSM?

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Chapter Four: Results and Findings
The purpose of this study was to add to the practitioner's knowledge and understanding of
SMPs college readiness and offer insight into aligning the mathematics instructional process that
will engage teachers and students in SMPs. It was critical first to understand the definition of
college readiness in mathematics, as defined by post-secondary mathematicians and educational
researchers. Next, the study highlighted the alignment between college readiness in mathematics
as defined by post-secondary mathematicians and researchers in education and the definition of
SMPs. The analytic process continued with connecting the cognitive verbs of the SMPs with
Revised Bloom’s Taxonomy to extrapolate the teachers’ and students’ actions tied to the
cognitive verbs (Anderson & Krathwohl, 2001). Additionally, O'Connell and SanGiovanni’s
(2013) teachers’ and students’ actions of the correlating factors of the SMPs were used to
determine the ways in which the two selected pedagogies, EDI—a teacher-centered approach,
and CGI—a student-centered approach, support the SMPs. This chapter presents the findings
related to the study's research question, which (a) seeks a systematic and structured process to
describe the alignment of college readiness expectations of the SMPs, and (b) to use the
structured process to identify whether the two selected pedagogies consist of teacher- and
students' actions that support the SMP’s expectations.
Statement of the Problem
College readiness has garnered the attention of researchers, state and national government
officials, parents, students, and various other stakeholders who understand the need to prepare
students with the skills needed for post-secondary education (Butrymowicz, 2017; Greene &
Winters, 2005). Many 2-year and 4-year colleges are placing students in remedial courses; a $7
billion dollar a year process that is a financial drain on students, colleges, and taxpayers

92
(Butrymowicz, 2017). This lack of student preparedness for college is especially evident in
national data from the ACT benchmark score for mathematics. Understanding students' lack of
preparedness for college-level mathematics is the first step in finding a solution (ACT, 2019;
Kaye et al., 2006).
What does it mean to be college ready in mathematics? Many have asked this question,
and it is a topic engulfed in a great deal of ambiguity (McCormick & Lucas, 2011). Conley
(2008) asserts that "college readiness can be defined operationally as the level of preparation a
student needs to enroll and succeed – without remediation – in a credit-bearing general education
course at a post-secondary institution that offers a baccalaureate degree or transfer to a
baccalaureate program" (p. 5). Researchers, colleges, universities, and other organizations have
worked to define college readiness in mathematics. The American Diploma Project (ADP)
(2004), funded by a $2.4 million grant, enlisted five states to examine why graduating seniors are
not ready for college. Their report noted that over 70 percent of high school student graduates
entering college never complete a degree. Additionally, more than 60 percent of employers rated
high school graduates' math skills as fair or poor. The ADP highlighted two areas of great
concern regarding high school graduates' knowledge: lack of foundational mathematical content
knowledge and inability to use cognitive process skills to problem solve. The ADP contends that
"algebra continues to be the most fundamental prerequisite for success in college mathematics"
(Ohio Board of Regents [OBR], 2007, p. 12).
Another group from Washington State called The Transition Mathematics Project (TMP)
(2006) collaborated with K-12 education, colleges, and universities to define college readiness.
The TMP findings concluded that high school graduates need "the abilities to communicate,
reason and problem-solve, make connections between mathematical concepts, and relate

93
mathematics to other disciplines and real-life situations” (McCormick & Lucas, 2011, p.6; TMP,
2006).
The importance of mathematics college readiness is further accentuated with the
following: (a) There is a high percentage of students who need remedial classes, (b) The cost of
remediation is insurmountable, (c) There is a growing number of Post-secondary graduates who
are unprepared for the changing workforce, and (d) The role of higher-level mathematics is
increasing in the post-secondary workplace (McCormick & Lucas, 2011). Despite essential and
varied reasons to support college readiness for all graduates graduating from high school is clear.
However, the pathway to preparing for college readiness in mathematics is a complex problem,
and the traditional definition of earning a high school diploma does not define college readiness.
The more meaningful definition of college readiness has less to do with college acceptance and
more to do with the students' ability to thrive and advance through the chosen field of study and
graduate from college on time (Kamin, 2016).
The CCSSM initiative promised to prepare students with the math content knowledge and
critical math processing skills students need to succeed in post-secondary education. However,
the CCSSM is not a prescriptive curriculum, teaching, or assessment method. The CCSSM was
meant to provide the local educational agencies the freedom to make curricular, instructional,
and assessment decisions unique to their communities (CCSSM, 2010). However, the lack of
structure in the CCSSM on how to teach left the mathematics communities challenged and
unable to produce good results in preparing students for college-level math (Marzano & Toth,
2014).

94
Research Question
In this chapter, the researcher presents the results from the structured process that
identified the ways in which the selected pedagogies consist of the teachers’ and students’
actions that support SMP's expectations. The researcher framed the results and data analysis to
increase teachers' knowledge and understanding of the SMPs, utilizing the following research
question: In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical
approaches support the SMPs’ college readiness in mathematics in the context of CCSSM?
Findings: Steps 1 & 2
Steps 1 & 2: Selection of Materials and Descriptive Analysis
In this comparative document analysis, extant documents related to the study variables'
existing principles and frameworks were selected and analyzed using the critical case sampling
technique. The study variables were organized into two groups: (a) Analytic Variables and (b)
Linking Variables. See Figure 20 for the Analytic and Linking Variable
Figure 20
Analytic and Linking Variables
Findings: Analytic Steps 1 & 2
Descriptions of the sample documents selected for the analysis.
Analytic Variables (AV) –
Preselected:
List of Linking Variables (LV) –
Selected in the research process:
AV 1: Standards for
Mathematical Practice
(SMPs) – analytic variable
representing college readiness
definition in mathematics.
AV 2: Explicit Direct
Instruction (EDI), a teacher-
centered pedagogy – analytic
variable to determine the
The researcher selected Linking Variables (LV) to connect the
similarity and contiguity aspects of the Analytic Variable (AV).
The researcher made the following selection decisions during
the initial stages of researching and analyzing the
comparability aspects of Analytic Variables.
LV 1: College Readiness Definition Alignment: Various
documents were selected related to college readiness in
mathematics definitions by the post-secondary
mathematicians and educational researchers.

95
ways in which it supports the
SMPs.
AV 3: Cognitively Guided
Instruction (CGI), a student-
centered pedagogy – analytic
variable to determine the
ways in which it supports the
SMPs.
LV 2: A Comprehensive Cognitive Process Framework:
Revised Bloom’s Taxonomy (RBT), was selected as a
comprehensive cognitive process framework, to analyze
the SMPs' cognitive process verbs and the correlating
teachers’ and students’ actions.
LV 3: Supporting Document: The researcher selected
another supporting document to understand the SMPs'
students' actions tied to each SMP.

Findings (Steps 1 & 2): Analytic Variable 1 (A V 1)
Analytic Variable 1 (Preselected): Standards for Mathematical Practices
The first step in the study began with three preselected analytic variables. The first
document, which is the Standards for Mathematical Practice (SMPs), was central to this study
and served as the definition of college readiness mathematics. The Common Core State
Standards for Mathematics (CCSSM) and the Standards for Mathematical Practice (SMPs) are
the standards documents of the standards movement. The critical importance of these standards
is the reason they are also the central documents for this study. The CCSSM content standards
"define what students should understand and be able to do in their study of mathematics," also,
the SMPs "describe varieties of expertise that mathematics educators at all levels should seek
to develop in their students" (CCCSM, 2010, Sections 1 & 4). In defining why, the SMPs exist,
it is inordinately clear that both teacher and student actions are at play within these standards.
This study deeply examined these teachers’ and students’ actions in light of two specific
pedagogies, EDI and CGI. As stated by the CBMS, “the features of mathematical practice
described in [the CCSSM] standards are not intended as separate from mathematical content.
Findings (Steps 1 & 2): Analytic Variables 2 and 3 (A V 2 and A V 3)
Analytic Variables 2 and 3 (Preselected): Explicit Direct Instruction (EDI) and Cognitively
Guided Instruction (CGI)

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The next two preselected documents were the pedagogies EDI and CGI, which were
selected due to their distinct characteristics representing teacher-centered and student-centered
pedagogies.
Linking Variables (Selected through Research): College Readiness Definitions
The first of the linking variables selected through research included various documents
related to college readiness in mathematics definitions by the post-secondary mathematicians and
educational researchers. In researching for college readiness definitions, the researcher sought to
(a) understand the interrelatedness of the various sectors' definitions of college readiness in
mathematics, (b) to determine if their definitions align with the SMPs, and (c) to verify how
post-secondary mathematicians and educational researchers believe students should be able to
engage with the cognitive processes that are appropriate for learning college-level mathematics.
The literature focused on college readiness in mathematics definition was reviewed to determine
whether there was alignment among the three sectors, post-secondary mathematicians,
educational researchers, and CCSSM.
Findings (Steps 1 & 2): Linking Variable 1 (LV 1)
Linking Variable 1 (Selected through Research): Revised Blooms Taxonomy (RBT)
The researcher required a comprehensive cognitive processing framework for this study
to identify teachers' and students' college readiness actions and students tied to categories and
levels of cognitive processes compared within the selected pedagogies, EDI, and CGI. In the
search for the most comprehensive cognitive processing framework, there were many
frameworks to consider, such as:
1. Revised Bloom’s Taxonomy (RBT);
2. Skemp’s Theory of Mathematical Understanding;
3. APOS Theory;
4. Process-object and the Notion of Reification;

97
5. The Notion of Procept;
6. Metacognitive Notions: Knowledge, Skills, and Experiences;
7. Schoenfeld’s Problem-Solving;
8. The SOLO Taxonomy; and
9. Tall’s Three Worlds of Mathematics.
The search for cognitive processing frameworks resulted in the selection of RBT, which
served as the Linking Variable 1 (LV1) used as the study’s comprehensive cognitive process
framework. The solidification of RBT as the selected framework was, in part, due to the
Radmehr and Drake (2018) study, which focused on comparing various one-dimensional
cognitive theories/frameworks, either leaning on the knowledge side or cognitive processing side
and addressing parts of the two-dimensional RBT. The researcher selected the
theories/frameworks used for comparison in the Radmehr and Drake study due to their extensive
use in mathematics education studies to explore students' mathematical understanding and
learning (Radmehr & Drake, 2018). The comparison process confirmed that each of the theories
and frameworks aligned to some degree with the elements of RBT; however, they were not as
comprehensive in their dimensions. For example, Skemp's Theory of Mathematical
Understanding incorporates RBT conceptual knowledge dimension and aligns procedural and
metacognitive knowledge dimensions. Skemp's theory does not demonstrate alignment with
RBT's factual knowledge dimension (Radmehr & Drake, 2018). See Table 20 for the theory and
framework comparisons to RBT.
Table 20
Revised Bloom’ s Taxonomy: The Knowledge Dimension
Revised Bloom’s Taxonomy: The Knowledge Dimension
Factual Knowledge Conceptual
Knowledge
Procedural Knowledge Metacognitive
knowledge

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Note. Source: Radmehr & Drake, 2018.
Further comparison of Skemp’s Theory of Mathematical Understanding with the
Cognitive Process Dimension reveals that Skemp’s Theory links to the remembering,
understanding, applying, and creating dimensions of RBT; it does not link to the evaluating
dimension. See Table 21 for further theory and framework comparisons to RBT.
Table 21
Revised Bloom’ s Taxonomy: The Cognitive Process Dimension
Skemp’s Theory of
Mathematical
Understanding
-
Incorporates

Aligns & Incorporates

Aligns
APOS Theory - Integrate - Integrate
Process-Object and
the Notion of
Reification
- Aligns Aligns -
The Notion of
Procept
Aligns Aligns Aligns -
Metacognitive
Knowledge, Skills,
and Experiences –
Various Authors
Embodies Embodies Embodies -
Schoenfeld’s
Problem Solving
Relates Relates Relates -
The SOLO
Taxonomy
Integrate Integrates Integrates
Tall’s Three Worlds
of Mathematics
Relates Relates Relates -
Revised Bloom’s Taxonomy: The Cognitive Process Dimension
Remembering Understanding Applying Evaluating Creating
Skemp’s Theory of
Mathematical
Understanding
Links

Links

Links

-
Links

APOS Theory - - - - -
Process-Object and
the Notion of
Reification
- - - - -
The Notion of
Procept
Aligns Aligns - - -
Metacognitive
Knowledge, Skills,
Embodies Embodies Embodies Embodies Embodies

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Note. Source: Radmehr & Drake, 2018.
Radmehr and Drake (2018) assert, "when exploring the teaching and learning of
mathematical concepts, it is common to use one or more conceptual framework, and Revised
Bloom’s Taxonomy seems to be the most detailed, comprehensive, and flexible framework, thus
providing researchers with the most potential for exploring the teaching, learning, and
assessment of mathematics'' (Radmehr & Drake, 2018; p.896). For this study, the researcher used
the Cognitive Process Dimension of the RBT to connect with the SMPs’ college readiness
cognitive process verbs that have been verified to align with the post-secondary mathematicians
and educational researchers. Then, the RBT’s cognitive process verbs that are found to have
connections with the SMPs are used within the RBT framework to identify and describe the
teachers’ and students’ actions, supporting the SMPs. Next, the final step is to determine whether
the specified actions are evident in the two selected study samples, teacher-centered, EDI, and
student-centered, CGI pedagogies.
The two-dimensionality of RBT addresses both the CCSSM content standards and the
SMPs. RBT’s Knowledge Dimension addresses the CCSSM content standards, and RBT’s
Cognitive Process Dimension addresses the SMPs. See Figure 21 for the alignment of CCSSM
and the two dimensions of RBT. This study targeted RBT’s Cognitive Processes Dimension to
focus on the SMPs and not the Content Standards. The researcher used the cognitive verbs and
and Experiences –
Various Authors
Schoenfeld’s
Problem Solving
- - Relates Relates -
The SOLO Taxonomy
Integrates

Integrates

Integrates

Integrates

Integrates

Tall’s Three Worlds
of Mathematics
Relates

Relates

- - -

100
the corresponding actions of teachers and students within RBT to connect the cognitive verbs of
SMPs within the study’s analytics categories.
Figure 21
Alignment of CCSSM and Two Dimensions of RBT

Findings from the Knowledge Dimension. The Knowledge Dimension in the RBT
contained four main categories. The first category is Factual Knowledge: Students must know
the basic elements to be acquainted with a discipline or solve problems in it. Factual knowledge
has two subtypes: knowledge of terminology and knowledge of details and elements (Anderson
& Krathwohl, 2001, p.46). Then the Conceptual Knowledge: The interrelationships among the
basic elements within a larger structure that enable them to function together. Conceptual
knowledge has three subtypes, including knowledge of classifications and categories; knowledge
of principles and generalizations; and knowledge of theories, models, and structures (Anderson
& Krathwohl, 2001, p. 46). Next, Procedural Knowledge: How to do something, methods of
inquiry, and criteria for using skills, algorithms, techniques, and methods. Procedural knowledge
has three subtypes, including knowledge of subject-specific skills and algorithms; knowledge of
subject-specific techniques and methods; and knowledge of criteria for determining when to use
Common Core State Standards -
Mathematics
Revised Bloom’s Taxonomy - Two
Dimensional
Content Standards
Knowledge Dimension
Factual Knowledge
Conceptual Knowledge
Procedural Knowledge
Metacognition Knowledge
Standards for Mathematical
Practice
Cognitive Process Dimension
Remembering
Understanding
Applying
Evaluating
Creating

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appropriate procedures (Anderson & Krathwohl, 2001, p.46). Finally, the Metacognitive
Knowledge: Knowledge of cognition in general and awareness and knowledge of one's cognition.
Metacognitive knowledge has three subtypes: strategic knowledge; knowledge about cognitive
tasks, including appropriate contextual and conditional knowledge; and self-knowledge
(Anderson & Krathwohl, 2001, p.46). Metacognitive knowledge has three subtypes: strategic
knowledge; knowledge about cognitive tasks, appropriate contextual and conditional knowledge;
and self-knowledge. See Table 22 for Revised Bloom's Taxonomy: Knowledge Dimension.
Table 22
Revised Bloom’ s Taxonomy: Knowledge Dimension Structure
Structure of the Knowledge Dimension
of the Revised Taxonomy
A
.
A. Factual Knowledge – The basic elements students must know or be acquainted with within a discipline or solve
problems.
Aa. Knowledge of terminology
Ab. Knowledge of specific details and elements
A. B
.
B. Conceptual Knowledge – The interrelationships among the basic elements within a larger structure that enable
them to function together.
Ba. Knowledge of classifications and categories
Bb. Knowledge of principles and generalizations
Bc. Knowledge of theories, models, and structures
C. C
.
C. Procedural Knowledge – How to do something; methods of inquiry, and criteria for using skills, algorithms,
techniques, and methods.
Ca. Knowledge of subject-specific skills and algorithms
Cb. Knowledge of subject-specific techniques and methods
Cc. Knowledge of criteria for determining when to use appropriate procedures
A. D
.
D. Metacognitive Knowledge – Knowledge of cognition in general and awareness and knowledge of one's
cognition.
Da. Strategic Knowledge
Db. Knowledge about cognitive tasks, including contextual and conditional knowledge
Dc. Self-knowledge
Note. Source: Anderson & Krathwohl, 2001, p. 46.
Findings from the Cognitive Process Dimension. RBT’s cognitive process dimension
has six hierarchical categories presented in verb form, from the low cognitive complexity
remembering to the high cognitive complexity creating. The six cognitive processes are divided
into 19 subcategories (Anderson & Krathwohl, 2001). The cognitive process dimension supports
the goals of retention and transfer. Anderson and Krathwohl point to May and Wittrock’s (1996)

102
definition of retention as “remembering material at a later time in much the same way it was
presented during instruction,” and transfer as “the ability to use what has been learned to solve
new problems, to answer new questions, or to facilitate learning new subject matter” (Anderson
& Krathwohl, 2001, p. 63).
Six cognitive process categories emphasize retention and transfer. The first category and
cognitive process is Remember, which emphasizes retention. Understand, Apply, Analyze,
Evaluate, and Create, the remaining five categories and cognitive processes all facilitate retention
but emphasize transfer. While the six cognitive processes are hierarchical, it does not imply that
any single process is higher than the others. However, it does, for example, "infer that before a
student can conduct an analysis, the first might need to know the methods of analysis, understand
the different elements to review, and consider which method to apply. It is only then that they
will be ready to conduct the analysis itself" (Center for Teaching Excellence, n.d., section 2). In
other words, "it was assumed that mastery of each simpler category was a prerequisite to the
mystery of the next more complex one" (Anderson & Krathwohl, 2001, p. 213). See Table 23 for
the Structure of the Cognitive Process Dimension of the Revised Bloom's Taxonomy.
Table 23
RBT – Cognitive Process Dimension
Structure of the Cognitive Process Dimension
of the Revised Taxonomy
1.0 Remember – Retrieving relevant knowledge from long-term memory.
1.1 Recognizing
1.2 Recalling
2.0 Understand – Determining the meaning of instructional messages, including oral, written, and graphic
communication.
2.1 Interpreting
2.2 Exemplifying
2.3 classifying
2.4 Summarizing
2.5 Inferring
2.6 Comparing
2.7 Explaining
3.0 Apply – Carrying out or using a procedure in a given situation

103
3.1 Executing
3.2 Implementing
4.0 Analyze – Breaking material into its constituent parts and detecting how the parts relate to one another and an
overall structure or purpose.
4.1 Differentiating
4.2 Organizing
4.3 Attributing
5.0 Evaluate – Making Judgements based on criteria and standards
5.1 Checking
5.2 Critiquing
6.0 Create – Putting elements together to form a novel, coherent whole or make an original product
6.1 Generating
6.2 Planning
6.3 Producing
Note. Source: Anderson & Krathwohl, 2001, p. 31.
Findings: Step 3
Categorizing and Connecting Strategy
In step three of this study, the researcher verified whether there was alignment
between the college readiness in mathematics definitions from the post-secondary
mathematicians and the educational researchers and the expectations of college readiness in
mathematics according to the SMPS. The SMP’s are the mathematical processes and practices in
which teachers will be expected to engage their students. (Ma, 1999; Ferrini-Mundy, 2001). The
CCSSM specify eight Standards for Mathematical Practice meant to support the acquisition and
application of content knowledge (NCTM, 2000). Maxwell and Chmiel (2014) assert that in
qualitative data analysis, "similarities and differences are generally used to define categories and
to group and compare data by category" (p. 22). In this study, the analytic categories were
created based on the organization that already existed within the SMPs. The organization with
the SMPs is the eight Standards of Mathematical Practices. Table 24 provides a list of the eight
SMPs, as they are organized in the CCSSM. The researcher categorized the SMPs into the
analytic categories, e.g., SMP 1 in Category 1, SMP 2 in Category 2, etc. The researcher then
initiated Similarity Relations to deconstruct and categorize the post-secondary mathematician's
and educational researcher's definitions of college readiness components in mathematics and

104
then grouped them into each study category. "Similarity-based relations involve resemblances or
common features; their identification is based on the comparison, which can be independent of
time and place" (Maxwell & Chmiel, 2014, p.22).
Table 24
Analytic Categories
Standards of Mathematical Practice Analytic Category
1. Make sense of problems and persevere in
solving them
1: Problem-Solving
2. Reason abstractly and quantitatively 2: Reasoning
3. Construct viable arguments and critique the
reasoning of others
3: Argumentation
4. Model with mathematics 4: Modeling
5. Use appropriate tools strategically 5: Tools
6. Attend to precision 6: Precision
7. Look for and make use of structure 7: Structure
8. Look for and express regularity in repeated
reasoning
8: Regularity

To analyze the definitions and SMPs, the researcher integrated Categorizing and
Connecting strategies in the Document Analysis process to make the connection between the
data found in the analytic categories, the SMPs, and the various documents defining college
readiness in mathematics used in the study, more specifically, college readiness in mathematics
definition according to the post-secondary mathematicians and the educational researchers and
Revised Bloom's Taxonomy. (Maxwell & Chmiel, 2014, p.29; Seuring & Gold, 2011). The
integration of categorizing and connecting strategies provided the researcher with the ability to
deconstruct the various selected seminal documents to highlight the text's underlying meanings.
The researcher then connected various interrelated documents that supported the research

105
question: In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical
approaches support the SMPs’ college readiness in mathematics in the context of CCSSM?
Figure 24 through 31 were the graphic organizer used in the analysis to verify each
SMP’s alignment of the college readiness mathematics definition with the college readiness
mathematics definition as defined by post-secondary mathematicians and educational researchers
through various documents in the research process. The left column contains the SMP definition
of college-ready mathematics for the specific SMP (e.g., SMP 1, SMP 2, etc.). The right column
contains the post-secondary mathematicians' and Researchers' definitions of college readiness in
mathematics. Both the left and right columns state the definitions of college-ready mathematics
cognitive skills students need to have to be successful at the college level mathematics. To verify
the definition within the K-12 college readiness definition as represented in SMPs, the researcher
abstracted the college-ready cognitive process verbs described by the post-secondary
mathematicians and educational researchers. See Figures 24 through Figures 31 for the college
readiness in mathematics definition alignment findings, Analytic Categories 1 – 8.
Findings (Step 3): Analytic Category 1 – Problem Solving
Figure 22
Definition Alignment – Analytic Category 1: Problem Solving
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 1: Problem Solving
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 1.
SMP 1: Problem Solving
(CCSSI 2010, p.6)
Post-secondary Mathematicians and
Educational Researchers
Definition:
• Mathematically proficient students start by explaining
the meaning of a problem and looking for entry points
to its solution.
• They analyze givens, constraints, relationships, and
goals.
Definition:
• Making sense of mathematical procedures for
relational understanding. (Kamin, 2016)
• Mathematical thinking is having the “skills necessary
to “understand ideas, discover relationships among
ideas, draw or support conditions about the idea and

106
• They make conjectures about the solution's form and
meaning and plan a solution pathway rather than
simply jumping into a solution attempt.
• They consider similar problems and try special cases
and simpler forms of the original problem to gain
insight into its solution.
• They monitor and evaluate their progress and change
course if necessary.
• Depending on the problem's context, older students
a) might transform algebraic expressions or
change the viewing window on their graphing
calculator to get the information they need.
b) Mathematically proficient students can explain
correspondences between equations, verbal
descriptions, tables, and graphs
c) or draw diagrams of important features and
relationships, graph data, and search for
regularity or trends.
• Younger students
a) might rely on using concrete objects or pictures
to help conceptualize and solve a problem.
b) Mathematically proficient students check their
answers to problems using a different method,
and they continually ask themselves, "Does this
make sense?"
c) They can understand the approaches of others
in solving complex problems and identify
correspondences between different
approaches.
their relationships, and solve problems involving
their ideas” (Lutfiyya 1998, p. 55-56)
• "College-ready students possess more than a
formulaic understanding of mathematics. They can
apply conceptual understandings to extract a
problem from a context, use mathematics to solve the
problem, and then interpret the solution back into
the context” (Conley, 2008, p. 15)
• “Without problem-solving, it is difficult to transition to
higher forms of mathematical thinking” (Corbishley &
Truxaw, 2010, p. 72).
• “[Students need] abstract skills such as understanding
basics, thinking and reasoning, conceptual
understanding, motivation and
attitude,...algebra,...and problem-solving skills” (Er,
2018, p. 950)
• "...pure mathematics centers on studying the
academic aspect of mathematics: its definitions,
relations, and structure. Upon mastery of these,
students may then be asked to apply their knowledge
to real-world situations that require this type of
mathematics knowledge for a solution” (Cogan,
Schmidt, & Guo, 2019, p. 532)

Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

• Explaining the problem
and Looking for entry
points to its solution
• Make sense of
mathematical procedures
for relational
understanding.

• Conceptualize and Solve
• Analyze
• Monitor and evaluate
• Transform

• Apply conceptual
understandings
• Extract a problem from a
context
• Using known strategies to
solve routine problems
• Use different methods
• Consider analogous
problems

• Use mathematics to solve,
then interpret the solution
back into the context.
• Creating novel strategies
to solve non-routine
problems
• Search for regularity or
trends
• Explain correspondences
(equations, verbal
• Discover relationships
among idea

107
descriptions, tables, and
graphs or draw diagrams
of important features and
relationships, graph data)
• Identify correspondences
between different
approaches.
• Understand the
approaches of others in
solving complex problems
• Draw or support
conditions about the idea
and their relationships
• Make conjectures

• Solve problems involving
their ideas

SMP 1 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:

• Subdomain: 3.1 Executing
o RBT’s Synonyms:
§ Carrying Out)
• Subdomain: 3.2 Implementing
o RBT’s Synonyms:
§ Using
• Subdomain: 2.7 Explaining
o RBT’s Synonyms:
§ Constructing Models)
• Subdomain: 2.6 Comparing
o RBT’s Synonyms:
§ Contrasting
§ Mapping
§ Matching
• Subdomain: 5.1 Checking
o RBT’s Synonyms:
§ Coordinating
§ Detecting
§ Monitoring
§ Testing

The results from the study’s Analytic Category 1 – Problem Solving in connection with
SMP 1 indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Conceptual understanding of math problems;

108
2. Using different methods and strategies to solve problems; and
3. Understanding correspondences and relationships in math problems.
Further analysis demonstrates a connection between the definition’s cognitive processing verbs
and the RBT’s cognitive processing verbs. The most prevalent RBT’s cognitive processing verbs
present in SMP 1 are Executing, Implementing, Explaining, Comparing, and Checking.
Findings (Step 3): Analytic Category 2 – Reasoning
Figure 23
Definition Alignment – Analytic Category 2: Reasoning
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 2: Reasoning
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 2.
SMP 2: Reasoning Abstractly and Quantitatively
(CCSSI 2010, p.6)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students make sense of
quantities and their relationships in problem situations.
• The students bring two complementary abilities to bear
on problems involving quantitative relationships:
a) the ability to decontextualize—to abstract a given
situation and represent it symbolically and to
manipulate the representing symbols as if they
have a life of their own, without necessarily
attending to their referents,
b) and the ability to contextualize, to pause as
needed during the manipulation process to
probing into the referents for the symbols
involved.
• Quantitative reasoning entails habits of creating a
coherent representation of the problem at hand,
• considering the units involved,
• attending to the meaning of quantities, not just how to
compute them, and
• knowing and flexibly using different properties of
operations and objects.
Definition:
• Analysis, interpretation, precision, and accuracy,
problem-solving, and reasoning (Conley, 2008, p.5)
• Analysis is the evaluation of data and other sources or
materials on the grounds of relevance and credibility,
among other things, and interpretation as accurately
describing events. (Kamin, 2016)
• Iteratively increasing the accuracy of approximations.
The accuracy portion relates to SMP 6. (Kamin, 2016)
• “Being able to adapt mathematical concepts and to
think in symbolic ways are two signs of higher-level
mathematical thinking” (Gray et al., 1999, in Corbishley
& Truxaw, 2010, p. 72).
• “…connecting and applying mathematical ideas have
been identified as necessary for building higher-level
mathematical thinking" (Gray et al., 1999, in Corbishley
& Truxaw, 2010, p. 79).
• “College-ready students possess more than a formulaic
understanding of mathematics… use mathematics to
solve the problem, and then interpret the solution
back into the context” (Conley, 2008, p. 15)
Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

• Decontextualize (represent it
symbolically and manipulate the
representing symbols, attending to
their referents)
• Adapt mathematical concepts and
to think in symbolic ways

109
• Creating a coherent representation
of the problem
• Knowing and flexibly using
different properties of operations
and objects.
• Analysis, interpretation, precision,
and accuracy, problem-solving, and
reasoning

• Attending to the meaning of
quantities
• Contextualize (probe into the
referents for the symbols involved)
• Evaluation of data and other
sources or materials
• Interpretation

• Make sense of quantities and their
relationships in problem situations
• Considering the units involved.
• Use mathematics to solve the
problem
• Connecting and applying
mathematical ideas
• Iteratively increasing the accuracy of
approximations.

SMP 2 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:
• Subdomain: 2.6 Comparing
o RBT’s Synonyms:
§ Contrasting
§ Mapping
§ Matching
• Subdomain: 4.1 Differentiating
o RBT’s Synonyms:
§ Discriminating
§ Distinguishing
§ Focusing
§ Selecting
• Subdomain: 4.2 Organizing
o RBT’s Synonyms:
§ Finding
§ Coherence
§ Integrating
§ Outlining
§ Parsing
§ Structuring
• Subdomain: 4.3 Attributing
o RBT’s Synonyms:
§ Deconstructing

The results from the study’s Analytic Category 2 – Reasoning in connection with SMP 2,
indicate an alignment between the college readiness process verbs and phrases within the

110
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Creating the representation of mathematics problems with precision, accuracy, problem-
solving, and reasoning; and
2. Representing mathematical problems symbolically.
Further analysis demonstrates a connection between the definition’s cognitive processing verbs
and the RBT’s cognitive processing verbs. The most prevalent RBT’s cognitive processing verbs
present in SMP 2 are Comparing, Differentiating, Organizing, and Attributing.
Findings (Step 3): Analytic Category 3 – Argumentation
Figure 24
Definition Alignment – Analytic Category 3: Argumentation
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 3: Argumentation
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 3.
SMP 3: Argumentation and Critique
(CCSSI 2010, p.6)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students understand and
use stated assumptions, definitions, and previously
established results in constructing arguments.
• They make conjectures and
• build a logical progression of statements to explore
the truth of their conjectures.
• They can analyze situations by breaking them into
cases and
• can recognize and use counterexamples.
• They justify their conclusions,
• communicate them to others,
• and respond to the arguments of others.
• They reason inductively about data,
• making plausible arguments that consider the
context from which the data arose.
• Mathematically proficient students can also compare
the effectiveness of two plausible arguments,
• distinguish correct logic or reasoning from that which
is flawed, and—if there is a flaw in an argument—
explain what it is. Elementary students
a) can construct arguments using concrete
referents such as objects, drawings, diagrams,
and actions. Such arguments can make sense
Definition:
• “Students need to be taught how to articulate sound
mathematical explanations and how to justify their
solutions. Encouraging the use of oral, written, and
concrete representations, effective teachers model
the process of explaining and justifying, guiding
students into mathematical conventions. They use
explicit strategies, such as telling students how they
are expected to communicate" (Hunter, 2005, p. 453).
• “When guiding students into ways of mathematical
argumentation, it is important that the classroom
learning community allows for disagreements and
enables conflicts to be resolved” (Chapin & O'Connor,
2007).
• Reasoning as constructing well-reasoned arguments,
as well as accepting and providing logical critique.
(Kamin, 2016) Using precision (SMP 6) to draw
accurate conclusions. (Kamin, 2016)
• Explaining and justifying why specific procedures and
techniques work involves engaging in mathematical
communication, Constructing, and critiquing
arguments. (Kamin, 2016)

111
and be correct, even though they are not
generalized or made formal until later grades.
b) Later, students learn to determine domains to
which an argument applies.
• Students in all grades
a) can listen or read others' arguments,
b) decide whether they make sense,
c) and ask useful questions to clarify or improve
the arguments.
Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

Construct arguments:
• Understand and use stated
assumptions, definitions, and
previously established results in
constructing arguments

• Make conjectures
• Build a logical progression of
statements
• Analyze situations and breaking
them into cases
• Recognize and use
counterexamples
• Justify their conclusions,
communicate them to others
• Mathematical argumentation
• Reasoning as constructing well-
reasoned arguments
• Mathematical communication

Respond to the arguments of
others:
• Reason inductively about data
• Compare the effectiveness of two
plausible arguments.
• Distinguish correct logic or
reasoning from that which is
flawed and Explain

• Draw accurate conclusions
• Explaining and justifying why
certain procedures and
techniques work

Support arguments and responses:
• Construct arguments using
concrete referents such as objects,
drawings, diagrams, and actions
• Determine domains to which an
argument applies.
• Listen or read others' arguments,
decide whether they make sense,
and ask useful questions to clarify
or improve the arguments.
• Articulate sound mathematical
explanations and how to justify
their solutions
• Use of oral, written, and concrete
representations
• Accepting and providing a logical
critique
• Constructing and critiquing
arguments

SMP 3 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the

112
Revised Bloom’s Taxonomy:
• Subdomain: 5.2 Critiquing
o RBT’s Synonyms:
§ Judging

The results from the study’s Analytic Category 3 – Argumentation in connection with SMP
3, indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Understand mathematical communication that uses reasoning to construct arguments;
2. Listen to others’ arguments and ask clarifying questions; and
3. Reason inductively to accurately draw conclusions, explain, and justify why a specific
procedure works.
Further analysis demonstrates a connection between the definition’s cognitive processing verbs
and the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent Revised
Bloom’s cognitive processing verb present in SMP 3 is Critiquing.
Findings (Step 3): Analytic Category 4 – Modeling
Figure 25
Definition Alignment – Analytic Category 4: Modeling
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 4: Modeling
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 4.
SMP 4: Modeling
(CCSSI 2010, p.6)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students can apply the
mathematics they know to solve problems in
everyday life, society, and the workplace.
• In early grades, this might be as simple as
writing an addition equation to describe a
situation.
• In middle grades, a student might apply
proportional reasoning to plan a school event or
analyze a community problem.
• By high school,
Definition:
• Analysis, interpretation, precision, and accuracy,
problem-solving, and reasoning (Conley, 2008, p.5)
• Analysis is the evaluation of data and other sources or
materials on the grounds of relevance and credibility,
among other things, and interpretation as accurately
describing events. (Kamin, 2016)

113
• a student might use geometry to solve a design
problem
• or use a function to describe how one quantity
of interest depends on another.
• Mathematically proficient students who can
apply what they know are comfortable making
assumptions and approximations to simplify a
complicated situation, realizing that these may
need revision later.
• They can identify important quantities in a
practical situation and map their relationships
using such tools as diagrams, two-way tables,
graphs, flowcharts, and formulas.
• They can analyze those relationships
mathematically to draw conclusions.
• They routinely interpret their mathematical
results in the context of the situation and reflect
on whether the results make sense, possibly
improving the model if it has not served its
purpose.
Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

• Apply the mathematics they know to
solve problems arising in everyday
life, society, and the workplace.
• Identify important quantities in a
practical situation.
• Writing an equation
• Problem-solving and reasoning

• Apply proportional reasoning, use
geometry, and functions
• Making assumptions and
approximations
• Analyze relationships mathematically
to draw conclusions
• Analysis
• Analysis as the evaluation

• Map relationships using tools:
diagrams, two-way tables, graphs,
flowcharts, and formulas.
• Precision and accuracy

• Interpret and make sense of their
results
• Interpretation - accurately describing
events

SMP 4 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:
• Subdomain: 2.1 Interpreting
o RBT’s Synonyms:

114
§ Clarifying
§ Paraphrasing
§ Representing
§ Translating
• Subdomain: 2.2 Exemplifying
o RBT’s Synonyms:
§ Illustrating
§ Instantiating
• Subdomain: 2.7 Explaining
o RBT’s Synonyms:
§ Constructing Models

The results from the study’s Analytic Category 4 – Modeling in connection with SMP 4,
indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Analyzing mathematical relationships;
2. Interpreting, making sense of, and describing the results; and
3. Apply mathematics to solve real-world problems in life, society, and the workplace.
Further analysis demonstrates a connection between the definition’s cognitive processing
verbs and the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent
Revised Bloom’s cognitive processing verbs present in SMP 4 are Interpreting, Exemplifying,
and Explaining.
Findings (Step 3): Analytic Category 5 – Tools
Figure 26
Definition Alignment – Analytic Category 5: Tools
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 5: Tools
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 5.
SMP 5: Tools
(CCSSI 2010, p.6)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students consider the
available tools when solving a mathematical
problem. These tools might include pencil and paper,
Definition:

115
concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical
package, or dynamic geometry software.
• Proficient students are sufficiently familiar with tools
appropriate for their grade or course to make sound
decisions about when each of these tools might be
helpful, recognizing both the insight to be gained and
their limitations.
• For example, mathematically proficient high school
students
a) analyze graphs of functions and solutions
generated using a graphing calculator.
b) They detect possible errors by strategically
using estimation and other mathematical
knowledge.
c) When making mathematical models, they
know that technology can enable them to
visualize the results of varying assumptions,
explore consequences, and compare
predictions with data.
• Mathematically proficient students at various grade
levels can identify relevant external mathematical
resources, such as digital content located on a
website, and use them to pose or solve problems.
They can use technological tools to explore and
deepen their understanding of concepts.
• “[Student] know when and how to estimate to
determine the reasonableness of answers and can use
a calculator appropriately as a tool, not a crutch”
(Conley, 2007, p. 15).

Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

Select the appropriate tools:
• Consider the available tools when
solving a mathematical problem
(pencil and paper, concrete
models, a ruler, a protractor, a
calculator, a spreadsheet, a
computer algebra system, a
statistical package, or dynamic
geometry software).
• Recognizing the usefulness and
limitations of the selected tools
• Analyze graphs of functions and
solutions generated using a
graphing calculator
• Use mathematical knowledge to
detect errors.
• Use technology to visualize and
deepen their understanding of
concepts.
• Identify relevant external
mathematical resources.
• Using mathematics knowledge
and external tools appropriately
to deepen learning.

SMP 5 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

116

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:

• Subdomain: 2.1 Interpreting
o RBT’s Synonyms:
§ Clarifying
§ Paraphrasing
§ Representing
§ Translating
• Subdomain: 5.1 Checking
o RBT’s Synonyms
§ Coordinating
§ Detecting
§ Monitoring
§ Testing

The results from the study’s Analytic Category 5 – Tools in connection with SMP 5, indicate
an alignment between the college readiness process verbs and phrases within the definitions of
the post-secondary mathematicians, educational researchers, and the SMPs. The most prominent
themes in the alignment across all of the analytic categories include:
1. Using mathematical knowledge and external tools to solve problems;
2. Recognizing the usefulness and limitations of the selected tools; and
3. Identify relevant external mathematical resources.
Further analysis demonstrates a connection between the definition’s cognitive processing
verbs and the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent
Revised Bloom’s cognitive processing verbs present in SMP 5 are Interpreting and Checking.
Findings (Step 3): Analytic Category 6 – Precision
Figure 27
Definition Alignment – Analytic Category 6: Precision
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 6: Precision
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 6.
SMP 6: Precision
(CCSSI 2010, p.6)
Post-secondary Mathematicians and Educational
Researchers

117
Definition:
• Mathematically proficient students try to
communicate precisely to others.
• They try to use clear definitions in discussion with
others and their reasoning.
• They state the meaning of the symbols they choose,
including using the equal sign consistently and
appropriately.
• They are careful about specifying measuring and
labeling axes to clarify the correspondence with
quantities in a problem.
• They calculate accurately and efficiently and express
numerical answers with precision appropriate for the
problem context.
• In the elementary grades,
• students give carefully formulated explanations to
each other.
• By the time they reach high school, they have learned
to examine claims and make explicit definitions.
Definition:
• Recognizing appropriate levels of precision for
various tasks. (Kamin, 2016)
• Using precision to draw accurate conclusions. (Kamin,
2016)
• Iteratively increasing the accuracy of approximations.
(Kamin, 2016)
• Accurately using mathematical terminology, symbols,
ideas, definitions, and arguments. (Kamin, 2016)

Analytic Findings:
Key cognitive process verbs and phrases

Analytic Findings:
Key cognitive process verbs and phrases

Precision in calculations and
performing math tasks:
• Calculate accurately and efficiently
• Express numerical answers with a
degree of precision appropriate for
the problem context
• Carefully formulated explanations
• Examine claims and make explicit use
of definitions
• Consistently and appropriately use
symbols and signs.
• Recognizing appropriate levels of
precision for various tasks
• Iteratively increasing the accuracy of
approximations.

Precision in communication:
• Communicate precisely
• Use clear definitions
• State the meaning of the symbols
• Careful about specifying units of
measure, and labeling axes
• Accurately using mathematical
terminology, symbols, ideas,
definitions, and arguments.
• Draw accurate conclusions

SMP 6 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:

118
• Subdomain: 5.1 Checking
o RBT’s Synonyms:
§ Coordinating
§ Detecting
§ Monitoring
§ Testing

The results from the study’s Analytic Category 6 – Precision in connection with SMP 6,
indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Precision in appropriately expressing and formulating explanations to math problems;
and
2. Accurately using mathematical terminology, symbols, ideas, definitions, and arguments.
Further analysis demonstrates a connection between the definition’s cognitive processing verbs
and the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent Revised
Bloom’s cognitive processing verb present in SMP 6 is Checking.
Findings (Step 3): Analytic Category 7 – Structure
Figure 28
Definition Alignment – Analytic Category 7: Structure
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 7: Structure
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 7.
SMP 7: Structure
(CCSSI 2010, p.8)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students look closely to
discern a pattern or structure.
• For example, young students
o might notice that three and seven more is the same
amount as seven and three more,
o or they may sort a collection of shapes according to
how many sides the shapes have.
• Later, students will see 7 × 8 equals the well-
remembered 7 × 5 + 7 × 3, in preparation for learning
about the distributive property.
Definition:
• Reasoning through the relationships between different
mathematical topics to understand the
interconnectedness of mathematics. (Kamin, 2016)
• Problem solving using known strategies to solve routine
problems and creating novel strategies to solve non-
routine problems. (Kamin, 2016)
• "...pure mathematics centers on studying the academic
aspect of mathematics: its definitions, relations, and
structure. Upon mastery of these, students may then be
asked to apply their knowledge to real-world situations

119
• older students
o In the expression x
2
+ 9x + 14, can see the 14 as 2 ×
7 and the 9 as 2 + 7.
o They recognize the significance of an existing line in
a geometric figure and can use the strategy of
drawing an auxiliary line for solving problems.
o They also can step back for an overview and shift
perspective.
o They can see complicated things, such as some
algebraic expressions, as single objects or as being
composed of several objects. For example, they can
see 5 - 3(x - y)
2
as 5 minus a positive number times a
square and use that to realize that its value cannot
be more than 5 for any real numbers x and y.

that require this type of mathematics knowledge for a
solution” (Cogan, Schmidt, & Guo, 2019, p. 532)

Analytic Findings:
Key cognitive process verbs and phrases
Analytic Findings:
Key cognitive process verbs and phrases
• Look closely to discern a pattern
or structure:
Discover flexibility of numbers:
o more is the same amount as
and can use the strategy
o recognize the significance of
an existing line
Discovering properties:
o preparation for learning
about the distributive
property
• Understand the
interconnectedness of
mathematics

• can step back for an overview
and shift perspective
• Using known strategies to solve
routine problems
• Creating novel strategies to
solve non-routine problems
• can see complicated things, such
as some algebraic expressions, as
single objects or as being
composed of several objects

• Apply definitions, relations, and
structure to solve problems

SMP 7 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:

• Subdomain: 4.2 Organizing
o RBT’s Synonyms:
§ Finding Coherence
§ Integrating
§ Outlining
§ Parsing

120
§ Structuring
• Subdomain: 2.3 Classifying
o RBT’s Synonyms:
§ Categorizing
§ Subsuming
• Subdomain: 2.4 Summarizing
o RBT’s Synonyms:
§ Abstracting
§ Generalizing
a) Subdomain: 2.5 Inferring
o RBT’s Synonyms:
§ Concluding
§ Extrapolating
§ Interpolating
§ Predicting

The results from the study’s Analytic Category 7 – Structure in connection with SMP 7,
indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Looking for patterns and understanding the interconnectedness in mathematics;
2. Using known strategies to solve routine problems and novel strategies to solve non-
routine problems; and
3. Applying definitions, relations, and structure to solve problems.
Further analysis demonstrates a connection between the definition’s cognitive processing
verbs and the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent
Revised Bloom’s cognitive processing verbs present in SMP 7 are Organizing, Classifying,
Summarizing, and Inferring.
Findings (Step 3): Analytic Category 8 – Regularity
Figure 29
Definition Alignment – Analytic Category 8: Regularity
College Readiness in Mathematics Definition Alignment – Step 3 of Analytic Category 8: Regularity
There is an alignment of the college readiness correlating cognitive process verbs and phrases within the
definitions of the Post-Secondary Mathematicians, Educational Researchers, and SMP 8.

121
SMP 8: Regularity
(CCSSI 2010, p.8)
Post-secondary Mathematicians and Educational
Researchers
Definition:
• Mathematically proficient students notice if
calculations are repeated
a) and look both for general methods and for
shortcuts.
• Upper elementary students
a) might notice when dividing 25 by 11 that they
are repeating the same calculations over and
over again and conclude they have a repeating
decimal. By paying attention to the calculation
of slope as they repeatedly check whether
points are on the line through (1, 2) with slope
3,
• Middle school students
a) might abstract the equation (y - 2)/(x - 1) = 3.
b) Noticing the regularity in the way terms cancel
when expanding (x - 1)(x + 1), (x - 1)(x
2
+ x + 1),
and (x - 1)(x
3
+ x2 + x + 1) might lead them to
the general formula for the sum of a geometric
series.
• As they work to solve a problem, mathematically
proficient students maintain oversight of the process
while attending to the details.
• They continually evaluate the reasonableness of their
intermediate results.
Definition:
• Students accustomed to looking for and finding these
consistencies and using them in problem-solving take
responsibility for their knowledge. (Kamin, 2016)
• Students learn to modify and adapt known problem-
solving techniques to solve novel problems through
structure and repeated reasoning. (Kamin, 2016)

Analytic Findings:
Key cognitive process verbs and phrases
Analytic Findings:
Key cognitive process verbs and phrases
• Notice general formulas through the
regularity in the repeated calculations

• Modify and adapt known problem-
solving techniques to solve novel
problems through structure and
repeated reasoning

• Maintain oversight of the process
• Attend to details
• Evaluate the reasonableness of
results
• Looking for and finding consistencies
and using them in problem-solving

SMP 8 College Readiness Definition is Aligned
and Cognitive Process Verbs in Connection with RBT

Correlating Cognitive Verbs from the
Revised Bloom’s Taxonomy:

• Subdomain: 2.5 Inferring
o RBT’s Synonyms:
§ Concluding
§ Extrapolating
§ Interpolating
§ Predicting

122

The results from the study’s Analytic Category 8 – Regularity in connection with SMP 8,
indicate an alignment between the college readiness process verbs and phrases within the
definitions of the post-secondary mathematicians, educational researchers, and the SMPs. The
most prominent themes in the alignment across all of the analytic categories include:
1. Looking for general methods of problem solving through the regularity in the
repeated calculations; and
2. Attending to details Modify and adapt known problem-solving techniques to solve
novel problems through structure and repeated reasoning. Further analysis
demonstrates a connection between the definition’s cognitive processing verbs and
the Revised Bloom’s Taxonomy’s cognitive processing verbs. The most prevalent
Revised Bloom’s cognitive processing verb present in SMP 8 is Inferring.
Findings: Step 4
Material Evaluation
In this section, the analysis juxtaposed and examined the components of SMPs, the
Revised Bloom's Taxonomy, Putting the Practice into Action, and the two selected pedagogies,
EDI, and CGI. The analysis examined teachers' and students' cognitive process actions to
determine the ways in which the two pedagogies support college readiness in mathematics
components of the SMPs. The preselected variables were: (a) the SMPs – college readiness in
mathematics definition central to the study, (b) EDI, a teacher-centered pedagogy, and (c) CGI, a
student-centered pedagogy. The researcher identified the preselected variables to conduct this
study of document analysis to answer the research question: In what ways do the teacher-
centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?
Also, the Revised Bloom's Taxonomy (2001) of teaching and learning was used. Revised
Bloom's Taxonomy is based on the original Engelhart and Bloom (1956), the taxonomy of
educational objectives: a cognitive domain. Engelhart and Bloom (1956) classified categories

123
and learning levels based on cognitive processes that learners engage with when they learn. The
Revised Bloom's Taxonomy (2001) provided the structured process for identifying and
understanding how teachers’ actions support the categories and levels of cognitive thinking for
students to be college ready in mathematics. See Figures 32 through Figures 39 for the teachers’
and students’ actions in the EDI and CGI comparison, Analytic Categories 1 – 8.
Findings (Step 4): Analytic Category 1 – Problem Solving
Figure 30
EDI and CGI Comparison –Analytic Category 1: Problem Solving
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 1: Problem Solving
Make sense of problems and persevere in solving them. (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection with SMP 1 (Anderson &
Krathwohl, 2001)

3.1 Executing – Teachers provide structured exercises.
Students:
• Carry out known skills, procedural knowledge, and algorithms to perform the task.

3.2 Implementing – Teachers engage students in problem-solving that require conceptual
understanding and the appropriate procedure to solve them.
Students:
• Identify procedures they will select and apply to solve the problems.
• Modify as necessary to the selected procedures and apply them to solve the problems.
• Construct procedures based on conceptual knowledge, theories, models, or structures
as guides and apply them to solve the problems.

2.7 Explaining – Teachers provide students with opportunities to construct a cause-and-effect
model to explain a given situation fully. Teachers describe a system for students to use in a cause-
and-effect model. Teachers provide opportunities for students to use Exampling as a cognitive
process in reasoning, troubleshooting, redesigning, and predicting.
• Explaining related to reasoning – students offer a reason to explain the given event.
• Explaining related to troubleshooting – students diagnose what might have gone wrong
in a situation that is not working.
• Explaining related to redesigning – students imagine altering one or more steps or
components to the system to achieve the intended results.
• Explaining related to predicting – students predict how a change in one part will affect a
difference in other parts of the system.

2.6 Comparing – Teachers provide opportunities for students to identify correlations, similarities,
and differences between the two ideas or objects.
Teachers provide students with concept mapping to show students how each part of one object,
idea, problem, or situation corresponds to each part of something else.
Students’ Actions:
• Abstract a rule from the more familiar situation.

124
• Apply the rule to a less familiar situation.
• Use an analogy in their thinking process.
• Identify correlations, similarities, and differences in the knowledge that is familiar to
them.

2.1 Checking – Teachers provide students with opportunities to monitor the consistency or the
inconsistency of the process, procedure, system, or plan.
Students’ Actions:
Students use appropriate tools and investigate: Is it working or not working? Is the conclusion
correct or not correct? If not working, where what, how is it not working? Is this where I should
be in light of what I've done so far?
Step 4c: Additional Teachers/Students Actions in Connection to SMP 1 (O’Connell & SanGiovanni, 2013)

Students’ Actions - Determine and articulate what the problem is asking.
Teachers’ Actions:
• Ask students to restate the problem in their own words.
• Have students turn to a partner to state the problem.

Students' Actions – Find a starting point by understanding mathematical situations.
Teachers’ Actions:
• Use diagrams to model math situations.
• Frequently ask, “What should I do first?” or “How should I get started?”

Students' Actions – Identify an appropriate way to solve the problem.
Teachers’ Actions:
• Discuss familiar problems (When have I seen something like this before? What did I
do?).
• Discuss the efficiency of various strategies (Will it work? Why does the strategy make
sense with this problem? Which strategy is more efficient?).

Students’ Actions – Connect problem situations to abstract representations of the problem (e.g.,
equations, visuals) to clarify the task.
Teachers’ Actions:
• Avoid simply circling keywords and focus on identifying the concepts or actions of the
operations.
• Consistently discuss building appropriate equations to solve problems (What equation
shows this situation?).
• Provides materials (manipulative, paper/pencil, etc.) to allow students to visualize
situations.

Students' Actions – Self-monitor their progress and change directions when necessary; adjust
strategies when having difficulty.
Teachers’ Actions:
• Think aloud to show students how the course is changed when needed during the
problem-solving process.
• Have students talk or write about how they got stuck, and they got unstuck when
problem-solving.

Students' Actions – Demonstrate perseverance and make adjustments until a problem is solved.
Teachers’ Actions:
• Think aloud and acknowledge that everyone feels like giving up at times.
• Share ways students can persevere or/and demonstrate patience when solving
problems (I'm getting frustrated, let me try something else.).
• Ask the student, "Are you getting stuck? What else can you do?"
• Affirm students who are showing perseverance and not giving up.

125
Students' Actions – Articulate the strategies they use to solve problems.
Teachers’ Actions:
• Provide opportunities for partner and group discussions while solving problems.
• Frequently ask students to articulate both orally and in writing how they solved
problems and why they chose their strategy.
• Orchestrate a class sharing time so students can show and talk about how they solved
problems.

Students' Actions – Identify or understand different ways to solve a problem.
Teachers’ Actions:
• Make classroom lists of possible strategies.
• Share alternate strategies when discussing how students solved a problem.
• Encourage students to show two ways to solve a problem.

Students' Actions – Articulate the reasonableness of strategies or solutions.
Teachers’ Actions:
• Frequently ask, “Does your answer make sense? Why?”
• Ask students to predict/estimate before computing the solution to a problem, and then
compare their answer to their estimate to check for reasonableness.
• Ask students to go back to the question and restate the question followed by their
answer. Does the answer work with the question?
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 1.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Concept Development
• The students are provided with different approaches to solve a
problem.
Skill Development/Guided Practice
• The students are asked process questions in which they need to
determine how they know which procedures to use in solving the
problems.
• The students are asked to interpret the solution to a problem.
• The students are asked to explain the correspondence between
equations, verbal descriptions, tables, and diagrams.
• The students are asked to check their answers to determine if they
make sense.
• The students are asked if they agree or disagree with the given
statements and asked to explain.
Periodic Reviews
The students are asked to solve unfamiliar problems.
Activate Prior Knowledge
Concept Development X
Skill Development/Guided Practice X
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews X
CGI – Examples of teachers’ and students’ actions supporting the SMP 1. Carpenter et al. (2015)
CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem
• Teachers pose a problem that can easily be directly modeled.
• Teachers use Unpacking Problems strategy to engage students in
making sense of the context and action or situation of the problem
to ensure that all students understand the problem.
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students Solving a
Problem
X

126

What Teachers do After Students
Solve a Problem
X o Teachers focus on students’ story comprehension.
o Teachers support students to attend to the connection
between the story and mathematics.
o Teachers support each student’s participation and access.
o Teacher support students to learn how to unpack problems
on their own.
o Teachers avoid doing the mathematical work of solving the
problem.
• Students look for the mathematical relationships that are a part of
the story and use them to get started on a solution
Teachers Observe Students Solving a Problem
• Students use strategies to understand the problem and solve, such
as Direct Modeling, Counting, Counting, Flexible Choice of
Strategies, and Derived Facts/Number Facts.
What Teachers do After Students Solve a Problem
• After students have solved a problem, teachers use Eliciting
Students Thinking strategies to make students mathematical
thinking explicit. Prompts used by the teachers: What did you do?
Tell me about your strategy.

Both EDI and CGI are found to positively support SMP 1. In the EDI, teacher-centered
approach, three out of eight components of the EDI lesson cycle were activated: Concept
development, Skill development/Guided practice, and Periodic reviews. In the CGI, student-
centered approach, three out of five components of the CGI Framework were activated: Teachers
pose a problem, Teachers observe students solving a problem, and What teachers do after
students solve a problem.
Findings (Step 4): Analytic Category 2 – Reasoning
Figure 31
EDI and CGI Comparison –Analytic Category 2: Reasoning
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 2: Reasoning
Reason abstractly and quantitatively (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 2 (Anderson &
Krathwohl, 2001)

2.6 Comparing – Teachers provide opportunities for students to identify correlations, similarities, and
differences between the two ideas or objects.
Teachers provide students with concept mapping to show students how each part of one object, idea,
problem, or situation corresponds to each part of something else.
Students’ Actions:
• Abstract a rule from the more familiar situation.
• Apply the rule to a less familiar situation.

127
• Use an analogy in their thinking process.
• Identify correlations, similarities, and differences in the knowledge that is familiar to them.

4.1 Differentiating – Teachers engage students to determine how parts fit into the whole or the overall
structure.
Students’ Actions:
• Distinguish relevant from extraneous materials.
• Use the broader context to determine what is relevant or irrelevant information in problem-
solving.

4.2 Organizing – Teachers engage students to determine how elements fit or function within a system.
Students’ Actions:
• Construct systematic and coherent connections between the different pieces of information.
• Identify the relevant or essential elements and then determine the overall structure within which
the parts fit.

4.3 Attributing – Teachers present materials for the students to determine the underlying view, values, or
intent.
• Ascertain the point of view, values, or intention underlying communications.
• Deconstruct and extend beyond the basic understanding to infer the underlying information from
the presented material.
Step 4c: Additional Teachers/Students Actions in Connection to SMP 2 (O’Connell & SanGiovanni, 2013)

Students' Actions – Make sense of quantities and their relationships in problem situations.
Teachers’ Actions:
• Ask students to identify and describe the data in the problem.
• Ask students to build equations to represent problems.

Students’ Actions – Decontextualize.
Teachers’ Actions:
• Discuss selecting appropriate operations to solve problems (e.g., Would it make sense to add,
subtract, multiply, or divide?).
• Model building appropriate equations to solve problems.
• Use diagrams to model math situations to make it easier to see what is happening in the problem.
Can students draw a diagram to show a word problem for 3 X 5?
• Frequently ask, “What operation makes sense?” or “How should we build an equation to match
this problem?”

Students’ Actions – Contextualize.
Teachers’ Actions:
• Ask students to write a word problem to go with a given equation.
• Consistently ask students to explain equations or diagrams, connecting them to the problem
scenario (e.g., What does the 6 represent in our equation 6 X 3 = 18?).
• Ask students to label answers by referring back to the problem to determine what the quantity
(solution) represents.
• Ask students if the quantity makes sense when referring back to the problem (e.g., Does 3.5714
buses make sense?).

Students’ Actions – Know and flexibly use different properties of numbers.
Teachers’ Actions:
• Model and discuss flexible use of numbers.
• Discuss building appropriate equations to solve problems (i.e., What equation shows this
situation? Is the more than one equation that would represent the problem? Would properties
like commutative, associative, or distributive allow you to create different equations for the same
problem?).

128
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 2.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Concept Development
• The students are presented with multiple representations of
the problem; they consider the units involved and attend to the
meaning of the quantities instead of purely manipulating them.
Skill Development/Guided Practice
• The students can represent a given situation symbolically and
manipulate the symbols to solve a problem.
Activate Prior Knowledge
Concept Development X
Skill Development/Guided Practice X
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews
CGI – Examples of teachers’ and students’ actions supporting the SMP 2. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem
Teachers Pose a Problem
• Students use strategies to understand and solve problems.
Teachers Observe Students Solving a Problem
• Students attend to the critical features of a problem and build
upon and extend the intuitive modeling skills to construct a
model of the situation specified in a problem.
What Teachers do After Students Solve a Problem
• Teachers find ways for each student to explain his/her thinking.
• Teachers prompt student with: Can you tell me how you solved
that? What did you do? Tell me what you did.
• Teachers ask about correct, incorrect, and incomplete
strategies.
• Teachers support students to work all the way through the
details of their strategies.
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students
Solving a Problem
X
What Teachers do After Students
Solve a Problem
X

Both EDI and CGI are found to support SMP 2. In the EDI, teacher-centered approach,
two out of eight components of the EDI lesson cycle were activated: Concept
development and Skill development/Guided practice. In the CGI, student-centered approach,
three out of five components of the CGI Framework were activated: Teachers pose a
problem, Teachers observe students solving a problem, and What teachers do after students
solve a problem.

129
Findings (Step 4): Analytic Category 3 – Argumentation
Figure 32
EDI and CGI Comparison Analytic Category 3: Argumentation
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 3: Argumentation
Construct viable arguments and critique the reasoning of others (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 3 (Anderson &
Krathwohl, 2001)

5.2 Critiquing – Teachers provide students with opportunities to detect inconsistencies in a result or
product by comparing it to the external standards or criteria.
Students Actions:
• Judge and determine the quality, effectiveness, and reasonableness by using a standard criterion.
• Detect the appropriateness of a procedure for a given task or problem.
• Judge which alternative methods is a more effective and efficient way to solve given problems.
Step 4c: Additional Teachers/Students Actions in Connection to SMP 3 (O’Connell & SanGiovanni, 2013)

Students' Actions – Use assumptions, definitions, examples, counterexamples, and previously established
results in constructing arguments.
Teachers’ Actions:
• Model effective arguments.
• Ask probing questions and supply wait time for students to elaborate on their arguments.
• Set high expectations for justifications (expect specificity, examples, and clear reasoning).
• Encourage students to construct arguments using concrete objects, diagrams, examples,
definitions, and data.
• Use, and expect students to use specific math vocabulary.

Students’ Actions – Justify conclusions, communicate them to others, and respond to the arguments of
others.
Teachers’ Actions:
• Set up ongoing opportunities for students to defend their arguments, encouraging them to
support their arguments with specific data and reasoning.
• Ask students to justify solutions, both orally and in writing, including the use of words, pictures,
and numbers.
• Set up opportunities for students to communicate their arguments to others through oral
presentations, partner discussions, or shared written work.

Students’ Actions – Reason inductively about data; make conjectures and build a logical progression of
statements to explore the truth of their conjecture.
Teachers’ Actions:
• Ask students to observe data (e.g., equations, diagrams, charts, tables, graphs) and draw
conclusions based on what is observed (e.g., display a row of parallelograms and ask students to
describe a parallelogram based on observations).
• Model and ask students to use if/then reasoning (e.g., if a polygon has 4 sides, then it is a
quadrilateral).

Students' Actions –Listen to or read others' arguments, decide whether they make sense, and ask useful
questions to clarify or improve the arguments.
Teachers' Actions:
• Frequently ask specific questions as students share their answers in class, including, “Do you
agree with…?” “Does that make sense?” Even though Jen did it differently, does her way work?”

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• Promote presentations and sharing sessions so students can share their arguments with others.
• Create student-to-student discussions (e.g., “Do you agree with Katie’s argument? Why or why
not?”).
• Encourage students to ask questions to clarify classmates’ arguments (e.g., “What did you mean
by…?”).
• Create a non-threatening classroom in which students feel safe to share their arguments and
know how to ask clarifying questions and how to disagree with others respectfully.

Students’ Actions – Distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in
an argument, explain what it is.
Teachers’ Actions:
• Give students opportunities to hear and critique each other’s’ thinking.
• Ask students to pinpoint the data or mathematical reasoning that strengthens their arguments.
• Ask students to identify data or reasoning that may be flawed and identify how that data impacts
the argument.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 3.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Concept Development/Skill Development/Guided Practice
• The students are provided with a definition of the concept that
contains critical and non-critical attributes. After the students
have had the opportunity to determine examples that meet the
definition, they are provided with counterexamples to
discriminate between the examples and counterexamples.
• The students are asked questions that clarify examples that
meet the critical attributes and the counterexamples that don't.
They are also asked to justify their reasoning by referring to the
definition of the concept.
Activate Prior Knowledge
Concept Development X
Skill Development/Guided Practice X
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews

CGI – Examples of teachers’ and students’ actions supporting the SMP 3. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem/Teachers Observe Students Solving a Problem/
What Teachers do After Students Solve a Problem
• Students are provided with opportunities to evaluate the
strategy to determine if they agree or disagree with the
reasoning of other students.
• Students look at another student’s strategy in relation to their
own ideas and make a judgement.
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students Solving a
Problem
X
What Teachers do After Students
Solve a Problem
X

Both EDI and CGI are found to support SMP 3. In the EDI, teacher-centered approach,
two out of eight components of the EDI lesson cycle were activated: Concept
development and Skill development/Guided practice. In the CGI, student-centered approach,
three out of five components of the CGI Framework were activated: Teachers pose a

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problem, Teachers observe students solving a problem, and What teachers do after students
solve a problem.
Findings (Step 4): Analytic Category 4 – Modeling
Figure 33
EDI and CGI Comparison Analytic Category 4: Modeling
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 4: Modeling
Model with mathematics (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 4 (Anderson &
Krathwohl, 2001)

2.1 Interpreting – Teachers provide students with opportunities to convert information from one
form of representation to another.
Students’ Actions:
• Provide information in one form for students to change the representation into another
form.
• Students draw a visual representation to model the problem.
• Students change numerical expressions or equations to verbal expressions.
• Students translate the number of sentences expressed in words into algebraic equations
or expressions.

2.2 Exemplifying – Teachers provide engagement activities that help students understand general
concepts or principles, and students can find specific examples to show their understanding.
Students’ Actions:
• Identify and define a concept or principle's features and use these features or parts to
connect it to a more significant meaning. Find specific examples or illustrations of a
concept or principle. (e.g., an equilateral must have three equal sides. Or a polygon with
three sides is a triangle.)

2.7 Explaining – Teachers provide students with opportunities to construct and use a
cause-and-effect model of a system.
Students’ Actions:
• State reasons to explain the given event.
• Diagnose what might have gone wrong.
• Imagine altering one or more steps or components to achieve the intended results.
• Predict how a change in one part will affect a difference in other parts of the system.
Step 4c: Additional Teachers/Students Actions in Connection to SMP 4 (O’Connell & SanGiovanni, 2013)

Students’ Actions – Make models to simplify a situation.
Teachers’ Actions:
• Model the use of diagrams and drawings to represent problems.
• Encourage students to create simple diagrams to show problems.
• Facilitate discussions in which students share multiple ways to model mathematics.
• Encourage students to revise diagrams as needed.

Students’ Actions – Identify models that are most efficient for solving specific problems or
representing specific math ideas.

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Teachers’ Actions:
• Discuss specific models and their value. (e.g., Why use base-ten blocks to show 27? Why
does a 10 X 10 grid work well to represent a decimal?).
• Discuss times when a specific model might be appropriate (e.g., Would a 10 X 10 grid
also be appropriate to model percents? Fractions? Why?).
• Ask students to explain why they chose a particular model.

Students’ Actions – Analyze models and draw conclusions based on what they see.
Teachers’ Actions:
• Consistently ask for their insights after looking at models.
• Ask students to interpret models for their classmates (i.e., describe or explain their
model).
• Have students write about what they learned from their model.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 4.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Independent Practice/Periodic Reviews
• The students are challenged with problems that arise in
everyday life, society, and the workplace.
• The students can identify the pertinent quantities in a given
situation. The students also map the quantities using diagrams,
two-way tables, graphs, flowcharts, and formulas.
• The students interpret their mathematical results in the context
of the situation and reflect if they make sense.
Activate Prior Knowledge
Concept Development
Skill Development/Guided Practice
Relevance
Skill/Concept Closure
Independent Practice X
Periodic Reviews X
CGI – Examples of teachers’ and students’ actions supporting the SMP 4. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem/Teachers Observe Students Solving a Problem/
What Teachers do After Students Solve a Problem
• Teachers first allow students to record answers in anyway
students chose, and then make the utility and clarity of
different ways of recording answers a specific focus of
discussion.
• Students use open number sentence to write as many
conjectures as they can that represent important mathematical
ideas.
• Teachers prompt students to answer, “Can you represent an
even number using a variable? How about an odd number?

Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students Solving a
Problem
X
What Teachers do After Students
Solve a Problem
X

The support for SMP 4 is evident in both EDI and CGI. In the EDI, teacher-centered
approach, two out of eight components of the EDI lesson cycle were activated: Independent
Practice and Periodic Reviews. In the CGI, student-centered approach, three out of five

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components of the CGI Framework were activated: Teachers pose a problem, Teachers observe
students solving a problem, and What teachers do after students solve a problem.
Findings (Step 4): Analytic Category 5 – Tools
Figure 34
EDI and CGI Comparison Analytic Category 5: Tools
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 5: Tools
Use appropriate tools strategically (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 5 (Anderson &
Krathwohl, 2001)

Teachers provide opportunities for the students to convert information from one form of
representation to another.
Teachers provide opportunities for the students to use appropriate tools to change the given
representation into another form.
Students’ Actions:
• Students draw a visual representation to model the problem.
• Students change numerical expressions or equations to verbal expressions.
• Students translate several sentences expressed in words into algebraic equations or
expressions.
Teachers provide opportunities for the students to monitor the consistency or the inconsistency
of the process, procedure, system, or plan.
Students’ Actions:
• Students use appropriate tools and investigate: Is it working or not working? Is the
conclusion correct or not correct? If not working, where what, how is it not working? Is
this where I should be in light of what I've done so far?
Step 4c: Additional Teachers/Students Actions in Connection to SMP 5 (O’Connell & SanGiovanni, 2013)

Students' Actions – Effectively use tools when solving a mathematical problem.
Teachers’ Actions:
• Model the use of tools as we solve problems (e.g., use grid paper to determine areas of
figures, use a ruler to find measures of length, use a calculator to determine products
for complex calculations).
• Make tools available during instruction.
• Provide opportunities for students to practice with tools.
• Discuss ways in which tools help simplify the problem-solving process.
• Ask students which tools might be helpful to solve specific problems.

Students' Actions – Make a sound decision about when certain tools might help recognize both
their advantages and limitations.
Teachers’ Actions:
• Familiarize our students with a variety of tools.
• Discuss why we select certain tools.
• Ask students to identify tools that make sense to solve a problem.
• Discuss the advantages and limitations of specific tools.

Students’ Actions – Detect possible errors by strategically using estimation and other
mathematical knowledge.

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Teachers Actions:
• Ask students to estimate before using tools and then use their estimate to check for
reasonableness.
• Discuss common errors (e.g., how to know which measure makes sense when using a
protractor to measure an angle).

Students' Actions – Use technological tools to explore and deepen their understanding of
concepts and recognize that technology can enable them to visualize and analyze math data.
Teachers’ Actions:
• Provide consistent exposure to technology tools (e.g., virtual manipulatives for students
at all levels, graphic calculators, and dynamic geometry software for middle grades
students).
• Include technology tools during instruction.
• Discuss potential insights gained from technology tools.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 5.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Concept Development
• The students are given precise definitions as tools to compare
mathematical quantities.
Skill Development/Guided Practice
• The students use the formula for perimeter or area as a tool to
solve different types of problems.
Periodic Reviews
• The students use the knowledge of setting-up the proportion
correctly as a tool to solve percent problems.
• The students use geometric software to verify the property
experimentally; Parallel lines are taken to parallel lines.
Activate Prior Knowledge
Concept Development X
Skill Development/Guided Practice X
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews
X
CGI – Examples of teachers’ and students’ actions supporting the SMP 5. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem
• Students use manipulatives, written representations, and problem-
solving strategies, such as Direct Modeling, Counting, Counting,
Flexible Choice of Strategies, and Derived Facts/Number Facts.
• Students begin with intuitive problem solving that makes sense to
them and then learn mathematical symbols as tools for
representing ideas that they already understand.

Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students
Solving a Problem

What Teachers do After Students
Solve a Problem

The support for SMP 5 is evident in both EDI and CGI. In the EDI, teacher-centered
approach, three out of eight components of the EDI lesson cycle were activated: Concept
Development, Skill Development/Guided Practice, and Periodic Reviews. In the CGI, student-

135
centered approach, one out of five components of the CGI Framework were activated: Teachers
pose a problem.
Findings (Step 4): Analytic Category 6 – Precision
Figure 35
EDI and CGI Comparison Analytic Category 6: Precision
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 6: Precision
Attend to precision (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 6 (Anderson &
Krathwohl, 2001)

Teachers provide students with opportunities to test whether or not a conclusion follows from
its premises, whether data support or disconfirm a hypothesis, or whether presented material
contains parts that contradict one another.
Students Actions:
• Use appropriate tools and investigate.
• Monitor the consistency or the inconsistency of the process, procedure, system, or
plan.
• Is it working or not working? If not working, where, what, how is it not working?
• Is the conclusion correct or not correct?
• Is this where I should be in light of what I've done so far?
Step 4c: Additional Teachers/Students Actions in Connection to SMP 6 (O’Connell & SanGiovanni, 2013)

Students' Actions – State the meaning of the symbols they choose.
Teachers’ Actions:
• Discuss and consistently ask students to explain the meaning of symbols.
• Explore varied ways in which symbols might be presented (e.g. 5 X 3 = 15 and 15 = 3 X
5).

Students Actions – Specify units of measure and label axes to clarify the correspondence with
quantities in a problem.
Teachers’ Actions:
• Ask students to label units, quantities, and graphs.
• Expect students to justify labels (e.g., why the area is labeled as square units and
volume is labeled as cubic units).

Students’ Actions – Calculate accurately and efficiently and express numerical answers with a
degree of precision appropriate for the problem context.
Teachers’ Actions:
• Expect accuracy, except in cases where we have asked for estimates or where
estimates make more sense.
• Expect precision as appropriate for the grade-level and content expectations (e.g.,
!
"

inches vs.
#
!$
inches; 5 vs. 5.2 or 5.24).

Students’ Actions – Formulate explanation to each other.
Teachers’ Actions:
• Model specific and thorough explanations.
• Discuss expectations and offer tips for formulating clear explanations.

136
• Allow students opportunities to work with partners to formulate explanations.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 6.

EDI Lesson Cycle “X” if Present Description
Lesson Objective X Throughout the lesson:
• The students are supported by academic and content language
throughout the lesson cycle. They are provided with pair-share and
share out to the whole class opportunities to communicate with
precision.
• The students are supported with procedural steps to check for
precision.
• Throughout the lesson, the teachers check for understanding
process questions for the students to attend to precision.
• Throughout the lesson, the students have opportunities to check
their answers and steps to attend to precision.
Skill/Concept Closure
• The students are given a word bank to allow them to communicate
with precision.
Activate Prior Knowledge X
Concept Development X
Skill Development/Guided Practice X
Relevance X
Skill/Concept Closure X
Independent Practice X
Periodic Reviews X
CGI – Examples of teachers’ and students’ actions supporting the SMP 6. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem/Teachers Observe Students Solving a
Problem/What Teachers do After Students Solve a Problem
• Teachers find ways for each student to explain his/her thinking.
Teacher prompt student with: Can you tell me how you solve that?
Teachers Observe Students Solving a Problem
• Teachers ask about correct, incorrect, and incomplete strategies.
• Teachers support students to work all the way through the details
of their strategies.
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students
Solving a Problem
X
What Teachers do After Students
Solve a Problem
X

The support for SMP 6 is evident in both EDI and CGI. In the EDI, teacher-centered
approach, eight out of eight components of the EDI lesson cycle were activated: Lesson
Objective, Activate Prior Knowledge, Concept Development, Skill Development/Guided
Practice, Relevance, Skill/Concept Closure, Independent Practice, Periodic Reviews. In the CGI,
student-centered approach, three out of five components of the CGI Framework were

137
activated: Teachers pose a problem, Teachers Observe Students Solving a Problem, What
Teachers do After Students Solve a Problem.
Findings (Step 4): Analytic Category 7 – Structure
Figure 36
EDI and CGI Comparison Analytic Category 7: Structure
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 7: Structure
Look for and make use of structure (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 7 (Anderson &
Krathwohl, 2001)

4.2 Organizing – Teachers involve imposing a structure on materials such as an outline, table,
matrix, or hierarchical diagram for students to learn and use appropriately. Teachers engage
students to determine how elements fit or function within a system.
Students Actions:
• Construct systematic and coherent connections between the different pieces of
information.
• Identify the relevant or essential elements and then determine the overall structure
within which the parts fit.

2.3 Classifying – Teachers begin from specific instances or examples that require the student to
find a general concept or principle.
Students’ Actions:
• Determine that something belongs to a category. (e.g., Determine the types to which
numbers belong).
• Detect relevant features or patterns that describe a specific event that connect to the
concept or principle.

2.4 Summarizing – Teachers present information from which the students construct a
representation of the information.
Students’ Actions:
• Determine a general theme or major points.

2.5 Inferring – Teachers present information form which for the students to draw a logical
conclusion.
Students’ Actions:
• Find a pattern within a series of examples or instances.
Step 4c: Additional Teachers/Students Actions in Connection to SMP 7 (O’Connell & SanGiovanni, 2013)

Students' Actions – Look closely to discern a pattern or structure.
Teachers Actions:
• Consistently ask students, “What do you notice?”
• Explore patterns in different ways using hundred charts, addition and multiplication
charts, ratio tables, place-value models, etc.
• Sort and classify shapes based on defining attributes (e.g., 3 sides, 3 vertices, similar
angle measurements).

Students' Actions – Understand math properties and apply them to computations.
Teachers’ Actions:

138
• Provide investigations in which students explore math properties.
• Facilitate discussions about math properties.
• Provide models for the exploration of properties.
• Ask students to construct an argument to prove math properties.
• Use an understanding of properties to help students perform computations with multi-
digit numbers, fractions, decimals, etc.

Students’ Actions – See the flexibility of numbers.
Teachers’ Actions:
• Continually model the composing and decomposing of numbers and shapes.
• Ask students to model the breaking apart and putting together numbers, including whole
numbers, fractions, and decimals, etc.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 7.

EDI Lesson Cycle “X” if Present Description
Lesson Objective Periodic Reviews
• The students answer questions such as, what did you discover
from constructing the triangles in problems 3 and 4? The
students look for and use structure to determine which angles
can be constructed and which angles cannot.
Activate Prior Knowledge
Concept Development
Skill Development/Guided Practice
Relevance
Skill/Concept Closure
Independent Practice
Periodic Reviews X
CGI – Examples of teachers’ and students’ actions supporting the SMP 7. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem
• Students detect and use a combination of strategies to solve a
problem that requires more than one step.
• Student use strategies such as directing modeling with ones and
tens to find the structure for base-ten concepts.
• Students use base-ten concepts to invent algorithms to solve
problems.
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students
Solving a Problem

What Teachers do After Students
Solve a Problem

The support for SMP 7 is evident in both EDI and CGI. In the EDI, teacher-centered
approach, one out of eight components of the EDI lesson cycle was activated: Periodic
Reviews. In the CGI, student-centered approach, one out of five CGI Framework components
were activated: Teachers pose a problem.

139
Findings (Step 4): Analytic Category 8 – Regularity
Figure 37
EDI and CGI Comparison Analytic Category 8: Regularity
Teachers’ and Students’ Actions in the EDI and CGI Comparison Tied to SMP and RBT Connection
Analytic Category 8: Regularity
Look for and express regularity in repeated reasoning (CCSSI 2010, p.6)
Step 4b: Teachers’/Students’ Actions Derived from RBT’s Cognitive Process Verbs in Connection SMP 8 (Anderson &
Krathwohl, 2001)

2.5 Inferring – Teachers present information from which the students' abstract patterns encode
the relevant features, understand the relationships, and solve problems.
• Students can compare the given information and draw a logical conclusion about the
pattern and describe the relationship of the pattern they see.
Step 4c: Additional Teachers/Students Actions in Connection to SMP 8 (O’Connell & SanGiovanni, 2013)

Students' Actions – Notice repetition.
Teachers’ Actions:
• Ask students to observe for repetition.
• Frequently ask, “What do you notice?” “Do you see any patterns?” “Have we seen this
before?”
• Pose problems that draw attention to repetition.

Students' Actions – Select appropriate strategies for solving familiar problems.
Teachers’ Actions:
• Ask students to think about how new problems are like previously solved problems.
• Ask students to use familiar problems as a way to decide on an appropriate strategy.

Students' Actions – Discover shortcuts or generalizations.
Teachers’ Actions:
• Set up investigations in which students gather data and observe for repetition to find
shortcuts.
• Encourage discoveries.
• Ask students to explain shortcuts (formulas, algorithms).
• Compare shortcuts to other methods to see how they are alike and different.
Step 4d: Final Pedagogical Comparison
In what ways do the teacher-centered, EDI and student-centered, CGI pedagogical approaches support the SMPs’ college
readiness in mathematics in the context of CCSSM?

0 Actions Identified = Does not support 1 or More Actions Identified = Support
EDI – Examples of teachers’ and students’ actions supporting the SMP 8.
EDI Lesson Cycle “X” if Present Description
Lesson Objective Skill Development/Guided Practice
• During activate prior knowledge, students are provided with
prior knowledge and skills the students can use for new
learning
• In concept development, students make the connection with
the previously learned knowledge and skills that were reminded
in the activate prior knowledge portion of the lesson.
Activate Prior Knowledge X
Concept Development X
Skill Development/Guided Practice
Relevance
Skill/Concept Closure X
Independent Practice
Periodic Reviews

140

• The students look for repeated measures to verify the
properties from the original and reflected figures.
CGI – Examples of teachers’ and students’ actions supporting the SMP 8. Carpenter et al. (2015)

CGI Framework “X” if Present Description
Teachers Choose a Problem Teachers Pose a Problem
• Students detect regularity in the types of problems and use a
combination of strategies to solve a problem that requires more
than one step.
• Teachers prompt students with questions: What do you notice?
Does this look familiar?
Teachers Choose the Students
Teachers Pose a Problem X
Teachers Observe Students
Solving a Problem

What Teachers do After Students
Solve a Problem

The support for SMP 8 is evident in both EDI and CGI. In the EDI, teacher-centered
approach, three out of eight components of the EDI lesson cycle was activated: Activate prior
Knowledge, Concept Development, and Skill/Concept Closure. In the CGI, student-centered
approach, one out of five CGI Framework components were activated: Teachers pose a problem
Summary of Findings
Three factors emerged from the findings of this qualitative comparative document
analysis which addressed the research question around the ways in which the teacher-centered,
EDI and student-centered, CGI pedagogies support the SMPs’ college readiness in mathematics.
A false dichotomy has been created around teacher-centered versus student-centered pedagogies
in mathematics. What is missing is strengthening teacher’s pedagogical skills to help them
become more effective in their classroom approaches to meet the intended student outcomes
based on the expectations of the SMPs. Teacher-centered, EDI and student-centered, CGI are
largely overlapping with strengths, and both prepare students to be college ready. Chapter Five
will focus on discussing the findings, their implications and recommendations.

141
Chapter Five: Discussion
The purpose of the study was to determine if the teacher-centered, EDI, and student-
centered CGI pedagogical approaches support the SMPs’ college readiness in mathematics in the
context of CCSSM. Using a comparative document analysis, the researcher analyzed the college
readiness factors of the SMPs and conducted a comparison of the two selected pedagogies, EDI
and CGI. This chapter presents a summary of the findings, the conclusions drawn from the
analysis and the researcher’s recommendations. Chapter 5 is organized using the following
sections. The first provides a statement of the problem, and a summary of the study. Next, the
chapter provides an overview of the sample selection, and data analysis, followed by the validity
and reliability. The next sections include the research question, a summary of the findings, a
discussion of the findings related to the literature, implications of the results for practice and the
recommendations for further research. Finally, the limitations of the study are discussed, and the
chapter ends with a conclusion of the study.
Statement of the Problem
Preparing students with the cognitive demands required to be successful in college-level
mathematics courses is critical and can be accomplished when teachers align their pedagogical
approach with college readiness expectations of the SMPs. A shift in mathematics pedagogy is
urgently needed in the face of the data showing no improvement in students’ mathematics
achievement data, and students are graduating high school underprepared for college-level
mathematics. Students’ lack of preparedness for college-level mathematics has lasting
implications that include: (a) an inability to access degree-granting postsecondary universities,
(b) an inability to enroll in credit-bearing college mathematics courses without first taking non-
college credit-bearing developmental math courses, and (c) narrowed opportunities in the job

142
market (Bureau of Labor Statistics, 2014; Conley & McGaughy, 2012). The urgency to resolve
this issue requires a call to action to address the barriers getting in the way of teachers providing
effective mathematics instruction and moving toward affecting the results of the CCSSM goal
for every high school graduate to be college, career, and life ready (CCSS About the Standard,
2019).
Summary of the Study
The goal of this comparative document analysis was to identify and examine specific
documents for the purpose of refining and expanding teachers’ knowledge and skills in the
alignment of their pedagogical approach with college readiness expectations in mathematics. The
need for a pedagogical shift in mathematics at the K-12 level is underscored with the
expectations in college readiness components of CCSSM-SMPs. The expectations call for an
increase in students’ critical thinking, problem-solving, and communication skills, which are
deeply affected by teachers’ knowledge, beliefs, and goals within their pedagogical approach.
For students to learn these skills, the teachers’ actions in the classroom are critical. Hattie (2010)
emphasizes the role of teachers “as the most powerful influences on learning” (p. 238).
The initial analysis of the SMPs, EDI, and CGI documents and the corresponding
research was focused on the definition of college readiness in mathematics as defined by both
mathematics instructors in post-secondary institutions and educational researchers. The
researcher needed to verify that there was alignment between the K-12 definition of college
readiness in mathematics and the post-secondary sector. The process began by researching the
college readiness expectations in the post-secondary mathematicians’ definitions while
simultaneously examining the educational researchers’ college readiness definition in
mathematics. Following the verification of the alignment between the K-12 and college

143
definitions, the researcher examined the college readiness expectations of the SMPs through a
structured process of Revised Bloom’s Taxonomy, a comprehensive cognitive process
framework. Through the structured framework of Revised Bloom’s Taxonomy, the researcher
connected the cognitive process verbs of the SMPs with the cognitive process verbs of Revised
Bloom’s Taxonomy. The rationale for linking the cognitive process verbs was to identify the
correlating teachers’ and students’ actions with the cognitive process skills within a framework
to ensure that these interactive actions are indeed the actions tied to the cognitive process verbs.
The study’s final step was to compare the correlating teachers’ and students’ actions with the two
selected pedagogies, EDI, and CGI. The goal was to locate the matching teachers’ and students’
actions within the two pedagogical spaces in the comparison to determine whether EDI and CGI
support the SMPs’ expectations.
The researcher viewed the study through a theoretical framework, Schoenfeld’s Theory
of Teaching in Context. The researcher used the framework to conceptualize the understanding
that students’ college preparedness in mathematics begins with teachers’ preparedness to teach
them the college readiness cognitive process skills in mathematics. This study’s research
findings exemplify how the two pedagogies, EDI and CGI, as a critical case comparison can help
teachers understand how to identify whether the pedagogical approaches support college
readiness in mathematics. The researcher conducted a comparative document analysis study to
increase the understanding of the SMPs' expectations and use the knowledge to bridge the
pedagogical misalignment between K-12 and post-secondary mathematics instruction.
Sample Selection
The researcher selected multiple documents used in the comparative document analysis
to provide a data source to answer the research question: In what ways do the teacher-centered,

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EDI, and student-centered, CGI pedagogical approaches support the SMPs’ college readiness in
mathematics in the context of the CCSSM. Extant documents related to the study variables’
existing principles and frameworks were selected and analyzed using the critical case sampling
technique. The variables were organized into two groups: (a) Analytic Variables, and (b) Linking
Variables. The analytic variables were preselected variables the researcher used to conduct the
comparative document analysis: (a) SMPs, (b) EDI, and (c) CGI. Conversely, the researcher
selected the linking variable during the study to link and compare the analytic variables’
comparability aspects.
Data Analysis
The data collection and data analysis were conducted within a systematic content analysis
of the documents related to college readiness in mathematics. The study’s initial data included
preselected documents, including the SMPS, EDI, and CGI documents. The SMPs were the
initial data defining college readiness in mathematics. The researcher used the SMPs’ definition
to verify the alignment with college readiness mathematics definitions by post-secondary
mathematicians and researchers, explicitly comparing the cognitive process verbs and phrases.
Once the researcher completed the alignment verification, the Revised Bloom’s Taxonomy
cognitive processes were analyzed and used to connect the comparability of the SMPs’ college
readiness and the two selected pedagogies. Using Revised Bloom’s Taxonomy as a linking
variable was a two-step process. First, the researcher connected the cognitive process verbs and
phrases of the SMPs and Revised Bloom’s Taxonomy. Secondly, the researcher identified the
correlating teachers’ and students’ actions tied to the identified cognitive process verbs within
Revised Bloom’s Taxonomy framework. The data from this analysis identified how teachers and
students should engage with the cognitive processes appropriate to each of the SMP analytic

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category’s college readiness components and used the correlating teachers’ and students’ actions
to conduct the final comparison that answers the research question. Therefore, the critical data
points analyzed in this study are cognitive process verbs and the correlating teachers’ and
students’ actions.
Validity and Reliability
Regarding validity, Leung (2005) states, “In assessing the validity of qualitative research,
the challenge can start from the ontology [how we come to know] and epistemology [how we
know] of the issue being studied” (Section 5). An in-depth analysis of how it is possible to reach
a point of knowing and understanding how it was known, in this study, was based on the selected
documents. Although limited in scope, the documents used in this study provided the data
needed to answer the research question. The documents used in this study, as Trochim (2005)
states, provided the best available approximation of the truth of a given proposition, inference, or
conclusion” (p.16).
Reliability is based on the idea that the researcher must be able to replicate any results
found through a research study (Leung, 2005). According to Krippendorff (2013), a reliable
procedure, when replicated, regardless of the conditions of the application, should produce the
same results. The researcher developed a structure of analysis that could be replicated to verify
whether any specific pedagogy would support the SMPs and college readiness in mathematics.
To strengthen the findings’ reliability, the researcher identified source documents representing a
wide range of researchers and mathematicians to define college readiness in mathematics. All
selected sources were formally published under editorial control or are official state documents
used in K-12 education. Data integrity was ensured by verifying accuracy in terms of form and
context with constant comparison by the researcher (Silverman, 2009).

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Research Question
The research in this study was used to answer the question: In what ways do the teacher-
centered, EDI and student-centered CGI pedagogical approaches support the SMPs college
readiness in mathematics in the context of CCSSM?
Discussion of Findings
The findings in Chapter 4 reveal that the two selected pedagogies, EDI and CGI support
the SMP’s expectations for college readiness in mathematics in the context of CCSSM. The key
findings from the study will discuss how the two pedagogies support each of the SMPs.
Key Findings: SMP 1
The focus of SMP 1 is on making sense of the problems and persevering in solving them.
Both EDI and CGI were similar in their support for SMP 1. Based on the findings, it is highly
evident that both pedagogical approaches provide opportunities for the students to model
problems and persevere in solving the problem. EDI provides students with different approaches
to solve problems, while CGI provides students the opportunity to think through the steps on
their own. EDI utilizes the steps approach to make comprehension possible for the students, and
with students learning different methods, they can select the method that best fits the problem.
For CGI, teachers provide students with problems they can easily model with manipulatives to
find their way of solving the problems. Both pedagogies provide students with opportunities to
ask clarifying questions and explain their thinking process. Students also have opportunities to
agree or disagree with other students' ideas and explain why they agree or disagree. Both
pedagogical approaches ensure that students are supported with vocabulary comprehension
related to the content, academic, and supporting words. Additionally, both pedagogies offer
opportunities for the students to check their solutions.

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Several components of Schoenfeld’s (1985) five-phase model of problem-solving were
evident in the teachers' and students’ actions identified in the findings related to SMP 1. One
example is when teachers ask clarifying questions, which relates to Schoenfeld's Analysis and
Exploration components. In these phases, teachers ensure students understand the problem by
simplifying or reformulating the problem based on students' answers to the clarifying questions.
Schoenfeld (2016) emphasizes problem-solving because it is through problem-solving that
students develop what he calls "patterns of productive thinking," which is critical to mathematics
learning (Schoenfeld, 2016: p. 507).
Key Findings: SMP 2
The focus of SMP 2 is on reason abstractly and quantitatively. Both EDI and CGI are
found to support SMP 2. In the EDI approach, the teachers’ actions show strongly in assisting
students with multiple representations of problems and then providing space for students to
symbolically manipulate them to solve a problem. In the CGI approach, the teachers’ actions are
strong in prompting students with questions that guide them to attend to the critical features and
build upon their intuitive problem-solving strategies. Evident in both pedagogical approaches is
the attention given to reason and manipulation of mathematical symbols. In both pedagogies,
students consider the units involved and attend to the meaning of the quantities and not purely
manipulate with numbers. In EDI, teachers are purposeful in providing the procedures and steps
related to the new learning. In CGI, teachers purposefully structure the problems so that students
can use their intuitive modeling skills to construct a model of the situation specified in a
problem.
Evident in the teachers’ and students’ actions identified in the findings related to SMP 2
is Galasso’ (2016) work where he emphasized the importance of decontextualizing – taking

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context and representing it abstractly, or contextualizing – taking abstract mathematical
representation and putting it into context. One example is when teachers guide students in
attending to the critical features and building upon their intuitive problem-solving strategies.
Also evident in the teachers’ actions is the work of Mason (1989), where the emphasis is on
developing abstract/quantitative reasoning and moving away from manipulating objects, e.g.,
students’ development of concrete to abstract thinking in mathematics. An example is when the
teachers focus students’ attention on attending to the meaning of the quantities and not purely
manipulate numbers.
Key Findings: SMP 3
The focus of SMP 3 is on constructing viable arguments and critiquing the reasoning of
others. Both EDI and CGI are found to support SMP 3. In both of the EDI and CGI approaches,
the teachers’ actions show strongly in teaching students how to determine the critical and non-
critical attributes within the definition of a concept or problem. This allows students to develop
skills in creating a viable argument, critiquing others’ arguments, and learning how to
communicate mathematically. In EDI, examples and counterexamples are strongly used to
provide students with opportunities to define and grasp the concepts. In CGI, students are
provided with opportunities to evaluate the strategy to determine if they agree or disagree with
the reasoning of other students. Both pedagogies provide opportunities to work with other
students, compare each other’s strategies, and make a judgment. There are opportunities in both
pedagogies for students to provide justifications for their reasoning. In EDI, students refer to the
definition and procedures of the concept to provide justifications. In CGI, students explain their
strategies to justify their ideas.

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Evident in the teachers’ and students’ actions identified in the findings related to SMP 3
is the work of Kilpatrick et al. (2001) and Rumsey and Langrall (2016), where the constructing
of viable arguments and the critiquing the reasoning of others is about developing mathematical
discourse through (a) constructing viable arguments grounded in mathematical discourse where
students explain and discuss their thinking processes aloud, (b) building networks of logical
statements, (c) discovering and deepening the understanding of mathematical content, and (d)
practice of discourse. One example is how the teachers provide opportunities for students to
agree or disagree and provide justifications for their reasoning.
Key Findings: SMP 4
The focus of SMP 4 is on modeling with mathematics. The support for SMP 4 is evident
in both EDI and CGI. While both EDI and CGI approaches provide opportunities for students to
model with mathematics, the approaches differ in how this plays out in the teachers’ and
students’ actions. Students are asked to model with mathematics in the EDI approach after they
have been taught a specific concept. In the CGI approach, students begin modeling after the
teacher poses a problem. In CGI, the students respond to the problem, and then the teacher bases
instruction on the information found in the responses to the mathematical problem. Both
pedagogies allow students to interpret their mathematical results in the context of the situation
given and reflect if their interpretations make sense. Students in both pedagogies are given
problems that require students to use strategies to convert the quantities into meaning-making
structures such as mapping, diagrams, formulas, and modeling.
Evident in the teachers’ and students’ actions identified in the findings related to SMP 4
is Ventak’s (2015) emphasis on developing students’ knowledge and skills in noticing patterns,
structure, and attending to variations, invariances, and mathematics connections. This is

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exemplified in how teachers ask students to model with mathematics and then reflect on their
interpretations within a given task.
Key Findings: SMP 5
The focus of SMP 5 is on the use of appropriate tools strategically. The support for SMP
5 is evident in both EDI and CGI. While both EDI and CGI approaches provide opportunities for
students to work with tools to solve mathematical problems, there is a difference in how the tools
are used. In the EDI approach, students are provided with instructions and parameters to use the
tools to solve different types of problems. In the CGI approach, students work with tools
independent of instructions and parameters. In CGI, students use their intuitive problem-solving
skills to direct how they will use the tools to solve mathematical problems.
Evident in the teachers’ and students’ actions identified in the findings related to SMP 5
is O'Connell & SanGiovanni’s (2013) work around the appropriate use of tools. The teachers’
actions are also evident in Sharma’s (2016) work where the emphasis is not on the tool, but
rather, on the questions teachers ask and the connections teachers make regarding the use of the
tool and the concept being taught.
Key Findings: SMP 6
The focus of SMP 6 is on attending to precision. The support for SMP 6 is evident in both
EDI and CGI. However, precision is especially apparent in EDI due to the call for precision from
the lesson objective's introduction through periodic reviews. Additionally, throughout the EDI
lesson cycle, there are consistent perioding steps where teachers check for understanding.
Students in an EDI class also have multiple opportunities to listen to the teacher and peers in
modeling and discussing mathematics. In the CGI approach, precision is evident in how the
teacher asks students about correct, incorrect, and incomplete strategies. Teachers base their

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questions and ongoing instruction around a specific problem by observing and listening to how
students explain their thinking. In EDI, students are supported with learning how to use academic
and content language in mathematics precisely. In EDI, students practice using newly learned
mathematics vocabulary throughout the lesson in the lesson's various engagement processes.
Also, students in EDI are given a word bank to support students' precision when they engage in
oral and written communication. In CGI, teachers support the students by encouraging them to
work through the details of their strategies. Students in CGI are asked to explain their thinking to
reflect on the accuracy of their work.
Evident in the teachers' actions identified in the findings related to SMP 6 is Sharma's
(2016) work that focuses on the teachers' role in clarifying process and outcomes in students'
problem-solving in mathematics. This is exemplified in how teachers organize and communicate
the steps of a mathematical process and the emphasis on oral and written communication in
mathematics.
Key Findings: SMP 7
The focus of SMP 7 is on looking for and making use of structure. The support for SMP 7
is evident in both EDI and CGI. Both EDI and CGI approaches allow students to look for and
make use of structure by facilitating discussions about mathematics problems and asking
students to use various strategies to solve a problem. In EDI, examples, and non-examples are
used to support students to detect structure. In CGI, students are provided with types of problems
within their levels that require them to use various intuitive ways of solving problems, by which
students detect the structure.
Evident in the teachers' and students’ actions identified in the findings related to SMP 7 is
Sharma's (2016) work focused on teaching students how to make use of a mathematical structure

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that goes beyond the information that is given in the problem, and to find the discerning
relationships between pieces of information either explicitly or implicitly. This is exemplified
through teachers facilitating discussions about mathematics problems and building on students'
intuitive problem-solving skills.
Key Findings: SMP 8
The focus of SMP 8 is on looking for and expressing regularity in repeated reasoning.
The support for SMP 8 is evident in both EDI and CGI. Both EDI and CGI approaches provide
students opportunities to look for and express regularity in repeated reasoning. How teachers
pose questions may slightly differ in the two pedagogies, which is how mathematical concepts
are introduced. In EDI, students are provided with the opportunity to activate their prior
knowledge and skills in learning new mathematical concepts. Students connect the regularity and
repeated measures in the mathematical problems in activating prior knowledge with the new
concept to be learned. In CGI, students detect regularity in repeated reasoning by using intuitive
ways of understanding math and using problem-solving strategies such as manipulative and
written representation to model the problems to more abstract ways of representing various types
of problems.
Evident in the teachers’ actions identified in the findings related to SMP 8 is O’Connell &
SanGiovanni’s (2013) focus on the prerequisite skills needed for developing the capacity to look
for and express regularity in repeated reasoning. Additionally, O’Connell & SanGiovanni
emphasize the use of questioning to assist students in developing reasoning skills. This is
exemplified in how teachers pose questions that activate prior knowledge around a specific
mathematical concept.

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Implications
The study identified whether the selected pedagogies consist of opportunities for
teachers’ and students’ actions within the pedagogical space to support the expectations of the
SMPs. The motivation for researching the college readiness in mathematics expectations of the
SMPs was due to the lack of students’ progress toward college readiness in mathematics. The
lack of student progress, which has been evident for decades, calls for a closer look at how
teachers approach mathematics instruction so that they develop instructional strategies that
effectively prepare students for post-secondary mathematics rigor. The declining trend among
high school students’ mathematics achievement data highlights the disconnect between K-12 and
post-secondary education sectors in mathematics expectations, which leads to a misalignment of
K-12 and college readiness mathematics instruction (Callan et al., 2006; Cohen, 2008). The
implications of not addressing the misalignment in K-12 and college readiness mathematics
instruction will continue the trend referenced by Peggy Carr, associate commissioner of the
National Center for Education statistics, when she pointed out, “Over the past decade, there has
been no progress in mathematics…” (Camera, 2019). Although the student achievement data
have remained flat across various national assessments, the CCSSM initiative has proven to be a
starting point where there is an agreement among the stakeholders around the college readiness
components of mathematics content and the process standards.
Theoretical Implications
The three components addressed in Schoenfeld’s model are related to the
interconnectedness of teachers’ knowledge, beliefs, and goals that impact their moment-by-
moment decisions in the classroom, ultimately affecting the quality of student achievement
outcomes. To further elaborate, the teachers’ knowledge component impacts their beliefs in what

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should be the priority when teaching, which then impacts their goals about what teachers will
persist in to ensure that students learn. In this section of theoretical implications, the researcher
will address all three components of Schoenfeld’s Model.
The first theoretical implication is that to ignore the knowledge component of teachers
teaching mathematics will result in a continued downward trend of student achievement in
mathematics. To highlight the knowledge component of Schoenfeld’s Teaching-in-Context
theoretical framework, teachers need both content and pedagogical knowledge (Schoenfeld,
2019). Schoenfeld (2019) asserted the need for teachers to have a deep understanding of subject
matter that ultimately becomes pedagogical knowledge. Regarding the CCSSM initiative, to
teach mathematics effectively and prepare students for college, teachers must have knowledge in
three areas: mathematics content knowledge, knowledge of the SMPs, and pedagogical
knowledge (CCSSM, 2010). Mathematician Keith Delvin (2014) warns against mathematics
instruction that is focused only on “algorithmic skills...a form of teaching that is hopelessly
inadequate for life in today’s world” (p.4). The mathematics needed in today’s world is centered
around the SMPs (CCSSM, 2010), which means that knowledgeable teachers in mathematics are
required (Schoenfeld, 2019). Keith Delvin (2014) states, “A mathematically knowledgeable
teacher given the goal of producing mathematically able students well-equipped for the Twenty-
First Century life and career(s)...would need to look no further than the eight guiding principles”
in the SMPs” (p. 4).
For the study analysis, the focus was on two specific pedagogies, EDI, and CGI.
However, it is worth mentioning teachers make pedagogical decisions in the classrooms no
matter the pedagogical preference. Therefore, within any pedagogies, to implement college
readiness thinking expectations as designed within the SMPs, teachers will make decisions based

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on their knowledge, beliefs, and goals around SMPs’ college readiness components. Schoenfeld
(1998) asserts that the beliefs teachers have are “mental constructs” (p.19) that are representative
of their experiences that shape the perceptions teachers have regarding certain circumstances and
the “goals they establish within the circumstances” (p.19). Schoenfeld (1998) describes how
teachers’ knowledge, beliefs, and goals, shape the instructional process stating, “teaching is a
dynamic act, responsive to what happens in interaction with the students. The teacher is
constantly monitoring what is taking place during instruction and acting based on perceptions of
what is taking place” (Schoenfeld, 1998, p.15). With the idea that teaching is a “response to what
happens in interaction,” one could apply this to a classroom setting in connection to the SMPs:
(a) teachers’ knowledge about the SMPs’ college readiness components, (b) teachers’ beliefs
about the importance of students developing the cognitive process thinking tied to the SMPs’
college readiness components, (c) teachers setting specific goals to achieve the implementations
of students developing the cognitive process thinking as designed by the SMPs’ framework, and
(d) teachers’ persistent actions to accomplish the goals daily throughout the lessons. These
interactions will shape how teachers implement the SMPs’ college readiness in mathematics in
the context of CCSSM and ultimately decide whether students are prepared or not prepared to
take college-level mathematics courses (Schoenfeld, 1998).
Practical Implications
This study’s practical implication is the shift from the previous federally mandated,
“college for all” to the current, “college for every student” will remain a goal not achieved in the
K-12 educational system if there is no explicit and targeted plan to achieve it. The plan for
“college for every student” can begin with a plan to target teachers’ knowledge around the
implementation of the SMPs. When Common Core began in 2010, teachers did not receive

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adequate training in the expectations of the CCSSM. Akkus (2016) underscores the mistake
made in K-12 education when the implementation of CCSS in general. Akkus (2016) argues that
the CCSS initiative was lumped together with other reform efforts in the minds of the teachers.
In the confusion of CCSS, inclusive of the CCSSM and SMPs, being absorbed into one of many
expectations to be carried out by the teachers, according to Reborn (2013), “the data suggest that
most teachers do not recognize the level of difficulty” when implementing the CCSS [CCSSM-
SMPs] (p. 51). Akkus (2016) asserts that teachers are flooded with competing signals regarding
content to teach, and state standards, state assessments, and textbooks provide conflicting
guidance and teachers receive neither the preparation nor the support they require to make
effective…[college-readiness in mathematics] decisions” (p.51).
The confusion teachers are having around the SMPs was not the intent of the CCSS
developers. On the contrary, the intent and development of the new standards released in 2010
were to bridge and clarify the standard definition of proficiency (CCSS Development, 2019).
Moreover, the college readiness standards’ design addresses students’ knowledge expectations to
graduate from high school (CCSS About the Standard, 2019). The CCSSM goal is to ensure that
the teaching and learning of mathematics at each grade level results in every student making
progress toward college readiness and that every student is college and career ready when they
leave high school (CCSS About the Standard, 2019).
This study highlights that teachers’ inability to adequately prepare students for college
mathematics has lasting implications for post-secondary students that include: (a) an inability
for students to access degree-granting postsecondary universities, (b) an inability for students to
enroll in credit-bearing college mathematics courses without first taking non-college credit-

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bearing developmental math courses, and (c) narrowing students’ opportunities in the job market
(Conley & McGaughy, 2012; The Bureau of Labor Statistics, 2014).
Future Implications and Recommendations
This study determined the alignment of the selected pedagogies, EDI, and CGI, with the
college readiness components of the SMPs. Future research is recommended to continue the
current study and include a longitudinal observation component of the study examining
mathematics pedagogical practices focused on the SMPs. Observing and supporting teachers’
pedagogical practices will offer insight into their knowledge, beliefs, and goals in implementing
effective classroom practices to meet the SMPs college readiness components in mathematics.
Following the teacher observations, the observations’ data could be analyzed using Schoenfeld’s
Theory of Teaching-in-Context (2006, 2011) to guide the findings’ practical application.
Schoenfeld (2006, 2011) regards the three variables of knowledge, goals, and beliefs in
Teaching-in-Context as critical to teachers’ decisions in numerous teaching situations.
Addressing three variables in the context of the SMPs’ college readiness in mathematics and
pedagogical frameworks’ expectations would significantly impact the students’ quality of
learning experiences toward college preparedness in mathematics (Schoenfeld, 1998).
The second recommendation is to ensure that the teachers understand how to effectively
implement the content and process standards within a pedagogical process to prepare students for
college-level mathematics courses. Teachers’ preparation is the first step in ensuring students’
college readiness in mathematics, as established by the National Mathematics Advisory Panel
(2008) in their review of studies related to teachers’ mathematical knowledge. The panel stated,
“it is clear that teachers’ knowledge of mathematics is positively related to student achievement”
(p. 37). Equally important is teachers’ pedagogical knowledge, central to delivering high-quality

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instruction to all students (Shulman, 1987). With teacher in-service support and professional
development as central factors in college readiness, this study’s recommendation is to develop a
mathematics in-service support and professional development for teachers that target what
Shulman calls general pedagogical knowledge and pedagogical content knowledge. The
professional development needs to be targeted so that teachers are developing their pedagogical
knowledge and skills in teaching mathematics to the level of the college readiness expectations
of the SMPs. Fullan and Quinn (2016) assert that targeted professional development provides
clarity and coherence for teachers in that “once the purpose and goals are identified, everyone
must perceive that there is a clear strategy for achieving them and be able to see their part in the
strategy” (p. 24). The target for professional development should be the expectations of the
SMPs. The clarity and coherence are the teachers’ understanding of how the targeted
professional development will support their pedagogical growth in teaching the college readiness
expectations of the SMPs. To accomplish targeted professional development that improves
teachers’ knowledge and skills in the SMPs, districts, and schools first need to examine the
pedagogical models implemented in the schools to determine whether the implemented
pedagogical model emphasizes teacher’s and students’ actions that prepare students to be college
ready in mathematics. This information is critical to developing a targeted professional
development program. Next, the targeted professional development needs to address teachers’
beliefs and attitudes about the SMPs and how the SMPs are not separate from the CCSSM,
which is often a belief held by many teachers. To counter this belief, targeted professional
development would address the CCSSM and SMPs in tandem.
The third recommendation is to afford opportunities in teacher preparation programs to
prepare teachers in how to engage their future students with the CCSSM content standards and

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standards for mathematical practice. The new teachers need time and practice to understand the
nuances of the CCSSM and SMPs. This will require the professors within the teacher preparation
programs to have knowledge of the SMPs and have the ability to provide new teachers with the
opportunity to engage in practices related to the SMPs. Max and Welder (2020) state that “the
features of mathematical practice described in the standards are not intended as separate from
mathematical content” (p. 846). Novice teachers need explicit and targeted training on
implementing the SMPs in the classroom to support content development.
Limitations
One limitation in this study was the lack of prior research on the alignment of specific
pedagogies with the SMPs. Most research around the topic of SMPs deals primarily with
instructional strategies. However, little research focused on explicitly describing where in the
pedagogical space lie the opportunities for the specific teachers’ and students’ actions that
connect with addressing the SMPs cognitive process verbs’ college readiness components. While
this was a limitation, it had little to no impact on the study, as the researcher was aware that there
was little prior research on the alignment of specific pedagogies with the SMPs.
Another possible limitation of the study is biased selectivity. Biased selectivity is present
when an incomplete collection of documents suggests ‘biased selectivity’ (Yin, 2003, p. 80). As
mentioned above, the resource documents were limited in various areas of the research. Possible
bias may have been present in selecting the documents containing the Mathematicians’ and
Researchers’ definitions of college readiness in mathematics. However, the research sampling
technique allowed for selectivity in documents based on critical case sampling. The researcher
sought to limit the selective bias by considering the subjectivity and possible biases the
document’s author may have had. Additionally, the researcher assessed the documents for their

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completeness and comprehensiveness (Bowen, 2009). While biased selectivity and limited
resource documents may have been a limitation, it had little to no impact on the study. If
anything, the researcher’s selectivity in the source documents supported the study’s findings.
According to Bowen (2009), “document analysis offers advantages that clearly outweigh the
limitations” (p. 32).
Conclusion
The CCSSM define a clear set of math skills and concepts through mathematical content;
however, the standards also include critical points of intersection with the SMPs that are the
foundation of mathematical thinking and practice (NGA & CCSSO, 2010). This study
understands the importance of both the content standards and the SMPs in a pedagogical space
so that mathematics instruction develops the critical thinking and problem-solving skills students
need in post-secondary mathematics. Devlin (2017) asserts that engaging students in
mathematics is much more than what is presented in the K-12 setting. “Mathematical thinking is
a whole way of looking at things, of stripping them down to their numerical, structural, or logical
essentials, and of analyzing the underlying patterns” (Devlin, 2017, p. 59). Teachers need to
understand this and they need support to know how to implement classroom strategies to
improve the instructional environment. Haber (2020) asserts that the development of students
who critically think is accomplished through “deliberate practice that specifically focuses on the
development of critical-thinking skills…explicit instruction on critical-thinking principles and
techniques, deliberate practice opportunities that put those techniques to work, encouraging
transfer between domains, and inspiring students to practice thinking critically on their own—all
represent high-leverage critical-thinking practices applicable to any domain” (section 3, para. 3;
section 4, para. 1).

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One of the critical findings in this research study is the verification of alignment between
the K-12 sector as described in the SMPs’ cognitive process verbs and the post-secondary’s
definition. See Figures 4.8 through Figures 4.15 for the college readiness in mathematics
definition alignment findings, Analytic Categories 1 – 8 in Chapter 4. The evidence is clear that
post-secondary institutions require students to have critical thinking skills beyond regurgitating
numbers devoid of understanding the meaning and principles behind the numbers (Hiebert &
Grouws, 2007; Kilpatrick, Swafford, & Findell, 2001). Students need high-level mathematical
thinking skills that involve using “formulas, algorithms, and procedures in ways that connect to
concepts, understanding, and meanings. Tasks that require students to think deeply about
mathematical ideas and connections encourage them to think for themselves” (Anthony &
Walshaw, 2009, p. 13).
Another critical finding is the problem of abstractness surrounding the SMPs is essential
to this study because the SMPs are the catalyst that determines whether the instruction in the
CCSSM implementation will lead students to reach college-level thinking in mathematics. The
critical problem in implementing the college readiness aspect of SMPs is the teachers’ lack of
knowledge on SMPs, and it does not support teachers developing beliefs around the importance
of the cognitive process levels described in the SMPs. (Lewis et al., 2016). Delvin (2014) asserts
that SMPs epitomize the aggregate of mathematical knowledge, skills, abilities, habits, and
attitudes, as the practices support the acquisition and application of mathematics content
knowledge. This study agrees with the belief of the Conference Board of Mathematical Sciences
(2012); there must be intentional opportunities for teachers to learn and fully understand the
implementation process of the SMPs in the context of the CCSSM to achieve the results the
CCSSM intended.

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References
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Abstract (if available)

Abstract Full Title: An examination of teacher-centered Explicit Direct Instruction and student-centered Cognitively Guided Instruction in the context of Common Core State Standards Mathematics college readiness: a document analysis. Despite many reform efforts in K-12 education, the declining trend among high school students' achievement in Mathematics and the lack of student success in post-secondary mathematics courses indicate a disconnect and misalignment in the mathematics instructions students receive in K-12 institutions. The mathematics instruction students receive in K-12 institutions fails to develop students' critical knowledge and skills to succeed in postsecondary mathematics courses (Callan et al., 2006; Cohen, 2008; Woods et al., 2018). This qualitative document analysis seeks to examine two pedagogical approaches: teacher-centered, Explicit Direct Instruction (EDI), and student-centered Cognitively Guided Instruction (CGI) to identify whether the two pedagogies support the college readiness skills represented in the Standards for Mathematical Practice (SMPs). The research question will reveal how the two pedagogies support the SMPsâ€™ college readiness in mathematics in the context of Common Core State Standards Mathematics. Theoretically, this study is viewed through the lens of Schoenfeld's (1998) Theory of Teaching-in-Context. Data will be collected and analyzed within a systematic content analysis of the documents related to college readiness in mathematics using Mayring's (2000) four-step model. The model will provide a structured process to determine how the two selected pedagogies, EDI and CGI, support college readiness in SMPs. The structured process provided a way to analyze the study documents, including preselected documents related to the SMPs, EDI, and CGI. The research identified additional documents critical to the document analysis, such as Revised Bloomâ€™s Taxonomy and various documents related to the definition of college readiness in mathematics. The study's findings revealed that the two selected pedagogies, EDI and CGI, support the SMP's expectations for college readiness in mathematics in the context of CCSSM.

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Asset Metadata

Creator Moore, Phoebe (author)

Core Title An examination of teacher-centered Explicit Direct Instruction and student-centered Cognitively Guided Instruction in the context of Common Core State Standards Mathematics…

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Degree Conferral Date 2021-05

Publication Date 05/10/2021

Defense Date 01/15/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Cognitively Guided Instruction,Explicit Direct Instruction,OAI-PMH Harvest,Standards for Mathematical Practice,student-centered instruction,teacher-centered instruction

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kaplan, Sandra (committee chair), Gallagher, Raymond (committee member), Ybarra, Sylvia (committee member)

Creator Email phoebe.moore@gmail.com,phoebemo@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC112720094

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